2 Elliptic Diﬀerential Equationsrohop/spring_08/Chapter2.pdf · For u ∈ C2(Ω), Taylor...

2 Elliptic Differential Equations

2.1 Classical solutions

As far as existence and uniqueness results for classical solutions are con-cerned, we restrict ourselves to linear elliptic second order elliptic differentialequations in a bounded domain Ω ⊂ Rd

(2.1.1) Lu := −d∑

i,j=1

aij uxixj +d∑

i=1

bi uxi + cu = f .

We assume that the data of the problem satisfies the following conditions:

There holds aij , bi, c ∈ C(Ω), 1 ≤ i, j ≤ d, f ∈ C(Ω) and(2.1.2a)

there exists a constant C > 0, such that for all x ∈ Ω

|aij(x)| , |bi(x)| , |c(x)| ≤ C , 1 ≤ i, j ≤ d ,

The functions aij are symmetric, i.e., for all 1 ≤ i, j ≤ d(2.1.2b)

and x ∈ Ω we have

aij(x) = aji(x) ,

and the matrix-valued function (aij)di,j=1 is uniformly

positive definite in Ω , i.e., there exists a constant α > 0 ,

such that for all x ∈ Ω und ξ ∈ Rd

d∑

i,j=1

aij(x)ξiξj ≥ α |ξ|2 .

For the sake of uniqueness of a solution of (2.1.1) we need to specify boundaryconditions on the boundary Γ = ∂Ω.

(2.1.3) Definition: Under the assumptions (2.1.2a),(2.1.2b) let L be the lin-ear second order elliptic differential operator given by (2.1.1), and assumef ∈ C(Ω) and uD ∈ C(Γ ). Then, the boundary value problem

Lu(x) = f(x) , x ∈ Ω ,(2.1.4a)

u(x) = uD(x) , x ∈ Γ(2.1.4b)

18 2 Elliptic Differential Equations

is called an inhomogeneous Dirichlet problem. If uD ≡ 0, (2.1.4a),(2.1.4b) issaid to be a homogeneous Dirichlet problem. A function u ∈ C2(Ω) ∩ C(Ω)is called a classical solution, if u satisfies (2.1.4a),(2.1.4b) pointwise.

In case c(x) ≥ 0 , x ∈ Ω, in (2.1.1), the uniqueness of a classical solutionof the Dirichlet problem (2.1.4a),(2.1.4b) follows from the Hopf maximumprinciple.

(2.1.5) Theorem: In addition to the conditions (2.1.2a),(2.1.2b) assumec(x) ≥ 0 , x ∈ Ω. Further, assume that the function u ∈ C2(Ω) satisfies

(2.1.6) Lu(x) ≤ 0 , x ∈ Ω .

Then, if there exists x0 ∈ Ω such that

(2.1.7) u(x0) = supx∈Ω

u(x) ≥ 0 ,

there holds

(2.1.8) u(x) = const. , x ∈ Ω .

Proof. We refer to Theorem 2.1.2 in Jost (2002). �

Corollary:Under the assumptions of Theorem (2.1.5) the Dirichlet problem(2.1.4a),(2.1.4b) admits at most one classical solution.

Proof. The proof is left as an exercise. �

For the prototype of a linear second order elliptic differential equation,the Poisson equation (1.1.14), we can prove the existence of a solution of thethe Dirichlet problem (2.1.4a),(2.1.4b) using Green’s representation formula.We assume Ω to be a C1 domain (see Definition (1.2.2)).

Green’s first and second formula are as follows:

(2.1.9) Theorem: Let Ω ⊂ Rd be a C1 domain and assume u, v ∈ C2(Ω).Further, let ν be the exterior unit normal vector on Γ . Then, there hold thefirst Green’s formula

(2.1.10)

∫

Ω

∆u v dx +

∫

Ω

∇u · ∇v dx =

∫

Γ

ν · ∇u v dσ

and the second Green’s formula

(2.1.11)

∫

Ω

(

∆u v − u ∆v)

dx =

∫

Γ

(

ν · ∇u v − u ν · ∇v)

dσ .

2.1 Classical solutions 19

Proof. The proof of (2.1.10) follows from Gauss’ theorem∫

Ω

∇ · q dx =

∫

Γ

ν · q dσ

applied to the vector-valued function q := v∇u. Exchanging the roles of uand v in (2.1.10) and subtracting the resulting equation from (2.1.10) implies(2.1.11). �

(2.1.12) Definition: A function u ∈ C2(Ω) is called a harmonic functionin Ω, if u satisfies the Laplace equation, i.e., if ∆u(x) = 0 , x ∈ Ω. Thefunction

(2.1.13) Γ (x, y) := Γ (|x− y|) :=

{12π ln(|x− y|) , d = 2

1d(2−d)|Bd1 |

|x− y|2−d , d > 2 ,

where x, y ∈ Rd , x 6= y, and |Bd1 | denotes the volume of the unit ball in Rd,

is called the fundamental solution of the Laplace equation. As a function ofx, the fundamental solution is a harmonic function in Rd \ {y}.

The notion ’fundamental solution’ is justified by Green’s representationformula:

(2.1.14) Theorem: Let Ω ⊂ Rd be a C1 domain and Γ (·, ·) the fundamentalsolution as given by (2.1.13). If u ∈ C2(Ω), in y ∈ Ω there holds

u(y) =

∫

Ω

Γ (x, y) ∆u(x) dx +(2.1.15)

+

∫

∂Ω

(

u(x) ν · ∇xΓ (x, y) − Γ (x, y) ν · ∇u(x))

dσ(x) ,

where ∇x denotes the gradient with respect to the differentiation in the vari-able x.

Proof. The proof follows from Green’s second formula (2.1.11). For detailswe refer to Theorem 1.1.1 in Jost (2002). �

The application of (2.1.15) to a test function ϕ ∈ C∞0 (Ω) results in

ϕ(y) =

∫

Ω

Γ (x, y) ∆ϕ(x) dx , y ∈ Ω .

If we denote by ∆x the Laplace operator with respect to the variable x andby δy Dirac’s delta function as a distribution with δy(ϕ) = ϕ(y), we caninterpret ∆xΓ (·, y) according to

∆xΓ (·, y) = δy .


(2.1.16) Definition: A function G(x, y), x, y ∈ Ω , x 6= y, is called Green’sfunction for Ω, if the following conditions are satisfied

G(x, y) = 0 , x ∈ ∂Ω ,(2.1.17a)

The function G(x, y) − Γ (x, y) is harmonic in x ∈ Ω .(2.1.17b)

For the domains Ω ⊂ Rd considered here, the existence of a Green’sfunction is guaranteed. Under certain assumptions, it enables the explicitrepresentation of the solution of the inhomogeneous Dirichlet problem forPoisson’s equation (1.1.14).

(2.1.18) Theorem: Under the assumptions of Theorem (2.1.14) let G(x, y)be a Green’s function for Ω. Then, the following holds true:(i) If u ∈ C2(Ω)∩C(Ω) is a classical solution of the inhomogeneous Dirichletproblem (2.1.4a),(2.1.4b) for Poisson’s equation (1.1.14), for y ∈ Ω we havethe representation

(2.1.19) u(y) =

∫

Ω

G(x, y) f(x) dx +

∫

∂Ω

uD(x) ν · ∇xG(x, y) dσ(x) .

(ii) If f is a Hölder continuous function in Ω and uD ∈ C(∂Ω), thenthe function u given by (2.1.19) is a classical solution of the inhomogeneousDirichlet problem (2.1.4a),(2.1.4b) for Poisson’s equation (1.1.14).

Proof. Assertion (i) follows readily from the application of Green’s secondformula (2.1.11) to the function v(x) = G(x, y) − Γ (x, y). The proof of (ii)requires additional results concerning the regularity of solutions of ellipticdifferential equations. We refer to chapters 9.1 and 10.1 in Jost (2002). �

2.2 Finite difference methods

We consider the boundary value problem (2.1.4a),(2.1.4b) under the assump-tions (2.1.2a),(2.1.2b) and c(x) > 0, x ∈ Ω. For the numerical solution weuse finite difference methods on the basis of an approximation of the partialderivatives in (2.1.4a) by difference quotients (cf. Chapter 3.5 in Stoer, Bu-lirsch (2002)). We begin by assuming that the domain Ω is a d-dimensionalcube.

(2.2.1) Definition: Let Ω := (a, b)d, a, b ∈ R, a < b, h := (b−a)/(N+1), N ∈N. Then, the set

Ωh := {(xi1 , · · · , xid)T | xij = a+ ijh , 0 ≤ ij ≤ N + 1 , 1 ≤ j ≤ d}

is called a grid-point set with step size h. It represents a uniform or equidistantgrid. A grid is called uniform or equidistant, if the grid-points have the samedistance from each other. We further refer to Ωh := Ωh ∩ Ω as the set ofinterior grid-points and to Γh := Ωh \Ωh as the set of boundary grid-points.

2.2 Finite difference methods 21

For the approximation of the first and second partial derivatives uxi , uxixiwe consider the following difference quotients:

(2.2.2) Definition: For u : Ω → R and i ∈ {1, · · · , d}, the difference quo-tients

D+h,iu(x) := h−1(u(x+ hei) − u(x)) ,(2.2.3a)

D−h,iu(x) := h−1(u(x) − u(x− hei)) ,(2.2.3b)

D1h,iu(x) := (2h)−1(u(x+ hei) − u(x− hei)) ,(2.2.3c)

where ei is the i-th unit vector in Rd, are called the forward, backward and

central difference quotients for the first partial derivatives uxi , 1 ≤ i ≤ d,.The difference quotients given by

(2.2.4) D2h,i,iu(x) := h−2(u(x+ hei) − 2u(x) + u(xi − hei))

are called central difference quotients for the second derivatives uxixi .

For u ∈ C2(Ω), Taylor expansion shows that the difference quotientsD±h,iu provide an approximation of uxi of order O(h), whereas for u ∈ C

4(Ω)

the central difference quotients D1h,iu and D2h,i,iu result in an approximation

of uxi and uxixi of order O(h2).

As far as the approximation of the mixed second partial derivatives uxixj , 1 ≤i 6= j ≤ d, in x ∈ Ωh is concerned, it is obvious to use a weighted sum offunctions values of u in points of the (xi, xj)-plane which are neighbors of x.This leads to

(2.2.5) D2h,i,ju(x) =∑

|α|,|β|≤1

κα,βu(x+ αhei + βhej) ,

where κα,β ∈ R , α, β ∈ Z. Assuming sufficient smoothness of u and re-quiring that these difference quotients approximate the mixed second partialderivatives of order O(h2), by Taylor expansion we obtain the conditions

κ0,0 = −(h2)−1γ1 ,(2.2.6)

κα,β = (2h2)−1(γ1 − sgn(α)γ2 − sgn(β)γ3) , |α| + |β| = 1 ,

κα,β = (4h2)−1(sgn(αβ) − γ1 + sgn(α)γ2 + sgn(β)γ3) , |α| = |β| = 1 ,

where γi ∈ R, 1 ≤ i ≤ 3, are free parameters. Hence, there exist severalpossibilities for the approximation of the mixed second partial derivatives oforder O(h2). We will discuss later which is the most suited one with regardto the finite difference approximation of (2.1.4a)-(2.1.4b).An appropriate tool to characterize difference quotients are difference stars.(see Fig. 1). Here, the points displayed by • mark those points used in thedifference quotient and the number next to a point refers to the correspondingweight.


κ−1,1

κ−1,0

κ−1,−1

κ0,1 κ1,1

κ0,0 κ1,0

κ0,−1 κ1,−1

Fig. 1. Difference star for mixed second partial derivatives

We also use the following characterization

(2.2.7)

κ−1,1 κ0,1 κ1,1κ−1,0 κ0,0 κ1,0κ−1,−1 κ0,−1 κ1,−1

.

The difference quotient given by (2.2.5) is also an example for a so-calledcompact difference approximation. A difference approximation

∑

|α|≤m

καu(x+ αh) , α ∈ Zd , m ∈ N ,

and the associated difference star are called compact, if besides the gridpoint x only its next neighbors are used in the difference approximation, i.e.,if m = 1.

An important special case of (2.1.4a) is Poisson’s equation (cf. (1.1.14)),

i.e., A = −∆ where ∆ is the Laplace operator ∆ =∑d

i=1 uxixi . In view of thedefinition of the central difference quotients for the second partial derivativesuxixi , we use the following difference approximation of the Laplace operator:

(2.2.8) Definition: Let D2h,i.i be given as in Definition (??). Then, the dif-ference operator

∆h :=

d∑

i=1

D2h,i,i

is called the discrete Laplace operator. In case u ∈ C4(Ω), ∆hu provides anapproximation of ∆u of order O(h2).For d = 2, the difference star associated with the difference operator −∆h isdisplayed in Fig. 2. Since exactly five grid points are involved in the differenceapproximation, the difference star is referred to as the five-point differencestar.

In order to derive compact difference approximations of −∆u in two spacedimensions, as in (2.2.5) we consider


− 1h24h2 − 1h2

− 1h2

− 1h2 −1

3h2

− 13h2

− 13h2

− 13h2 −1

3h2

83h2 − 13h2

− 13h2 −1

3h2

Fig. 2. Five-point and nine-point difference stars for the approximation of thedifferential operator −∆ in two space dimensions

∑

|α|,|β|≤1

κα,βu(x+ αhei + βhej) , α, β ∈ Z

and try to determine the weights κα,β such that we obtain a difference appro-ximation of order O(h2) for sufficiently smooth u. This leads to the conditions

κ1,1 = κ1,−1 = κ−1,1 = κ−1,−1 ,

κ1,0 = κ0,1 = κ0,−1 = 4κ−1,0 ,

κ1,1 + 4κ1,0 + κ0,0 = 0 ,

2κ1,1 + κ1,0 = −1 .

For κ1,1 = 0 and κ1,0 = −1/h2 we get the five-point difference star. Forκ1,1 = κ1,0 = −1/(3h

2) we obtain the nine-point difference star displayedin Fig. 2 (right). However, it is not possible to obtain the order O(h3). Onthe other hand, using non-compact difference approximations one can realizeapproximations of higher order than 2.

In case of a more general bounded domain Ω ⊂ Rd with boundary Γ , thedefinition of the set of interior grid points has to be renewed.

(2.2.9) Definition: Let Ω ⊂ Rd be a bounded, simply connected domain withboundary Γ and Rdh, h > 0 the grid-point set given by

Rdh := {x = (x1, · · · , xd)

T | xi = ih , i ∈ Z} .

Then, the set Ωh := Rdh∩Ω is called the set of interior grid points. A grid point

x ∈ Γ is called a boundary grid point, if there exists an interior grid pointx∗ ∈ Ωh such that x = x

∗ +αhei for some i ∈ {1, · · · , d} and α ∈ R, |α| < 1.In this case, x∗ is said to be next to the boundary. All interior grid points thatare not next to the boundary are called far from the boundary. We denote byΓh, Ω

Γh and Ω

oh the set of boundary grid points, the set of grid points next to

the boundary, and the set of grid points far from the boundary, respectively(see Fig. 3). We set Ωh := Ωh ∪ Γh.


The line between two neighboring grid points along a grid-line is called anedge, and the two grid points are called the vertices of the edge. We calla grid-point set discretely connected, if any two grid points in Ωh can beconnected by a path consisting of a finite number of edges with vertices inΩh.

Ω

xΓ

ν

x∗Γ

x1

x2

Fig. 3. Grid point next to the boundary • and boundary grid points ◦ in a do-main with curved boundary (left) and approximation of the normal derivative in aboundary grid point (right)

Remark.Observe that an interior grid point x can be next to the boundaryalthough x± hei ∈ Ωh, 1 ≤ i ≤ d.

The definition of difference quotients and of the discrete Laplace operatoralso has to be modified. This leads to the Shortley-Weller approximation.

(2.2.10) Definition: Let Ωh ⊂ Rd be a grid-point set as in Definition (2.2.9)

and x ∈ Ωh, x ± h±i ei ∈ Ωh, h

±i = α

±i h, |α

±i | ≤ 1, 1 ≤ i ≤ d. For the

approximation of uxi in x, the forward and backward difference quotientsare given by

D+h,iu(x) :=(

u(x+ h+i ei) − u(x))

/h+i ,

D−h,iu(x) :=(

u(x) − u(x− h−i ei))

/h−i .

The central difference quotient for the second derivatives uxixi is given by

D2h,iu(x) :=2

h+i + h−i

(u(x+ h+i ei) − u(x)

h+i+u(x− h−i ei) − u(x)

h−i

)

.

For the mixed second derivatives uxixj , 1 ≤ i 6= j ≤ d, we may proceed as in(2.2.5).We define the discrete Laplace operator ∆h as in Definition (2.2.8). For d = 2and x = (x1, x2) ∈ Ωh such that (x1 ± α

±1 h, x2 ± α

±2 h) ∈ Ωh, we obtain the

explicit representation


∆hu(x) :=2

h2

“ 1

α+1 (α+1 + α

−1 )

u(x1 + α+1 h, x2) +

1

α−1 (α+1 + α

−1 )

u(x1 − α−1 h, x2) +

+1

α+2 (α+2 + α

−2 )

u(x1, x2 + α+2 h, ) +

1

α−2 (α+1 + α

−2 )

u(x1, x2 − α−2 h) −

−(1

α+1 α−1

+1

α+2 α−2

)u(x1, x2)”

.

Remark.(i) In grid points next to the boundary, the central difference quo-tient for the second derivatives and the discrete Laplace operator in generalonly provide an approximation of uxixi and ∆u of order O(h).(ii) The definition of the difference quotients and the discrete Laplace opera-tor in (2.2.10) can be easily generalized to non-equidistant grids.

For the discretization of the normal derivative uν = ν · ∇u in xΓ ∈ Γh,where ν denotes the exterior normal unit vector in xν , let x

∗Γ ∈ Ωh be the

first intersection in Ωh of the normal through xΓ and a grid line (cf. Fig. 3(right)). Let xi, 1 ≤ i ≤ 2, be those grid points on that grid line next to x∗Γassuming that the grid is sufficiently fine such that xi ∈ Ωh, 1 ≤ i ≤ 2. Werefer to ũ(x∗Γ ) as the value obtained by linear interpolation with respect tou(x1) and u(x2). Then, an approximation of the normal derivative ν · ∇u inxΓ is given by

(2.2.11) ∂hν u(xΓ ) :=u(xΓ ) − ũ(x∗Γ )

|xΓ − x∗Γ |.

This approach can also be used for directional derivatives uµ = µ · ∇u withrespect to a non-tangential vector µ in xΓ .

We compute approximations within the linear space of functions definedon a grid-point set.

(2.2.12) Definition: Let Ωh be a grid-point set as in (2.2.10). A functionuh : Ωh → R is called a grid function We refer to C(Ωh) as the linear spaceof grid functions. The space C(Ωh) will be equipped with the norm

(2.2.13) ‖uh‖h := ‖uh‖ℓ∞(Ωh) = maxx∈Ωh

|uh(x)| .

We first consider a finite difference method for the approximation ofPoisson’s equation with inhomogeneous Dirichlet boundary conditions bymeans of the discrete Laplace operator. Given grid functions fh ∈ C(Ωh)and uDh ∈ C(Γh), we are looking for uh ∈ C(Ωh) such that

−∆huh = fh in Ωh ,(2.2.14a)

uh = uDh auf Γh .(2.2.14b)

The existence and uniqueness of a solution of (2.2.14a),(2.2.14b) can be showneither using a discrete maximum principle, a discrete Green’s function (cf.,


e.g., Jost (2002) or by means of the investigation of the linear algebraic system(2.2.14a),(2.2.14b). Here, we will follow the latter approach. For simplicity,we will restrict ourselves to the case d = 2 and Ω = (a, b)2 and assume auniform grid Ωh. The unknown values of the grid function uh in the interiorgrid points x ∈ Ωh can be interpreted as the components of a vector. Thestructure of the resulting linear algebraic system depends on the ordering ofthe grid points.

(2.2.15) Definition: Assume Ω = (a, b)2, a, b ∈ R, a < b, and let Ωh be auniform grid of step size h := (b − a)/N,N ∈ N, with interior grid pointsx = (xi, xj)

T , 1 ≤ i, j ≤ N, where xi = a+ ih. An ordering of the grid pointsfrom ’bottom’ to ’top’ and from ’left’ to ’right’ according to

x(i−1)N+j = (a+ ih, a+ jh)T , 1 ≤ i, j ≤ N ,

is called a lexicographic ordering. If we first count the grid points with eveni+ j and then those with odd i+ j, the ordering is said to be a checkerboardordering.

If we order the interior grid points according to Definition (2.2.15), thevalues of the grid function in the interior grid points can be assigned a vectorwith nh := N

2 components. For simplifying the notation, this vector willbe denoted in the same way as the grid function, i.e., we define uh ∈ R

nh

according to uh,i := uh(xi), 1 ≤ i ≤ nh. Then, (2.2.14a),(2.2.14b) correspondsto a linear algebraic system

(2.2.16) Ahuh = bh

with the coefficient matrix Ah ∈ Rnh×nh and a vector bh ∈ Rnh whose

components are given by the function values of fh and uDh .

In case of a lexicographic ordering of the grid points, Ah is the block tridia-gonal matrix

(2.2.17) Ah = h−2 tridiag(−I, T,−I) ,

where the nh diagonal blocks T ∈ RN×N are tridiagonal matrices of the form

(2.2.18) T = tridiag(−1, 4,−1)

and the off-diagonal blocks represent the negative N ×N unit matrix.In view of (2.2.17) and (2.2.18) it is easily seen that Ah is a symmetric positivedefinite matrix. with the eigenvalues

(2.2.19) λij(Ah) = 4h−2

(

sin2(iπh/2) + sin2(jπh/2))

, 1 ≤ i, j ≤ N .

The components of the associated orthonormal eigenvectors x(ij) ∈ Rnh , 1 ≤i, j ≤ N, are given by


x(ij)kℓ =

h

2sin(ihkπ) sin(jhℓπ) , 1 ≤ k, ℓ ≤ N .

The structure of the linear algebraic system (2.2.17) in case of a checkerboardordering of the grid points is left as an exercise.

The properties of Ah imply the unique solvability of the linear system(2.2.16) and hence of the finite difference method (2.2.14a),(2.2.14b). Theproperties of Ah further imply the convergence of uh to the solution u ofPoisson’s equation provided fh → f and uDh → u

D for h → 0. We wantto address the issue of the convergence of finite difference methods in amore general framework. For that purpose we consider such a method forthe approximate solution of the boundary value problem (2.1.4a),(2.1.4b) inΩ = (a, b)2 with respect to a uniform grid Ωh with step size h > 0. Forthe approximation of the second partial derivatives uxixi and the first partialderivatives uxi we use central difference quotients. The mixed second partialderivatives uxixj , i 6= j, are approximated by difference quotients of the form(2.2.5),(2.2.6). This leads to the finite difference method

Lhuh = fh in Ωh ,(2.2.20a)

uh = uDh on Γh .(2.2.20b)

Here, Lh stands for the difference operator

Lhuh(x) := −2∑

i,j=1

aij(x)D2h,i,juh(x) +

2∑

i=1

bi(x)D1h,iuh(x) + c(x)uh(x)

and fh ∈ C(Ωh), uDh ∈ C(Γh) are given grid functions.

Assuming the existence and uniqueness of a solution uh, we define the conver-gence and the order of convergence of the finite difference method as follows:

(2.2.21) Definition: Given Ω := (a, b)2, a, b ∈ R, a < b, and a uniform gridΩh with step size h > 0, let u : Ω → R be the classical solution of theboundary value problem (2.1.4a),(2.1.4b) and let uh ∈ C(Ωh) be the solutionof the finite difference method (2.2.20a),(2.2.20b). Then, the finite differencemethod is said to be convergent, if

‖u− uh‖h = maxx∈Ωh

|u(x) − uh(x)| → 0 for h→ 0 .

It is said to be convergent of order p, if for sufficiently small h there holds

‖u− uh‖h = maxx∈Ωh

|u(x) − uh(x)| ≤ C hp ,

where C > 0 is a constant independent of h.

Sufficient conditions for the convergence can be stated in terms of the con-sistency with the boundary value problem (2.1.4a),(2.1.4b) and the stability


of the finite difference method. The consistency is measured by means of thelocal discretization error which is obtained by evaluating the finite differencemethod for the solution of the boundary value problem.

(2.2.22) Definition: Let u be the classical solution of the boundary valueproblem (2.1.4a),(2.1.4b) and let Lh and fh, u

Dh be the difference opera-

tor and the grid functions from (2.2.20a),(2.2.20b). Then, the grid functionτh ∈ C(Ωh) given by

τh(x) :=

{Lhu(x) − fh(x) , x ∈ Ωhu(x) − uDh (x) , x ∈ Γh

is called the local discretization error. The finite difference method (2.2.20a),(2.2.20b) is said to be consistent with the boundary value problem (2.1.4a),(2.1.4b), if

maxx∈Ωh

|τh(x)| → 0 as h→ 0 .

It is called consistent of order p, if there exists C > 0, independent of h suchthat

maxx∈Ωh

|τh(x)| ≤ C hp .

(2.2.23) Lemma: Assume that the boundary value problem (2.1.4a),(2.1.4b)has a classical solution u such that u ∈ C4(Ω), Further, suppose that thereholds

maxx∈Ωh

|f(x) − fh(x)| = O(h2) , max

x∈Γh|uD(x) − uDh (x)| = O(h

2) .

Then, the finite difference method (2.2.20a),(2.2.20b) is consistent with theboundary value problem (2.1.4a),(2.1.4b) of order p = 2.

Proof. Under the assumptions on fh and uDh the proof follows easily

taking into account the properties of the difference quotients used in thedefinition of the difference operator Lh. �

Remark.The result of Lemma (2.2.23) can be generalized without difficul-ties to higher dimensions. In case of more general bounded domains and theuse of the Shortley-Weller approximation, we can only expect the order ofconsistency p = 1.

For linear finite difference methods the stability corresponds to theasymptotically uniform boundedness of the inverse of the difference operator.

(2.2.24) Definition: The finite difference method (2.2.20a),(2.2.20b) is calledstable, if there exists hmax > 0 such that for h ≤ hmax the operator Lh isinvertible and there is a constant C > 0 independent of h such that

‖L−1h ‖ ≤ C .


(2.2.25) Theorem: Assume that (2.2.20a),(2.2.20b) is stable and consistentwith the boundary value problem (2.1.4a),(2.1.4b). Then, the finite differencemethod is convergent and the order of convergence is the same as the orderof consistency.

Proof. The proof is an immediate consequence of

Lh(u(x) − uh(x)) = τh(x) , x ∈ Ωh ,

and u(x) − uh(x) = τh(x), x ∈ Γh. �

If the finite difference method is consistent with the boundary value pro-blem, a sufficient condition for stability is the L0-matrix property of thematrix associated with the finite difference method.

(2.2.26) Definition: A matrix A = (aij)ni,j=1 is called an L0-matrix, if aij ≤

0, 1 ≤ i 6= j ≤ n. An L0-matrix is said to be an L-matrix, if aii > 0, 1 ≤ i ≤ n.An L0-matrix is called an M-matrix, if it is regular and A

−1 ≥ 0.

We have the following characterization of M-matrices.

(2.2.27) Theorem: An L0-matrix A ∈ Rn×n is an M-matrix if and only ifthere exists a vector z ∈ Rn such that z > 0 and Az > 0. Moreover, thereholds

(2.2.28) ‖A−1‖ ≤‖z‖∞

min1≤i≤n

(Az)i,

where ‖ · ‖ refers to the row-sum norm.

Proof. Let A be an M-matrix. Then A−1 ≥ 0 and we may choose z =A−1e, where e = (1, · · · , 1)T .

Conversely, assume z > 0 and Az > 0, i.e.,∑n

j=1 aijzj > 0, 1 ≤ i ≤ n.Since aijzj ≤ 0 for i 6= j, we must have aii > 0, 1 ≤ i ≤ n. Hence, the matrixDA := diag(A) is invertible. We set P := D

−1A (DA − A) and hence, we have

A = DA(I − P ). Obviously, P ≥ 0 and (I − P )z = D−1A Az > 0, whence

Pz < z. For the special norm

‖x‖z := max1≤i≤n

|xi|

zi

we thus obtain

‖P‖z = max‖x‖z=1

‖Px‖z = ‖Pz‖z = max1≤i≤n

(Pz)izi

< 1.

Consequently, I − P is invertible with (I − P )−1 =∑∞

i=0 Pi ≥ 0. We then

conclude A−1 = (I − P )−1D−1A ≥ 0.


In order to prove (2.2.28), for b ∈ Rn let x ∈ Rn be the solution of Ax = b.Then, there holds

(2.2.29) ±x = ±A−1b ≤ ‖b‖inftyA−1e.

Moreover, Az ≥ min1≤i≤n(Az)ie, whence

(2.2.30) A−1e ≤z

min1≤i≤n

(Az)i.

From (2.2.29) and (2.2.30) we conclude

‖A−1b‖∞ = ‖x‖∞ ≤ ‖b‖∞‖z‖∞

min1≤i≤n

(Az)i,

which gives the assertion. �We apply the previous result to the finite difference approximation under

consideration.

(2.2.31) Lemma: Assume that (2.2.20a),(2.2.20b)is consistent with the bound-ary value problem (2.1.4a),(2.1.4b), and let Ah ∈ R

nh×nh be the coefficientmatrix of the linear algebraic system associated with the finite differencemethod. If Ah is an L0-matrix, then the finite difference method is stable.

Proof. In view of Theorem (2.2.27) it suffices to find zh ∈ Rnh such that

zh > 0, Ahzh > 0 and the upper bound in

(2.2.32) ‖A−1h ‖ ≤‖zh‖∞

min1≤i≤nh

(Ahzh)i

is asymptotically uniformly bounded. Without restriction of generality wemay assume that uD = 0 in (2.1.4b). Let u∗ be the solution of (2.1.4a),(2.1.4b)for f = 1 and uD = 0. According to the maximum principle due to Hopf(2.1.5) we have u∗(x) > 0, x ∈ Ω. Moreover, Lu∗(x) ≥ 1, x ∈ Ω. We definethe vector zh according to zh := (u

∗(x1), · · · , u∗(xnh))

T . In view of the con-sistency, for sufficiently small h it follows that Ahzh ≥ 1/2. Hence, (2.2.32)implies the asymptotically uniform boundedness of A−1h and thus the stabilityof the finite difference method. �

According to Lemma (2.2.31) the stability of the finite difference methodfollows from the L0-matrix property of Ah ∈ R

nh×nh . It is easily shown thatfor a12(x) = 0, x ∈ Ω, the difference operator Lh gives rise to the differencestar

1

h2

0 h2 b2 − a22 0−h2 b1 − a11 2(a11 + a22)

h2 b1 − a11

0 −h2 b2 − a22 0

.


Assuming h|bi(x)| < 2aii(x), 1 ≤ i ≤ 2, Ah turns out to be an M-matrix.However, for non-vanishing a12 we have to choose the difference quotientsD2h,i,juh(x), 1 ≤ i 6= j ≤ 2, from (2.2.5) with (2.2.6) depending on the sign ofa12(x). For γ1 = ±1, γ2 = γ3 = 0 in (2.2.6) we obtain the difference stars

1

2h2

−1 1 01 −2 10 1 −1

,1

2h2

0 −1 1−1 2 −1

1 −1 0

.

We denote by D2,−h,i,j and by D2,+h,i,j the difference operators associated with

the left and right difference star. Then, if ±a12(x) > 0 and we use the differ-ence quotient D2,±h,i,juh(x) in the definition of Ah, it is easily shown that thecondition

(2.2.33) h |bi(x)| < 2 (aii(x) − |a12(x)|) , 1 ≤ i ≤ 2 , x ∈ Ω

implies the L0-matrix property of Ah.

(2.2.34) Lemma: Let Ah be the difference operator from (2.2.20a) such thatD2h,i,juh(x) = D

2,±h,i,juh(x), 1 ≤ i 6= j ≤ 2, if ±a12(x) > 0, x ∈ Ω, and suppose

that (2.2.33) holds true. Then, the finite difference method (2.2.20a),(2.2.20b)is stable.

Proof. Condition (2.2.33) implies the M-matrix property of the matrixassociated with the difference operator and hence, the assertion follows fromLemma (2.2.31). �

In summary we obtain the following result:

(2.2.35) Theorem: Under the assumptions of Lemma (2.2.23) and Lemma(2.2.34) the finite difference method (2.2.20a),(2.2.20b) is convergent of orderp = 2.

Proof. Lemma (2.2.23) proves the consistency of order p = 2 and Lemma(2.2.34) guarantees the stability of the finite difference method. Theorem(2.2.25) allows to conclude. �

Remark.(i) For general bounded domains and the Shortley-Weller approxi-mation the stability of finite difference methods can be shown under appro-priate conditions on the coefficient functions of the difference operator. Ac-cording to Remark (10.3.2.26) the order of convergence reduces to p = 1.(ii) If the coefficient function bi significantly dominates the coefficient func-tion aii in absolute value, the step size h has to be chosen unrealistically smallin order to guarantee condition (2.2.33). Such a problem is called convection-dominated, since for diffusion-convection-reaction problems as described by(2.1.4a) the term b · ∇u represents the convective part of the process. Forsuch problems, the discretization of the first partial derivatives is done by


the forward or the backward difference quotient depending on the sign of bi(upwind discretization).

The convergence analysis presented in this section relies on smoothnessassumptions with respect to the classical solution of the boundary value pro-blem. For many problems, such assumptions do not hold true, e.g., in caseof singularities due to inconsistent boundary conditions or due to the geome-try of the domain. As far as the development, analysis and implementation offinite difference methods for such problems is concerned, we refer to Samarskii,Lazarov and Makarov (1987).

2.3 Weak solutions

We consider a boundary value problem for a linear second order elliptic dif-ferential operator in selfadjoint form with homogeneous Dirichlet boundaryconditions

Lu = f in Ω,(2.3.1a)

u = 0 on Γ,(2.3.1b)

whereLu := −∇

(

a∇u)

+ cu.

We assume Ω ⊂ Rd to be a bounded Lipschitz domain with boundary Γ =∂Ω. We further suppose that a = (aij)

di,j=1 with aij ∈ L

∞(Ω), 1 ≤ i, j ≤ d,is a uniformly positive definite matrix-valued function, i.e., there exists aconstant α > 0 such that for almost all x ∈ Ω

(2.3.2)d∑

i,j=1

aij(x)ξiξj ≥ α|ξ|2 , ξ = (ξ1, · · · , ξd)

T ∈ Rd.

Moreover, we assume c ∈ L∞(Ω) such that c(x) ≥ 0 for almost all x ∈ Ω andf ∈ L2(Ω).Multiplication of (2.3.1a) by a function v ∈ C∞0 (Ω), integration over Ω andthe application of Green’s formula yields

(2.3.3)

∫

Ω

(

a∇u · ∇v + cuv)

dx =

∫

Ω

fv dx.

Obviously, the integrals in (2.3.3) are well defined for functions u and v in theSobolev space H10 (Ω) (see section (1.2)). Therefore, we introduce the notionof weak (generalized) solution.

(2.3.4) Definition: A function u ∈ H10 (Ω) is called a weak or generalizedsolution of (2.3.1a),(2.3.1b), if (2.3.3) is satisfied for all v ∈ H10 (Ω).

2.3 Weak solutions 33

In case f ∈ C(Ω) it is obvious that a classical solution of (2.3.1) in thesense of Definition 2.1.3 also represents a solution in the weak sense. On theother hand, if u ∈ H10 (Ω) is a weak solution such that u ∈ H

2(Ω) , theapplication of Green’s formula to (2.3.3) yields

∫

Ω

(

Lu− f)v dx = 0 , v ∈ H10 (Ω),

and hence, Lu(x) = f(x) for almost all x ∈ Ω.

(2.3.5) Definition: A function u such that Lu ∈ L2(Ω), which satisfies(2.3.1a) for almost all x ∈ Ω, is called a strong solution.

If f ∈ C(Ω) and if a weak solution satisfies u ∈ Hm(Ω) for suf-ficiently large m such that Hm(Ω) →֒ C2(Ω) ∩ C0(Ω), it follows thatLu(x) = f(x), x ∈ Ω, and u(x) = 0, x ∈ Γ , i.e., u is a solution in theclassical sense.We observe that a weak solution is a solution in the strong or in the classicalsense only if it has additional smoothness properties. The study of the regu-larity of weak solutions is a central issue in the theory of partial differentialequations (cf. e.g, Evans (2000)).

The left-hand side in (2.3.3) is a bilinear form on H10 (Ω)×H10 (Ω), whereas

the right-hand side represents a bounded linear functional on H10 (Ω). There-fore, we study the existence and uniqueness of weak solutions within theframework of variational equations in Hilbert spaces:

(2.3.6) Definition: Let V be a real Hilbert space with inner product (·, ·)Vand associated norm ‖ · ‖V , let a(·, ·) : V × V → R be a bilinear form andℓ ∈ V ∗. The problem to find a function u ∈ V such that

(2.3.7) a(u, v) = ℓ(v) , v ∈ V,

is called a variational equation.

As we shall see now, the existence and uniqueness of a solution of the vari-ational equation (2.3.7) essentially requires the boundedness and V -ellipticityof the bilinear form.

(2.3.8) Definition: A bilinear form a(·, ·) : V ×V → R is said to be bounded,if there exists a constant C > 0 such that

(2.3.9) |a(u, v)| ≤ C ‖u‖V ‖v‖V , u, v ∈ V.

(2.3.10) Definition: A bilinear form a(·, ·) : V ×V → R is called V - elliptic,if there exists a constant α > 0 such that

(2.3.11) |a(u, u)| ≥ α ‖u‖2V , u ∈ V.


The following celebrated Lemma of Lax-Milgram provides an existenceand uniqueness result for variational equations of the form (2.3.7).

(2.3.12) Theorem: Let V be a Hilbert space with dual V ∗ and suppose thata(·, ·) : V × V → R is a bounded, V -elliptic bilinear form and ℓ ∈ V ∗. Then,the variational equation (2.3.7) has a unique solution u ∈ V .

Proof. We denote by A : V → V ∗ the operator associated with thebilinear form according to

(2.3.13) Au(v) = a(u, v) , u, v ∈ V.

The operator A is a bounded linear operator, since in view of (2.3.9) thereholds

‖Au‖V ∗ = supv 6=0

|Au(v)|

‖v‖V≤ C ‖u‖V ,

and hence,

‖A‖ = supu6=0

‖Au‖V ∗

‖u‖V≤ C.

We will reformulate the variational equation (2.3.7) as an operator equationin V by means of the Riesz mapping τ : V ∗ → V , ℓ 7−→ τℓ which is givenby

(2.3.14) ℓ(v) = (τℓ, v)V , v ∈ V.

It can be easily verified that the Riesz mapping is an isometry, i.e.,

(2.3.15) ‖τℓ‖V = ‖ℓ‖V ∗ , ℓ ∈ V∗.

Indeed, using the Riesz mapping, the variational equation (2.3.7) can beequivalently written as the operator equation

(2.3.16) τ(Au− ℓ) = 0 .

We prove the existence and uniqueness of a solution of (2.3.16) by an ap-plication of Banach’s fixed point theorem. For that purpose we consider themapping T : V → V defined by means of

(2.3.17) Tv := v − ρ (τ(Av − ℓ)) , ρ > 0.

Obviously, u ∈ V solves (2.3.16) if and only if u ∈ V is a fixed point of theoperator T . For the proof of the existence and uniqueness of a fixed point inV it suffices to show that the operator T is a contraction on V , i.e., thereexists a constant 0 ≤ q < 1 such that

2.4 Conforming finite element methods 35

‖Tv1 − Tv2‖V ≤ q ‖v1 − v2‖V , v1, v2 ∈ V.(2.3.18)

Setting w := v1 − v2 and taking (2.3.13) as well as (2.3.15) into account itfollows that

‖Tw‖2V = ‖w − ρ τ Aw‖2V =

= ‖w‖2 − 2 ρ (τAw,w)V + ρ2 ‖τAw‖2V =

= ‖w‖2 − 2 ρ a(w,w) + ρ2‖Aw‖2V ∗ ≤

≤(

1 − 2 ρ α+ ρ2 C2)

‖w‖2V .

We thus obtain(

1 − 2 ρ α+ ρ2 C2)

< 1 ⇐⇒ ρ <2 α

C2,

which proves the assertion. �

Example. We consider the elliptic boundary value problem (2.3.1). Theassociated bilinear form a(·, ·) : H10 (Ω) ×H

10 (Ω) → R is given by

a(u, v) :=

∫

Ω

(

a∇u · ∇v + cuv)

dx , u, v ∈ H10 (Ω).

in view of aij ∈ L∞(Ω), 1 ≤ i, j ≤ d, and c ∈ L∞(Ω) we easily deduce theboundedness. On the other hand, (2.3.2) and c(x) ≥ 0 f.a.a. x ∈ Ω readilyimply

a(u, u) ≥ α|u|21,Ω , u ∈ H10 (Ω).

Since on H10 (Ω) the |·|1,Ω-norm is equivalent to the ‖·‖1,Ω-norm, the previousinequality proves the H10 (Ω)-ellipticity. Finally, for f ∈ L

2(Ω) the linearfunctional ℓ(v) :=

∫

Ωfvdx is bounded on H10 (Ω). The Lax-Milgram lemma

(Theorem 2.3.12) implies the existence and uniqueness of a weak solution of(2.3.1).

2.4 Conforming finite element methods

We assume that V ⊆ H1(Ω) is a Hilbert space, a(·, ·) : V × V → R abounded, V-elliptic bilinear form and ℓ : V → R a bounded linear functional.The Lemma of Lax-Milgram (2.3.12) implies that the variational equation

(2.4.1) a(u, v) = ℓ(v) , v ∈ V

admits a unique solution u ∈ V . The Galerkin method consists in the approx-imation of u ∈ V by an approximate solution uh ∈ Vh in a finite dimensionalsubspace Vh ⊂ V,dim Vh = nh, which is the solution of the variational equa-tion (2.4.1) restricted to Vh, i.e.,


(2.4.2) a(uh, vh) = ℓ(vh) , vh ∈ Vh .

Another application of the Lemma of Lax-Milgram reveals the existence anduniqueness of uh ∈ Vh.

Remark.If the bilinear form a(·, ·) : V × V → R is symmetric, the Galerkinmethod is also referred to as the Ritz-Galerkin method.

Specifying a basis (ϕ(i)h )

nhi=1 of Vh, it is easily seen that the Galerkin method

leads to linear algebraic system. Since the solution uh ∈ Vh of (2.4.2) can bewritten as a linear combination of the basis functions according to

(2.4.3) uh =

nh∑

j=1

αj ϕ(j)h

uh satisfies the variational equation (2.4.2) if and only if

(2.4.4)

nh∑

j=1

a(ϕ(j)h , ϕ

(i)h ) αj = ℓ(ϕ

(i)h ) , 1 ≤ i ≤ nh .

Obviously, (2.4.4) represents a linear algebraic system

(2.4.5) Ah αh = bh

in the unknown vector αh = (α1, ..., αnh)T . The stiffness matrix Ah ∈ R

nh×nh

and the load vector bh = (bh,1, · · · , bh,nh)T ∈ Rnh are given by

(2.4.6) Ah :=

a(ϕ(1)h , ϕ

(1)h ) · · · a(ϕ

(nh)h , ϕ

(1)h )

· · · · ·· · · · ·

a(ϕ(1)h , ϕ

(nh)h ) · · · a(ϕ

(nh)h , ϕ

(nh)h )

and

(2.4.7) bh,i := ℓ(ϕ(i)h ) , 1 ≤ i ≤ nh .

Remark. We remark that the notions ’stiffness matrix’ and ’load vector’ arestemming from mechanical applications.

With regard to the appropriate choice of finite dimensional subspaces, thereare essentially two aspects, namely the

– efficient numerical solution of the linear algebraic system (2.4.5) and the– accuracy of the approximation of the solution u ∈ V of (2.4.1) by the

solution uh ∈ Vh of (2.4.2).


The latter aspect will be addressed in Céa’s Lemma. Céa’s Lemma statesthat under the assumptions of the Lax-Milgram Lemma the accuracy of theapproximate solution corresponds to the best approximation of the solutionu ∈ V by a function in Vh. Hence, the issue is reduced to a problem inapproximation theory. Céa’s Lemma is based on the observation that theerror u− uh is a-orthogonal with respect to Vh, i.e.,

(2.4.8) a(u− uh, vh) = 0 , vh ∈ Vh .

This property is called the Galerkin orthogonality.

(2.4.9) Definition: If the bilinear form a(·, ·) is symmetric and V-elliptic,it defines an inner product (·, ·)a := a(·, ·) and an associated norm ‖ · ‖a :=a(·, ·)1/2 on V which is called the energy norm. Galerkin orthogonality (2.4.8)means that the solution uh ∈ Vh of (2.4.2) is the projection of the solutionu ∈ V of (2.4.1) onto Vh. Therefore, the approximate solution is referred toas the elliptic projection.

(2.4.10) Theorem: Under the assumptions of the Lemma of Lax-Milgram(2.3.12) let u ∈ V and uh ∈ Vh be the unique solutions of (2.4.1) and (2.4.2).Then, there holds

(2.4.11) ‖u− uh‖V ≤C

αinf

vh∈Vh‖u− vh‖V .

Proof. Using the V-ellipticity and the boundedness of a(·, ·) as well asthe Galerkin orthogonality (2.4.8), for an arbitrary vh ∈ Vh we have

α ‖u− uh‖2V ≤ a(u− uh, u− uh) =

= a(u− uh, u− vh) + a(u− uh, vh − uh)︸︷︷︸

= 0

≤

≤ C ‖u− uh‖V ‖u− vh‖V ,

which implies (2.4.11). �

The efficient numerical solution of the linear algebraic system (2.4.5) re-sulting from the Galerkin method depends on the structure of the stiffnessmatrix Ah. In particular, we are interested in sparsely populated matriceswith a specific sparsity pattern which can be realized by an appropriate choiceof the basis functions of Vh ⊂ V with minimal support. In case of conformingfinite element methods, Vh is constructed on the basis of triangulations of thecomputational domain Ω ⊂ Rd.

(2.4.12) Definition: Let Ω ⊂ Rd be a bounded Lipschitz domain. A triangu-lation Th(Ω) of Ω is a partition of Ω into a finite number of subsets T ,calledfinite elements such that


Ω =⋃

T∈Th(Ω)

T ,(2.4.13a)

T = T , int(T ) 6= ∅ , T ∈ Th(Ω) ,(2.4.13b)

int(T1) ∩ int(T2) = ∅ for all T1, T2 ∈ Th(Ω) , T1 6= T2 ,(2.4.13c)

∂T , T ∈ Th(Ω) is Lipschitz continuous .(2.4.13d)

We distinguish between two types of finite elements: simplicial elementsand quadrilateral elements.

(2.4.14) Definition: A simplex T in Rd is the convex hull of d + 1 pointsaj = (aij)

di=1 ∈ R

d:

T = { x =d+1∑

j=1

λj aj | 0 ≤ λj ≤ 1 ,d+1∑

j=1

λj = 1 } .

A simplex T is called non-degenerate, if every point x ∈ Rd can be uniquelyrepresented according to

x =

d+1∑

j=1

λj aj , λj ∈ R ,d+1∑

j=1

λj = 1 .

In R2, a simplex is a triangle, in R3 a tetrahedron.

Remark. The non-degeneracy of a simplex is related to the unique solv-ability of the linear algebraic system

a11 · · · a1,d+1· · · · ·· · · · ·ad1 · · · ad,d+11 · · · 1

︸︷︷︸

=: A

λ1··λdλd+1

=

x1··xd1

.(2.4.15)

Obviously, T is non-degenerate if and only if the matrix A is non-singular.

(2.4.16) Definition: The points aj , 1 ≤ j ≤ d + 1, of a simplex T are calledvertices. An m-dimensional face of a simplex T , 0 ≤ m ≤ d, is a simplexin Rm, whose vertices are vertices of T in D. A one-dimensional face is saidto be an edge. For a subset D ⊆ Ω we denote by Fh(D) and Eh(D) the setsof the (d − 1)-dimensional faces and one-dimensional edges of Th(Ω). Thesimplex T̂ with the vertices â1 = (0, ..., 0)

T and âi+1 = ei, 1 ≤ i ≤ d, is calledsimplicial reference element.

(2.4.17) Definition: The barycentric coordinates λj , 1 ≤ j ≤ d+1, of a pointx ∈ Rd with respect to the d+1 vertices aj of a non-degenerate simplex T are


the components of the unique solution of the linear algebraic system (2.4.15).The center of gravity xS of a non-degenerate simplex T is the point satisfying

λj(xS) =1

d+ 1, 1 ≤ j ≤ d+ 1 .

(2.4.18) Lemma: Any non-degenerate simplex T ⊂ Rd is the image of thesimplicial reference element T̂ under an affine transformation

FT : Rd → Rd

x̂ 7−→ FT (x̂) = BT x̂+ bT

with a non-singular matrix BT ∈ Rd×d and a vector bT ∈ R

d.


(2.4.19) Definition: A triangulation Th(Ω) of a polyhedral domain Ω ⊂ Rd

is called a simplicial triangulation, if any of its elements is a non-degeneratesimplex in Rd.

An alternative to simplicial triangulations is to use quadrilateral elements.

(2.4.20) Definition: A quadrilateral element T in Rd is the tensor productof d intervals [ci, di], ci ≤ di, 1 ≤ i ≤ d, i.e.,

T =d∏

i=1

[ci, di] = { x = (x1, ..., xd)T | ci ≤ xi ≤ di , 1 ≤ i ≤ d } .

A quadrilateral element in R2 is a rectangle, in R3 cube. A quadrilateralelement T is said to be non-degenerate, if ci < di, 1 ≤ i ≤ d.The quadrilateral element T̂ := [0, 1]d (unit cube) is called the quadrilateralreference element in Rd. The points

aj = (aj1 , ..., ajd)T , aji = ci or aji = di , 1 ≤ i ≤ d,

of a quadrilateral element T are called vertices. An m-dimensional face ofa quadrilateral element T, 1 ≤ m ≤ d − 1, is a quadrilateral element in Rm

whose vertices are vertices of T . A one-dimensional face is said to be an edge.For D ⊆ Ω we refer to Fh(D) and Eh(D) as the sets of (d− 1)-dimensionalfaces and one-dimensional edges of Th(Ω) in D.

(2.4.21) Lemma: Any non-degenerate quadrilateral element T ⊂ Rd is theimage of the quadrilateral reference element T̂ under a diagonal-affine trans-formation

FT : Rd → Rd

x̂ 7−→ FT (x̂) = BT x̂+ bT

with a non-singular diagonal matrix BT = (bTii)

di=1 and a vector bT ∈ R

d.



(2.4.22) Definition: A triangulation Th(Ω) of a quadrilateral domain Ω ⊂Rd is called a quadrilateral triangulation, if any of its elements T is a quadri-lateral element.

Given a triangulation Th(Ω), conforming finite element functions will bespecified locally for the elements T ∈ Th(Ω) and then composed such thatthe resulting global function belongs to the underlying function space V .

(2.4.23) Definition: Let Th(Ω) be a triangulation of Ω ⊂ Rd and T ∈ Th(Ω).

Moreover, let PT be a linear space of functions p : T → R such that dim PT =nT . The elements of PT are called local trial functions. For linear boundedfunctionals ℓi : PT → R, 1 ≤ i ≤ nT , we define

(2.4.24) ΣT := { ℓi(p) | p ∈ PT , 1 ≤ i ≤ nT } .

The elements of the set ΣT are called degrees of freedom.

(2.4.25) Definition: Let Th(Ω) be a triangulation of Ω ⊂ Rd, T ∈ Th(Ω)

and PT , ΣT as in Definition (2.4.23). Then, the triple (T, PT , ΣT ) is called afinite element.A finite element (T, PT , ΣT ) is said to be unisolvent, if any p ∈ PT is uniquelydetermined by the degrees of freedom in ΣT .

Remark. We remark that the notion ’finite element’ is used for T ∈ Th(Ω)as well as for the triple (T, PT , ΣT ).

The numerical computation of the elements of the stiffness matrix and thecomponents of the load vector is greatly facilitated in case of affine equivalentfinite elements.

(2.4.26) Definition: Let Th(Ω) be a triangulation of Ω ⊂ Rd, T ∈ Th(Ω) and

PT , ΣT as in Definition (2.4.23). Moreover, let (T̂ , P̂T̂ , Σ̂T̂ ) be a referenceelement. Then, the finite elements (T, PT , ΣT ), T ∈ Th(Ω) are called affineequivalent, if there exists an invertible affine mapping FT : R

d → Rd suchthat for all T ∈ Th(Ω)

T = FT (T̂ ) ,(2.4.27a)

PK = {p : T → R | p = p̂ ◦ F−1T , p̂ ∈ P̂T̂ } ,(2.4.27b)

ΣT = {ℓi : PT → R | ℓi = ℓ̂i ◦ F−1T , ℓ̂i ∈ Σ̂T̂ , 1 ≤ i ≤ nT } .(2.4.27c)

(2.4.28) Definition: Let Th(Ω) be a triangulation of Ω ⊂ Rd, T ∈ Th(Ω) and

PT , ΣT as in Definition (refdef:10.3.4.25). Then,

(2.4.29) Vh := { vh : Ω → R | vh|T ∈ PT , T ∈ Th(Ω) }

is said to be a finite element space.


The following result provides sufficient conditions for a finite elementspace Vh to be a subspace of H

1(Ω).

(2.4.30) Theorem: Let Vh be a finite element space and assume that thereholds

PT ⊂ H1(T ) , T ∈ Th(Ω) ,(2.4.31a)

Vh ⊂ C(Ω) .(2.4.31b)

Then, we have

(2.4.32) Vh ⊂ H1(Ω) .

Proof. Let vh ∈ Vh. Obviously, we have vh ∈ L2(Ω). It remains to be

shown that vh has weak first derivatives wαh ∈ L

2(Ω), |α| = 1, i.e.,

(2.4.33)

∫

Ω

vh Dαz dx = (−1)|α|

∫

Ω

wαh z dx , z ∈ C∞0 (Ω) .

Since vh|T ∈ H1(T ), we apply Green’s formula elementwise and obtain

∫

Ω

vh Dαz dx =

∑

T∈Th(Ω)

∫

T

vh Dαz dx =(2.4.34)

= −∑

T∈Th(Ω)

∫

T

Dαvh z dx +∑

T∈Th(Ω)

∫

∂T

vh ναDαz dσ =

= −∑

T∈Th(Ω)

∫

T

Dαvh z dx +∑

F∈Fh(Ω)

∫

F

[vh] ναDαz dσ ,

where [vh] denotes the jump [vh] := vh|T1 − vh|T2 across F = T1 ∩ T2, Ti ∈Th(Ω), 1 ≤ i ≤ 2,. But vh ∈ C(Ω) and hence, [vh] = 0 in (2.4.34) which shows(2.4.33) with wαh |T := D

αvh|T , T ∈ Th(Ω). �

Corollary:Let Vh be a finite element space and assume that

PT ⊂ H1(T ) , T ∈ Th(Ω) ,(2.4.35a)

Vh ⊂ C0(Ω) .(2.4.35b)

Then, there holds

(2.4.36) Vh ⊂ H10 (Ω) .

Proof. The proof is an immediate consequence of Theorem (2.4.30). �

In the following we will be concerned with the construction of simplicialLagrangian finite elements. The local trial functions are chosen as polynomials


of degree k: Let T be a simplex in Rd. For k ∈ N0 we define Pk(T ) as denthe linear space of polynomials of degree ≤ k in T , i.e., p ∈ Pk(T ), if

p(x) =∑

|α|≤k

aα xα , aα ∈ R , |α| ≤ k ,

where

xα :=d∏

i=1

xαii , α := (α1, ..., αd) ∈ Nd0 , |α| :=

d∑

i=1

αi .

We note that

(2.4.37) dim Pk(T ) =

(k + d

d

)

=(k + d)!

d! k!.

(2.4.38) Definition: Let T be a simplex in Rd. Denoting by ai, 1 ≤ i ≤ d+1,the vertices of T , the set

(2.4.39) Lk(T ) := { x =d+1∑

i=1

λi ai | λi ∈ {0,1

k, ...,

k − 1

k, 1} ,

d+1∑

i=1

λi = 1 }

is called the principal lattice of order k of T . We have

card Lk(T ) =

(k + d

d

)

.

The elements of the principal lattice of order k are called nodal points. ForD ⊆ Ω we denote by Nh(D) the set of nodal points in D.

Defining ΣT as the set of degrees of freedom as given by the values of p ∈Pk(T ) in the nodal points, we obtain a simplicial Lagrangian finite element.

(2.4.40) Definition: Let Th(Ω) be a simplicial triangulation of Ω ⊂ Rd. The

triple (T, Pk(T ), Lk(T )), T ∈ Th(Ω), is called a simplicial Lagrangian finiteelement of type (k).

(2.4.41) Lemma: A simplicial Lagrangian finite element of type (k) is uni-solvent.

Proof. The proof is left as an exercise. �D

(2.4.42) Lemma: Simplicial finite elements of type (k) are affine-equivalentto the simplicial reference element (T̂ , P̂T̂ , Σ̂T̂ ).

Proof. Let FT : Rd → Rd be the invertible affine mapping such that T =

FT (T̂ ). Then, there holds Pk(T ) = {p = p̂◦F−1T | p̂ ∈ Pk(T̂ )}. Since x ∈ Lk(T )

iff x = FT (x̂), x̂ ∈ Lk(T̂ ), we also have (2.4.27c). �

Remark. In case d = 2, a simplicial Lagrangian finite element of type (1) iscalled Courant’s triangle.


(2.4.43) Definition: The finite element space Vh consisting of simplicial La-grangian finite elements of type (k) is called simplicial Lagrangian finite ele-ment space and will be denoted by Sk(Ω, Th(Ω)).

In order to guarantee the conformity of Sk(Ω, Th(Ω)) we have to assumethat the simplicial triangulation Th(Ω) is geometrically conforming.

(2.4.44) Definition: A simplicial triangulation Th(Ω) is said to be geometri-cally conforming, if the intersection of two distinct elements of the triangu-lation is either empty or consists of a common face or a common edge or acommon vertex.

(2.4.45) Lemma: Let Th(Ω) be a geometrically conforming triangulation.Then, there holds

(2.4.46) Sk(Ω, Th(Ω)) = {vh ∈ C(Ω) | vh|T ∈ Pk(T ), T ∈ Th(Ω)} ⊂ H1(Ω).

Proof. Since Pk(T ) ⊂ H1(T ) , T ∈ Th(Ω), and in view of Theorem (2.4.30)

we have to show that Sk(Ω, Th(Ω)) ⊂ C(Ω). The proof will be given in cased = 2 and k = 2: Let Ti ∈ Th(Ω), 1 ≤ i ≤ 2, be two adjacent elementssuch that E = T1 ∩ T2 ∈ Eh(Ω) with nodal points aj ∈ Nh(E), 1 ≤ j ≤ 3.Moreover, assume pi ∈ P2(Ki), 1 ≤ i ≤ 2. Then pi|E ∈ P2(E). Since p1(aj) =p2(aj), 1 ≤ j ≤ 3, we conclude that p1|E ≡ p2|E . �

Corollary:We define the subspace

Sk,0(Ω, Th(Ω)) := {vh ∈ Sk(Ω, Th(Ω)) |(2.4.47)

vh|∂T∩∂Ω = 0, T ∈ Th(Ω), T ∩ ∂Ω 6= ∅} .

Under the same assumptions as in Lemma (2.4.45) there holds

(2.4.48) Sk,0(Ω, Th(Ω)) ⊂ H10 (Ω).

It remains to specify the nodal basis functions:

(2.4.49) Definition: Let Nh(Ω) = {xj | 1 ≤ j ≤ nh}. The function ϕi ∈Sk(Ω, Th(Ω)) as given by

(2.4.50) ϕi(xj) = δij , 1 ≤ i, j ≤ nh ,

is called the nodal basis function associated with the nodal point xi. The set ofthese functions is a basis of the Lagrangian finite element space Sk(Ω, Th(Ω)).

The representation of nodal basis functions by means of barycentric co-ordinates as well as a geometric characterization are left as exercises.

We will now deal with the construction of quadrilateral Lagrangian finite

elements: Let T :=d∏

i=1

[ci, di] ⊂ Rdbe a quadrilateral element. For k ∈ N0 we


denote by Qk(T ) the linear space of polynomials of degree ≤ k in each of thed variables xi, 1 ≤ i ≤ d, i.e., p ∈ Qk(T ) is of the form

(2.4.51) p(x) =∑

αi≤k

aα1...αd xα11 x

α22 ...x

αdd ,

whence

(2.4.52) dim Qk(T ) = (k + 1)d .

(2.4.53) Definition: Let L[k](T ) denote the point set

L[k](T ) := {x = (xi1 , ..., xid)T | xiℓ := cℓ +

iℓk

(dℓ − cℓ) ,(2.4.54)

iℓ ∈ {0, 1, ..., k} , 1 ≤ ℓ ≤ d}

and let Σ(T ) be given by

(2.4.55) ΣT := {p(x) | x ∈ L[k](T )} , p ∈ Qk(T ) .

The element (T,Qk(T ), ΣT ) is called a quadrilateral finite element of type[k]. and will be denoted by S[k](K). We refer to x ∈ L[k](T ) as the associatednodal points.

(2.4.56) Lemma: The Lagrangian finite element of type [k] is based on atensor product polynomial interpolation of Lagrangian type. The polynomialsp ∈ Qk(T ) interpolating in x ∈ L[k](T ) admit the representation

(2.4.57) p(x) =∑

y∈L[k](T )

L(x) p(y) ,

where L ∈ Qk(T ) stands for the Lagrangian fundamental polynomials

(2.4.58) L(x) =

d∏

ℓ=1

Liℓ(x) .

Here, Li, 1 ≤ i ≤ d, are the one-dimensional Lagrangian fundamental poly-nomials (cf. (2.1.1.3) in Stoer and Bulirsch (2002)).

Proof. Assume p ∈ Qk(T ) and let e ⊂ T be an edge of T according

e := [cj , dj ] ×d∏

j 6=ℓ=1

{gℓ} , gℓ ∈ {cℓ, dℓ} .

Then, p|e ∈ Pk(e) is uniquely determined by its values in xi := cj +ik (dj −

cj) ∈ e, i ∈ {0, 1, ..., k}, and has the Lagrangian representation

p|e(x) =k∑

i=0

Li(x) p|e(xi) , x ∈ e .


(2.4.59) Lemma: A Lagrangian finite element of type [k] is unisolvent.

Proof. The proof is an immediate consequence of Lemma 2.4.56. �

(2.4.60) Lemma: Let T̂ := [0, 1]d be the quadrilateral reference element and

P̂T̂ := Qk(T̂ ) , k ∈ N ,(2.4.61a)

Σ̂T̂ := {p̂(x̂) | x̂ ∈ L[k](T̂ ) , p̂ ∈ Q̂T̂ } .(2.4.61b)

The Lagrangian finite elements of type [k] are affine equivalent to the refer-ence element (T̂ , Q̂T̂ , Σ̂T̂ ).


(2.4.62) Definition: A quadrilateral triangulation Th(Ω) is called geometri-cally conforming, if the intersection of two distinct elements is either emptyor consists of a common face or a common edge or a common vertex.

(2.4.63) Definition: Let Ω ⊂ Rd be the union of a finite number of quadri-laterals in Rd and let Th(Ω) be a quadrilateral triangulation. The finiteelement space Vh consisting of quadrilateral finite elements of type [k] iscalled a quadrilateral Lagrangian finite element space and will be denotedby S[k](Ω, Th(Ω)), i.e., we have

S[k](Ω, Th(Ω)) := {vh : Ω → R | vh|T ∈ Qk(T ), T ∈ Th(Ω)} .

We further define the subspace

S[k],0(Ω, Th(Ω)) := {vh ∈ S[k](Ω, Th(Ω)) | vh|∂T∩∂Ω = 0, T ∈ Th(Ω), T∩∂Ω 6= ∅} .

(2.4.64) Lemma: Let Th(Ω) be a geometrically conforming triangulation ofthe computational domain Ω. Then, there holds

S[k](Ω, Th(Ω)) ⊂ H1(Ω) ,(2.4.65a)

S[k],0(Ω, Th(Ω)) ⊂ H10 (Ω) .(2.4.65b)


(2.4.66) Definition: Let Nh(Ω) = {x ∈ L[k](T ) | T ∈ Th(Ω)} , card(Nh(Ω)) =nh. The function ϕi ∈ S[k](Ω, Th(Ω)) as given by

(2.4.67) ϕi(xj) = δij , 1 ≤ i, j ≤ nh ,

is called the nodal basis function associated with the nodal point xi ∈ Nh(Ω).The set of these functions is a basis of S[k](Ω, Th(Ω)).


The notion ’Lagrangian finite element’ stems from the underlying polyno-mial interpolation of Lagrangian type. Another type of polynomial interpo-lation is given by Hermite interpolation which is based on the interpolationof point values and derivatives of a function (cf. Section 2.1.5 in Stoer andBulirsch (2002)). As far as the construction of Hermitian finite elements andassociated Hermitian finite element spaces is concerned, we refer to chapter2.3 in Ciarlet (2002).

The accuracy of finite element approximations uh ∈ Vh of the solutionu ∈ V of (2.4.1) can be determined by a priori error estimates of the globaldiscretization error u − uh within the framework of interpolation in Sobolevspaces.

(2.4.68) Definition: Let Th(Ω) be a simplicial triangulation of Ω ⊂ Rd, and

let (T, PT , ΣT ), T ∈ Th(Ω), where PT = Pk(T ) such that dim(PT ) = nk andΣT = {p(xi | xi ∈ Lk(T ), 1 ≤ i ≤ nk} be a simplicial Lagrangian finite

element of type (k). Moreover, let ϕ(i)T , 1 ≤ i ≤ nk, be the basis functions

associated with the nodal points xi. Then, IT : V ∩ C(T ) → PT as given by

IT v :=

nk∑

i=1

v(xi) ϕ(i)T , v ∈ V ∩ C(T )

is called the local interpolation operator. If Vh is the associated simplicialLagrangian finite element space, the operator Ih : V ∩ C(Ω) → Vh given by

Ihv|T := IT v|T , T ∈ Th(Ω)

is called the global interpolation operator.

Now, if u ∈ V is the solution of (2.4.1) such that u ∈ V ∩ Hk+1(Ω) forsufficiently large k ∈ N such that Hk+1(Ω) ⊂ C(Ω), then Ihu is well definedand Céa’s Lemma (cf. Theorem 2.4.10) implies

(2.4.69) ‖u− uh‖1,Ω ≤C

α‖u− Ihu‖1,Ω .

In order to derive an upper bound for the interpolation error ‖u−Ihu‖1,Ω , weuse that in view of the definition of Ih and due to PT = Pk(T ) ⊂ H

1(T ), T ∈Th(Ω), there holds

(2.4.70) ‖u− Ihu‖1,Ω =( ∑

T∈Th(Ω)

‖u− ITu‖21,T

)1/2

,

i.e., we obtain an estimate of the global discretization error via an estimate ofthe global interpolation error by means of estimates of the local interpolationerror ‖u − ITu‖1,T , T ∈ Th(Ω). Due to the affine equivalence of simplicialLagrangian finite elements (cf. Lemma (2.4.42)), it suffices to provide suchan estimate for the reference element T̂ . Moreover, we need estimates of affinetransformed functions in Sobolev spaces.


(2.4.71) Lemma: Let T̂ be the simplicial reference element and T = FT (T̂ )such that FT (x̂) = BT x̂ + bT , BT ∈ R

d×d, bT ∈ Rd, and let v ∈ Hm(T ),m ∈

{0, 1}. For v̂ := v ◦ FT there holds v̂ ∈ Hm(T̂ ), and there exists a constant

C > 0 only depending on d and m such that

|v̂|m,T̂ ≤ C ‖BT ‖m |det(BT )|

−1/2 |v|m,T ,(2.4.72a)

|v|m,T ≤ C ‖B−1T ‖

m |det(BT )|1/2 |v̂|m,T̂ .(2.4.72b)

Further, denoting by hT the diameter of T and by ρT the diameter of thelargest ball contained in T , i.e.,

ρT := sup{diam(K) | K is a ball such that K ⊂ T} ,

there holds

‖BT ‖ ≤hTρT̂

, ‖B−1T ‖ ≤hT̂ρT

,(2.4.73a)

|det(BT )| ≤|T |

|T̂ |.(2.4.73b)

Proof. The proof of (2.4.72a) by taking into account

|Dαv̂(x̂)| ≤ ‖Dv̂(x̂)‖ = sup‖ξ‖≤1

|Dv̂(x̂)ξ| ,

where Dv̂(x̂) stands for the total differential of v̂, and by means of the chainrule

Dv̂(x̂)ξ = Dv(x)(Bξ)

as well as the transformation rule for multiple integrals

∫

T̂

‖D(v(Fx̂))‖2 dx̂ = |det(B−1T )|

∫

T

‖Dv(x)‖2 dx .

The proof of (2.4.72b) can be done analogously. The proofs of (2.4.73a),(2.4.73b)are left as exercises. �

The following Lemma of Bramble-Hilbert is an important tool in the apriori error analysis of finite element methods.

(2.4.74) Theorem: Let Ω ⊂ Rd be a simply connected Lipschitz domain. Fork ∈ N let ℓ ∈ Hk(Ω)∗ such that Pk(Ω) ⊂ Ker(ℓ). Then, there exists a positiveconstant C depending on Ω such that for all u ∈ Hk+1(Ω)

(2.4.75) |ℓ(u)| ≤ C ‖ℓ‖Hk(Ω)∗ |u|k+1,Ω .


Proof. For p ∈ Pk(Ω) there holds

|ℓ(u)| = |ℓ(u+ p)| ≤ ‖ℓ‖Hk(Ω)∗ ‖[u]‖k+1,Ω ,

where [u] := {w ∈ Hk+1(Ω) | w − u ∈ Pk(Ω)}. We claim the existence of apositive constant C depending on Ω such that

(2.4.76) ‖[u]‖k+1,Ω ≤ C |u|k+1,Ω ,

which together with the previous inequality proves the assertion. For theproof of (2.4.76) let nk := dim(Pk(Ω)) and let ℓi, 1 ≤ i ≤ nk, be a basis ofthe dual space of Pk(Ω) such that

(2.4.77) ℓi(p) = 0 , 1 ≤ i ≤ nk , p ∈ Pk(Ω) ⇐⇒ p = 0 .

The Hahn-Banach extension theorem implies the existence of functionals ℓ̃i ∈Hk+1(Ω)∗, 1 ≤ i ≤ nk, with ℓ̃i|Pk(Ω) = ℓi. By a contradiction argument weshow

(2.4.78) ‖u‖k+1,Ω ≤ C(Ω)(

|u|k+1,Ω +nk∑

i=1

|ℓ̃i(u)|)

, u ∈ Hk+1(Ω) .

Indeed, if (2.4.78) does not hold true, there exists a sequence {uk}N, uk ∈Hk+1(Ω), k ∈ N, such that

(2.4.79) ‖uk‖k+1,Ω = 1 , k ∈ N , limk→∞

(

|uk|k+1,Ω +nk∑

i=1

|ℓ̃i(uk)|)

= 0 .

Since the sequence {uk}N is bounded in Hm+1(Ω) and since Hk+1(Ω) is

compactly embedded in Hk(Ω), there exist a subsequence N′ ⊂ N and afunction u ∈ Hm(Ω) such that

(2.4.80) ‖uk − u‖k,Ω → 0 (k ∈ N′ , k → ∞) .

On the other hand, due to (2.4.79) we have

‖uk‖k+1,Ω → 0 (k ∈ N′ , k → ∞) .

Hence, in view of (2.4.80) it follows that uk → u (k ∈ N′, k → ∞) in Hk+1(Ω)

such that

‖Dαu‖0,Ω = limk→∞

‖Dαuk‖0,Ω = 0 , |α| = k + 1 .

Consequently, we have Dαu = 0, |α| = k + 1. The fact that Ω is simplyconnected implies u ∈ Pk(Ω). Due to (2.4.79) we obtain

ℓ̃i(u) = limk→∞

ℓ̃i(uk) = 0 , 1 ≤ i ≤ nk .

Since ℓ̃i(u) = ℓi(u), (2.4.77) implies u = 0 thus contradicting (2.4.79). �

The combination of Lemma (2.4.71) and the Lemma of Bramble-Hilbert(2.4.74) results in the following estimate of the local interpolation error:


(2.4.81) Lemma: Let (T̂ , Pk(T̂ ), Σ̂T̂ ) be the simplicial Lagrangian reference

element of type (k), where k ∈ N is such that Hk+1(T̂ ) is continuously embed-ded in C(T̂ ). Then, for all elements (T, Pk(T ), ΣT ) that are affine equivalentto (T̂ , Pk(T̂ ), Σ̂T̂ ) there holds

(2.4.82) |v − IT v|m,T ≤ Chk+1TρmT

|v|k+1,T , v ∈ Hk+1(T ) , m ∈ {0, 1} ,

where C is a positive constant depending only on the reference element.

Proof. The proof relies on the polynomial preserving property of the localinterpolation operator ÎT̂ , i.e., ÎT̂ = p̂. Denoting by Î the continuous embed-

ding Î : Hk+1(T̂ ) → H1(T̂ ), it follows that for v̂ ∈ Hk+1(T̂ ) and p̂ ∈ Pk(T̂ )there holds

v̂ − ÎT̂ v̂ = (Î − ÎT̂ )(v̂ + p̂) .

Hence, the Lemma of Bramble-Hilbert (2.4.74) implies

|v̂ − ÎT̂ v̂|m,T̂ ≤ C |v̂|k+1,T̂ .

Moreover, due to Lemma (2.4.71) and in view of v̂− ÎT̂ v̂ =̂v − IT v we obtain

|v − IT v|m,T ≤ C ‖B−1T ‖

m |det(BT )|1/2 |v̂ − ÎT̂ v̂|m,T̂

and|v̂|k+1,T̂ ≤ C ‖BT ‖

k+1 |det(BT )|−1/2 |v|k+1,T .

Taking account of (2.4.73a),(2.4.73b), the combination of the previous esti-mates gives the assertion. �

Now, let H be a null sequence of positive real numbers and let {Th(Ω)}Hbe a family of geometrically conforming simplicial triangulations of Ω. Then,(2.4.82) shows that the estimate (2.4.70) of the global interpolation errordepends on the triangulations in such a way that the upper bound in (2.4.70)is unbounded, if hT /ρT → ∞, T ∈ Th(Ω) as h → 0. In order to exclude thiscase, we have to impose a further condition on the family of triangulations{Th(Ω)}H.

(2.4.83) Definition: The family {Th(Ω)}H of geometrically conforming sim-plicial triangulations is called regular, if there exists a constant σ > 0, inde-pendent of hT , T ∈ Th(Ω), such that for all T ∈ Th(Ω), h ∈ H

(2.4.84)hTρT

≤ σ .

We are now in a position to provide an a priori estimate of the globaldiscretization error in the ‖ · ‖1,Ω norm.


(2.4.85) Theorem: Assume that {Th(Ω)}H is a regular family of geometri-cally conforming simplicial triangulations of Ω. Further, suppose that {Vh}His the associated family of finite element spaces on the basis of simplicial La-grangian finite elements of type (k) where k ∈ N0 is such that the embeddingsHk+1(Ω) → C(Ω) and Hk+1(T̂ ) → C(T̂ ) are continuous. If u ∈ V is thesolution of (2.4.1) such that u ∈ V ∩ Hk+1(Ω) and if uh ∈ Vh, h ∈ H, arethe solutions of (2.4.2), then there exists a constant C > 0, depending onlyon the local geometry of the triangulations, such that

(2.4.86) ‖u− uh‖1,Ω ≤ C hk |u|k+1,Ω .

Proof. The proof is an immediate consequence of (2.4.69),(2.4.70),(2.4.82)and (2.4.84). �

For so-called (k + 1)-regular variational equations we obtain upper boundsthat only depend on h and on the data of the problem.

(2.4.87) Definition: The variational equation (2.4.1) with

ℓ(v) = (f, v)0,Ω , f ∈ L2(Ω)

is said to be k-regular, k ≥ 2, if its solution satisfies u ∈ V ∩Hk+1(Ω) andif there exists a constant C > 0 such that

(2.4.88) ‖u‖k,Ω ≤ C ‖f‖0,Ω .

Corollary: Under the assumptions of Theorem (2.4.85) suppose that thesolution u of (2.4.1) is (k + 1)-regular in the sense of Definition (2.4.87).Then, there exists a constant C > 0, depending only on the local geometry ofthe triangulations, such that

(2.4.89) ‖u− uh‖1,Ω ≤ C hk ‖f‖0,Ω .

Proof. The proof follows directly from (2.4.86) and (2.4.88). �

In view of ‖v‖0,Ω ≤ ‖v‖1,Ω , v ∈ V, and under the assumptions of Theorem(2.4.85), it follows readily from (2.4.86) that ‖u− uh‖0,Ω = O(h

k). However,the interpolation estimate ‖Ihu− u‖0,Ω = O(h

k+1) from Lemma 2.4.81 sug-gests that this estimate is not optimal. Indeed, under additional assumptionsthe optimal order O(hk+1) can be verified. The theoretical background forsuch an optimal a priori estimate of the global discretization error in the L2-norm will be provided by the Lemma of Aubin-Nitsche, which is also knownas Nitsche’s trick.

(2.4.90) Theorem: Let V be a Hilbert space with inner product (·, ·)V andlet a(·, ·) : V × V → R be a bounded, V -elliptic bilinear form and ℓ ∈ V ∗.Moreover, let H be another Hilbert space with inner product (·, ·)H such that


V is continuously and densely embedded in H and let Vh ⊂ V . Finally, letu ∈ V and uh ∈ Vh be the unique solutions of (2.4.1) and (2.4.2). Then,there holds:

(i) For any g ∈ H the adjoint variational equation

(2.4.91) a(v, zg) = (g, v)H , v ∈ V

has a unique solution zg ∈ V .

(ii) There exists a positive constant C such that for the global discretizationerror u− uh there holds

(2.4.92) ‖u− uh‖H ≤ C ‖u− uh‖V(

supg∈H

1

‖g‖Hinf

ϕh∈Vh‖zg − ϕh‖V

)

.

Proof. In view of (g, v)H = g(v), g ∈ H ⊂ V ∗, the Lemma of Lax-Milgram(2.3.12) implies that the adjoint variational equation (2.4.91) admits a uniquesolution which proves (i). For the proof of (ii) let v = u− uh ∈ V in (2.4.91).Then, we have

a(u− uh, zg) = (g, u− uh)H .

On the other hand, choosing v = vh = ϕh ∈ Vh in (2.4.1),(2.4.2), we obtain

a(u− uh, ϕh) = 0 , ϕh ∈ Vh .

The combination of both equations yields

a(u− uh, zg − ϕh) = (g, u− uh)H ,

whence|(g, u− uh)H | ≤ C ‖u− uh‖V inf

ϕh∈Vh‖zg − ϕh‖V .

Finally, the estimate (2.4.92) results from

‖u− uh‖H = supg∈H,g 6=0

|(g, u− uh)H |

‖g‖H.

�

The following a priori estimate of the global discretization error in the L2

norm is a direct consequence of the Lemma of Aubin-Nitsche (2.4.90):

(2.4.93) Theorem: Under the assumptions of Theorem (2.4.85) suppose thatthe adjoint variational equation (2.4.91) is (k+1)-regular. Then, there existsa positive constant C, depending only on the local geometry of the triangula-tions, such that

(2.4.94) ‖u− uh‖0,Ω ≤ C hk+1 |u|k+1,Ω .


Proof. The proof follows from the Lemma of Aubin-Nitsche (2.4.90) withH = L2(Ω). In particular, due to the Corollary of Theorem 2.4.85 and the(k + 1) regularity of the adjoint variational equation we have

infϕh∈Vh

‖zg − ϕh‖1,Ω ≤ ‖zg − Ihzg‖1,Ω ≤ C h ‖zg‖k+1,Ω ≤ C h ‖g‖0,Ω .

Hence, from (2.4.92) we deduce

‖u− uh‖0,Ω ≤ C h ‖u− uh‖1,Ω .

Theorem (2.4.85) allows a further estimate of the right-hand side which im-plies (2.4.94). �

In Lemma (2.4.81), the error with respect to the approximation of aSobolev space function by its finite element interpoland is measured in acoarser Sobolev norm than the norm associated with the Sobolev space wherethe function lives in. In general, the converse does not hold true, i.e., it isnot possible to estimate the norm of a Sobolev space function from above ina coarser norm. However, for finite element functions such estimates can beprovided which are known as inverse estimates.

(2.4.95) Theorem: Let {Th(Ω)}H be a regular family of geometrically con-forming simplicial triangulations and let {Vh}H be the associated family ofsimplicial Lagrangian finite element spaces based on simplicial Lagrangian fi-nite elements of type (k) where k ∈ N0 is such that H

k+1(Ω) is continuouslyembedded in C(Ω). For ℓ ∈ N such that 0 ≤ k+ 1 ≤ ℓ there exists a constantC > 0, depending on k and ℓ as well as on σ in (2.4.84), such that for vh ∈ Vhthere holds

(2.4.96) ‖vh‖ℓ,Ω ≤ C hk+1−ℓ ‖vh‖k+1,Ω .

Proof. In a first step, for the reference element we show

(2.4.97) |v̂|ℓ,K̂ ≤ C |v̂|k+1,K̂ , v̂ ∈ P̂K̂ .

We assume P̂K̂ = span{p̂1, ..., p̂nk} and choose (f̂i)nki=1 as a basis of the dual

space, i.e.,

f̂i : P̂K̂ → R , 1 ≤ i ≤ nk ,

p̂j 7−→ f̂i(p̂j) = δij , 1 ≤ i, j ≤ nk .

Then, we have the equivalence of the ‖ · ‖k+1,K̂ norm and the norm given by

|||v̂|||k+1,K̂ := |v̂|k+1,K̂ +nk∑

i=1

|f̂i(v̂)| .

Due to the finite dimension of Vh this also holds true for the norms ‖ · ‖k+1,K̂and ‖ · ‖ℓ,K̂ . We thus obtain the estimate


|v̂|ℓ,K̂ ≤ |v̂ − Π̂K̂ v̂|ℓ,K̂ + |Π̂K̂ v̂|ℓ,K̂︸︷︷︸

= 0

≤

≤ ‖v̂ − Π̂K̂ v̂‖ℓ,K̂ ≤ C ‖v̂ − Π̂K̂ v̂‖k+1,K̂ =

= C(

|v̂ − Π̂K̂ v̂|k+1,K̂ +nk∑

i=1

|f̂i(v̂ − Π̂K̂ v̂)|

︸︷︷︸

= 0

)

≤

≤ C(

|v̂|k+1,K̂ + |Π̂K̂ v̂|k+1,K̂︸︷︷︸

= 0

)

,

which proves (2.4.97). Finally, the inverse estimate follows by a scaling argu-ment according to

|v|ℓ,K ≤ C ‖B−1K ‖

ℓ |det(Bk)|1/2 |v̂|ℓ,K̂ ≤

≤ C ‖B−1K ‖ℓ |det(Bk)|

1/2 |v̂|k+1,K̂ ≤

= C ‖B−1K ‖ℓ ‖BK‖

k+1|v|k+1,K ≤

≤ C hk+1−ℓK |v|k+1,K .

�

As far as a priori estimates in the L∞ norm are concerned, using aninverse estimate we prove the following result which, however, is not optimal:

(2.4.98) Theorem: Under the assumptions of Theorem (2.4.95) there existsa positive constant C, depending only on the local geometry of the triangula-tions, such that

(2.4.99) ‖u− uh‖∞,Ω ≤ C hk |u|k+1,Ω .

Proof. Using the global interpolation operator Ih, we split the error accordingto

‖u− uh‖∞,Ω ≤ ‖u− Ihu‖∞,Ω + ‖Ihu− uh‖∞,Ω

and estimate the two terms on the right-hand side separately.

(i) In order to estimate ‖u − Ihu‖∞,Ω we use the affine equivalence. For

v̂ ∈ Hk+1(K̂), the Lemma of Bramble-Hilbert (2.4.74) implies

‖v̂ − ÎK̂ v̂‖∞,K̂ ≤ C |v̂|k+1,K̂ .

A scaling argument yields

‖u− IKu‖∞,K ≤ ‖û− ÎK̂ û‖∞,K̂ ≤ C |û|k+1,K̂ ≤ C hk |u|k+1,K ,

whence


‖u− Ihu‖∞,Ω = maxK∈Th

‖u− IKu‖∞,K ≤(2.4.100)

≤ C hk maxK∈Th

|u|k+1,K ≤ C hk |u|k+1,Ω .

(ii) For the estimation of ‖Ihu − uh‖∞,Ω , the Sobolev embedding theorem(1.2.11) and the inverse inequality (2.4.96) imply

‖uh − Ihu‖∞,Ω ≤ C ‖uh − Ihu‖1,Ω ≤ C h−1 ‖uh − Ihu‖0,Ω .

Taking advantage of the a priori estimate (2.4.94) and the interpolationestimate‖u− Ihu‖0,Ω ≤ chk+1|u|k+1,Ω it follows that

‖uh − Ihu‖0,Ω ≤ C ‖uh − u‖0,Ω + ‖u− Ihu‖0,Ω ≤ C hk+1 |u|k+1,Ω ,

and hence,

‖uh − Ihu‖0,Ω ≤ C hk |u|k+1,Ω .(2.4.101)

The estimates (2.4.100) and (2.4.101) allow to conclude. �

The derivation of an optimal a priori estimate in the L∞ norm requiresthe use of weighted Sobolev norms. (cf. Chapter 3.3. in Ciarlet (2002)).

(2.4.102) Theorem: Assume that the conditions of Theorem (2.4.98) holdtrue for k = 1. Further, suppose that the solution u of (2.4.1) satisfies u ∈V ∩W 2,∞(Ω). Then, there exists a positive constant C, depending only onthe local geometry of the triangulations, such that

(2.4.103) ‖u− uh‖∞,Ω ≤ C h2 |log h|3/2 ‖D2u‖∞,Ω .

We note that the estimate (2.4.103) is optimal in h.

Proof. We refer to Theorem 3.3.7 in Ciarlet (2002). �

2.5 Numerical quadrature and nonconforming methods

We consider the variational equation

(2.5.1) a(u, v) = ℓ(v) , v ∈ V ⊂ H1(Ω) ,

and assume that the assumptions of the Lemma of Lax-Milgram (2.3.12) aresatisfied. In particular, (2.5.1) admits a unique solution u ∈ V . Nonconfor-mity in the finite element approximation of (2.5.1) may occur due to severalreasons. One is the construction of approximations uh in subspaces Vh ⊂ Vby means of bilinear forms ah(·, ·) : Vh×Vh → R and functionals ℓh : Vh → Rthat are not simply the restrictions of a(·, ·) and ℓ(·) to Vh. Another oneis to compute approximations uh in finite dimensional spaces Vh which are

2.5 Numerical quadrature and nonconforming methods 55

not subspaces of V . In both cases, besides the approximation error we haveadditionally to consider a consistency error. A priori estimates of the globaldiscretization error are addressed by Strang’s first and second lemma.

We suppose that H is a null sequence of positive real numbers and that{Vh}h∈H is a family of conforming finite element spaces Vh ⊂ V, h ∈ H. Weapproximate the bilinear form a(·, ·) : V × V → R and the functional ℓ(·) :V → R in (2.5.1) by bounded bilinear forms and bounded linear functionals

(2.5.2) ah(·, ·) : Vh × Vh → R , ℓh(·) : Vh → R , h ∈ H .

On this basis, we approximate (2.5.1) by the finite dimensional variationalequation

(2.5.3) ah(uh, vh) = ℓh(vh) , vh ∈ Vh , h ∈ H .

(2.5.4) Definition: The sequence {ah(·, ·)}h∈H of bilinear forms ah(·, ·) :Vh × Vh → R is said to be uniformly Vh-elliptic, if there exists a positiveconstant α such that uniformly in h ∈ H there holds

(2.5.5) ah(uh, uh) ≥ α ‖uh‖2V , uh ∈ Vh .

Under the assumption of uniform Vh-ellipticity, the variational equations(2.5.3) admit unique solutions uh ∈ Vh. The following result, known asStrang’s first lemma, can be viewed as a generalization of Céa’s Lemma(2.4.10).

(2.5.6) Theorem: Let {ah(·, ·)}h∈H be a uniformly Vh-elliptic sequence ofbilinear forms and let u ∈ V and uh ∈ Vh, h ∈ H be the unique solutionsof (2.5.1) and (2.5.3). Then, there exists a constant C > 0, independent ofh ∈ H, such that for h ∈ H

‖u− uh‖V ≤ C(

infvh∈Vh

(

‖u− vh‖V +(2.5.7)

+ supwh∈Vh

|a(vh, wh) − ah(vh, wh)|

‖wh‖V

)

+ supwh∈Vh

|ℓ(wh) − ℓh(wh)|

‖wh‖V

)

.

Proof. Using the uniform Vh-ellipticity, for vh ∈ Vh we obtain

α ‖uh − vh‖2V ≤ ah(uh − vh, uh − vh) ± a(u− vh, uh − vh) =

= a(u− vh, uh − vh) +(

a(vh, uh − vh) − ah(vh, uh − vh))

+

+(

ℓh(uh − vh) − ℓ(uh − vh))

.

In view of the boundedness of the bilinear forms a(·, ·), i.e., |a(u, v)| ≤C‖u‖V ‖v‖V , division by ‖uh − vh‖V results in


α‖uh − vh‖2V ≤(2.5.8)

≤ C‖uh − vh‖V +|a(vh, uh − vh) − ah(vh, uh − vh)|

‖uh − vh‖V+

+|ℓh(uh − vh) − ℓ(uh − vh)|

‖uh − vh‖V≤

≤ C ‖uh − vh‖V + supwh∈Vh


‖wh‖V+

+ supwh∈Vh

|ℓh(wh) − ℓ(wh)|

‖wh‖V.

The triangle inequality yields

‖u− uh‖V ≤ ‖u− vh‖V + ‖uh − vh‖V .

We conclude by estimating the second term on the right-hand side by meansof (2.5.8). �

Strang’s first lemma shows that the upper bound for the global discretiza-tion error consists of two parts, the approximation error

(2.5.9) infvh∈Vh

‖u− vh‖V ,

and the consistency error

infvh∈Vh

supwh∈Vh


‖wh‖V,(2.5.10a)

supwh∈Vh

|ℓ(wh) − ℓh(wh)|

‖wh‖V.(2.5.10b)

We now assume that {Vh}h∈H is a sequence of finite dimensional linearspaces equipped with grid dependent norms ‖·‖h, h ∈ Th, on Vh +V . Further,we suppose that the functional ℓ is well defined on Vh + V .We consider a sequence of approximating bilinear forms

(2.5.11) ah(·, ·) : (Vh + V ) × (Vh + V ) → R .

We assume that the sequence {ah(·, ·)}h∈H is uniformly Vh-elliptic, i.e.,

(2.5.12) ah(vh, vh) ≥ α̃ ‖vh‖2Vh

, vh ∈ Vh , h ∈ H ,

and uniformly bounded on Vh + V , i.e.,

(2.5.13) |ah(u, v)| ≤ C ‖u‖h ‖v‖h , u, v ∈ Vh + V , h ∈ H ,

where α and C are positive constants independent of h ∈ H.We approximate (2.5.1) by the finite dimensional variational equation

(2.5.14) ah(uh, vh) = ℓ(vh) , vh ∈ Vh , h ∈ H .

Strang’s second lemma provides an upper bound for the global discretizationerror u− uh in the grid dependent norm ‖ · ‖h.


(2.5.15) Theorem: Let {ah(·, ·)}h∈H be a uniformly bounded and uniformlyVh-elliptic sequence of bilinear forms and assume that u ∈ V and uh ∈ Vh, h ∈H, are the unique solutions of the variational equations (2.5.1) and (2.5.14).Then, there exists a constant C > 0, independent of h ∈ H, such that

(2.5.16) ‖u−uh‖h ≤ C(

infvh∈Vh

‖u− vh‖h + supwh∈Vh

|ah(u,wh) − ℓ(wh)|

‖wh‖h

)

.

Proof. The uniform Vh-ellipticity implies that for vh ∈ Vh

α ‖uh − vh‖2h ≤ ah(uh − vh, uh − vh) =

= ah(u− vh, uh − vh) +(

ℓ(uh − vh) − ah(u, uh − vh))

,

and hence, by the uniform boundedness we obtain

α ‖uh − vh‖h ≤ ah(uh − vh, uh − vh) ≤(2.5.17)

≤ C ‖u− vh‖h +|ℓ(uh − vh) − ah(u, uh − vh)|

‖uh − vh‖h≤

≤ C ‖u− vh‖h + supwh∈Vh

|ℓ(wh) − ah(u,wh)|

‖wh‖h.

Using the triangle inequality

‖u− uh‖h ≤ ‖u− vh‖h + ‖uh − vh‖h ,

the assertion follows from (2.5.17). �

As in case of Strang’s first lemma, the upper bound for the global dis-cretization error consists of the approximation error

(2.5.18) infvh∈Vh

‖u− vh‖h

and the consistency error

(2.5.19) supwh∈Vh

|ℓ(wh) − ah(u,wh)|

‖wh‖h.

As an application of Strang’s first lemma, we consider the approximationof (2.5.1) for

a(u, v) :=

d∑

i,j=1

∫

Ω

aij∂u

∂xi

∂v

∂xjdx , u, v ∈ V ⊂ H1(Ω) ,(2.5.20a)

ℓ(v) :=

∫

Ω

fv dx , v ∈ V ,(2.5.20b)


evaluating the integrals in (2.5.20a) and (2.5.20b) by numerical integrationusing a quadrature formula. We assume Ω ⊂ Rd to be a bounded polyhedraldomain. As far as the data of the problem are concerned, we suppose thatfor some p ≥ 2 and k ≥ d/p there holds

(2.5.21) aij ∈Wk,∞(Ω) , 1 ≤ i, j ≤ d , f ∈W k,p(Ω) ,

and that (aij)dij=1 is a symmetric and uniformly positive definite matrix-

valued function. Moreover, we assume that the solution u of (2.5.1) satisfies

(2.5.22) u ∈ V ∩Hk+1(Ω) .

(2.5.23) Definition: Let Th(Ω) be a geometrically conforming triangulationof Ω and

(2.5.24) bTi ∈ T , ωTi ∈ R , T ∈ Th(Ω) , 1 ≤ i ≤ L , L ∈ N .

Given a function ϕ : Ω → R, for the approximation of the integral∫

Tϕdx,

we refer to

(2.5.25) QT (ϕ) :=

L∑

i=1

ωTi ϕ(bTi )

as a quadrature formula. The points bTi and the numbers ωTi , 1 ≤ i ≤ L, are

called the nodes and the weights of the quadrature formula and the quantity

(2.5.26) ET (ϕ) :=

∫

T

ϕ dx − QT (ϕ)

is said to be the quadrature error.

If Vh ⊂ V is a simplicial Lagrangian finite element space of type (k), theaffine equivalence of the finite elements allows the construction of (2.5.25)based on a quadrature formula

(2.5.27) Q̂T̂ (ϕ̂) :=

L∑

i=1

ω̂T̂i ϕ̂(b̂T̂i )

for the reference element T̂ with nodes b̂T̂i ∈ T̂ and weights ω̂T̂i , 1 ≤ i ≤ L.

(2.5.28) Lemma: Let Q̂T̂ be a quadrature formula for the reference element

T̂ of the form (2.5.27). For an element T ∈ Th(Ω) such that T = FT (T̂ ),where FT (x̂) = BT x̂ + bT , x̂ ∈ T̂ , we obtain a quadrature formula QT withnodes and weights given by

(2.5.29) bTi = FT (b̂T̂i ) , ω

Ti = det(BT ) ω̂

T̂i , 1 ≤ i ≤ L ,

which satisfies

(2.5.30) ET (ϕ) = det(BT ) ÊT̂ (ϕ̂) , ϕ̂ = ϕ ◦ FT .


Proof. The assertions readily follow from∫

T

ϕ dx = det(BT )

∫

T̂

ϕ̂ dx̂ .

�

On the basis of a quadrature formula of the form (2.5.25) we define bilinearforms ah(·, ·) : Vh × Vh → R and linear functionals ℓh(·) : Vh → R, h ∈ H,according to

ah(uh, vh) :=∑

T∈Th(Ω)

L∑

ℓ=1

ωTℓ

d∑

i,j=1

(aij∂uh∂xi

∂vh∂xj

)(bTℓ ) ,(2.5.31a)

ℓh(vh) :=∑

T∈Th(Ω)

L∑

ℓ=1

2 Elliptic Diﬀerential Equationsrohop/spring_08/Chapter2.pdf · For u ∈ C2(Ω), Taylor...

Documents

Transcript of 2 Elliptic Diﬀerential Equationsrohop/spring_08/Chapter2.pdf · For u ∈ C2(Ω), Taylor...