Finite Elements · Finite Elements Colin Cotter March 12, 2019 Colin Cotter FEM. I Consider a...

Finite Elements

Colin Cotter

March 12, 2019

Colin CotterFEM

I Consider a triangulation T with recursively refinedtriangulations Th and corresponding finite element spaces Vh.

I Given stable finite element variational problems, we have asequence of solutions uh as h→ 0, satisfying theh-independent bound

‖uh‖H1(Ω) ≤ C .

What are these solutions converging to?

I We need to find a Hilbert space that contains all Vh ash→ 0, that extends the H1 norm to the closure of finiteelement functions.

Colin CotterFEM

Our first task is to define a derivative that works for all finiteelement functions, without reference to a mesh. This requiressome preliminary definitions.

Definition 1 (Compact support on Ω)

A function u has compact support on Ω if there exists ε > 0 suchthat u(x) = 0 when miny∈∂Ω |x − y | < ε.

Definition 2 (C∞0 (Ω))

We denote by C∞0 (Ω) the subset of C∞(Ω) corresponding tofunctions that have compact support on Ω.

Colin CotterFEM

First we will define a space containing the generalised derivative.

Definition 3 (L1loc)

For triangles K ⊂ int(Ω), we define

‖u‖L1(K) =

∫K|u| dx ,

andL1K =

u : ‖u‖L1(K) <∞

.

ThenL1loc =

f : f ∈ L1(K ) ∀K ⊂ int(Ω)

.

Colin CotterFEM

Definition 4 (Weak derivative)

The weak derivative Dαw f ∈ L1

loc(Ω) of a function f ∈ L1loc(Ω) is

defined by∫ΩφDα

w f dx = (−1)|α|∫

ΩDαφf dx , ∀φ ∈ C∞0 (Ω).

Colin CotterFEM

Proposition 5

Let V be a C 0 finite element space. Then, for u ∈ V , the finiteelement (first) derivative of u is equal to the weak (first) derivativeof u.

Similar results hold for C 1 finite element space with secondderivatives and so on.

Colin CotterFEM

Proof.

Taking any φ ∈ C∞0 (Ω), we have∫Ωφ∂

∂xi|FEu dx =

∑K

∫Kφ∂u

∂xidx ,

=∑K

(−∫K

∂φ

∂xiu dx +

∫∂Kφniu dS

),

= −∑K

∫K

∂φ

∂xiu dx = −

∫Ω

∂φ

∂xiu dx ,

as required.

Colin CotterFEM

Proposition 6

Then, for u ∈ C |α|(Ω), the usual “strong” derivative Dα of u isequal to the weak derivative Dα

w of u.

Proof.

Exercise. [very similar to previous proof]

Due to these equivalences, we do not need to distinguish betweenstrong, weak and finite element derivatives. All derivatives areassumed to be weak from now on.

Colin CotterFEM

We are now in a position to define a space that contains all C 0

finite element spaces.

Definition 7 (Sobolev spaces)

H1(Ω) is the function space defined by

H1(Ω) =u ∈ L1

loc : ‖u‖H1(Ω) <∞.

More generally, Hk(Ω) is the function space defined by

Hk(Ω) =u ∈ L1

loc : ‖u‖Hk (Ω) <∞.

We conventionally write H0 = L2.

Colin CotterFEM

Proposition 8 (Hk spaces are Hilbert spaces)

The space Hk(Ω) is closed.

Proof

Let ui be a Cauchy sequence in Hk . Then Dαui is a Cauchysequence in L2(Ω) (which is closed), so ∃vα ∈ L2(Ω) such thatDαui → vα for |α| ≤ k . If wj → w in L2(Ω), then for φ ∈ C∞0 (Ω),∫

Ω(wj − w)φ dx ≤ ‖wj − w‖L2(Ω)‖φ‖L∞ → 0. (1)

Colin CotterFEM

Proof (Cont.)

We use (1) to get∫Ωvαφ dx = lim

i→∞

∫ΩφDαui dx ,

= limi→∞

(−1)|α|∫

ΩuiD

αφ dx ,

= (−1)|α|∫

ΩvDαφ dx ,

i.e. vα is the weak derivative of u as required.

Colin CotterFEM

We quote the following much deeper results without proof.

Theorem 9 (H = W )

Let Ω be any open set. Then Hk(Ω) ∩ C∞(Ω) is dense in Hk(Ω).

The interpretation is that for any function u ∈ Hk(Ω), we can finda sequence of C∞ functions ui converging to u. This is very usefulas we can compute many things using C∞ functions and take thelimit.

Colin CotterFEM

Theorem 10 (Sobolev’s inequality)

Let Ω be an n-dimensional domain with Lipschitz boundary, let kbe an integer with k > n/2. Then there exists a constant C suchthat

‖u‖L∞(Ω) = ess supx∈Ω|u(x)| ≤ C‖u‖Hk (Ω).

Further, there is a C 0 continuous function in the L∞(Ω)equivalence class of u.

The interpretation is that if u ∈ Hk then there is a continuousfunction u0 such that the set of points where u 6= u0 has zeroarea/volume.

Colin CotterFEM

Corollary 11 (Sobolev’s inequality for derivatives)

Let Ω be a n-dimensional domain with Lipschitz boundary, let k bean integer with k −m > n/2. Then there exists a constant C suchthat

‖u‖Wm∞(Ω) :=

∑|α|≤m

‖Dαu‖L∞(Ω) ≤ C‖u‖Hk (Ω).

Further, there is a Cm continuous function in the L∞(Ω)equivalence class of u.

Proof.

Just apply Sobolev’s inequality to the m derivatives of u.

Colin CotterFEM

We can now consider linear variational problems defined on Hk

spaces, by taking a bilinear form b(u, v) and linear form F (v),seeking u ∈ Hk (for chosen Hk) such that

b(u, v) = F (v), ∀v ∈ Hk .

Colin CotterFEM

Since Hk is a Hilbert space, the Lax-Milgram theorem theexistence of a unique solution to an Hk linear variational problem.

Proposition 12 (Well-posedness for (modified) Helmholtz))

The Helmholtz variational problem on H1 satisfies the conditionsof the Lax-Milgram theorem.

Proof.

The proof for C 0 finite element spaces extends immediately toH1.

Colin CotterFEM

Functions in Hk make boundary conditions hard to interpret sincethey are not guaranteed to have defined values on the boundary.We make the following definition.

Definition 13 (Trace of H1 functions)

Let u ∈ H1(Ω) and choose ui ∈ C∞(Ω) such that ui → u. Wedefine the trace u|∂Ω on ∂Ω as the limit of the restriction of ui to∂Ω.

Colin CotterFEM

Proposition 14 (Trace theorem for H1 functions)

Let u ∈ H1(Ω) for a polygonal domain Ω. Then the trace u|∂Ω

satisfies‖u‖L2(∂Ω) ≤ C‖u‖H1(Ω). (2)

Proof.

Adapt the proof for C 0 finite element functions, choosingu ∈ C∞(Ω), and pass to the limit in H1(Ω).

Interpretation: if u ∈ H1(Ω) then u|∂Ω ∈ L2(∂Ω).

Colin CotterFEM

Proposition 15

Let u ∈ H2(Ω), v ∈ H1(Ω). Then∫Ω

(−∇2u)v dx =

∫Ω∇u · ∇v dx −

∫∂Ω

∂u

∂nv dS .

Proof.

First note that u ∈ H2(Ω) =⇒ ∇u ∈ (H1(Ω))d . Then

‖v∇u‖H1(Ω) ≤ ‖v‖H1(Ω)‖∇u‖H1(Ω) =⇒ v∇u ∈ H1(Ω).

Then, take vi ∈ C∞(Ω) and ui ∈ C∞(Ω) converging to v and u,respectively, and vi∇ui ∈ C∞(Ω) converges to v∇u. These satisfythe equation; we obtain the result by passing to the limit.

Colin CotterFEM

Proposition 16

For f ∈ L2, let u ∈ H2(Ω) solve

u −∇2u = f ,∂u

∂n= 0 on ∂Ω,

in the L2 sense, i.e. ‖u −∇2u − f ‖L2 = 0. Then u solves thevariational form of the Helmholtz equation.

Colin CotterFEM

Proof.

u ∈ H2 =⇒ ‖u‖H2 <∞ =⇒ ‖u‖H1 <∞ =⇒ u ∈ H1.Multiplying by test function v ∈ H1, and using the previousproposition gives∫

Ωuv +∇u · ∇v dx =

∫Ωfv dx , ∀v ∈ H1(Ω),

as required.

Colin CotterFEM

Proposition 17

Let f ∈ L2(Ω) and suppose that u ∈ H2(Ω) solves the variationalHelmholtz equation on a polygonal domain Ω. Then u solves thestrong form Helmholtz equation with zero Neumann boundaryconditions.

Colin CotterFEM

Proof

Using integration by parts for u ∈ H2, v ∈ C∞0 (Ω) ∈ H1, we have∫Ω

(u −∇2u − f )v dx =

∫Ωuv +∇u · ∇v − vf dx = 0.

It is a standard result that C∞0 (Ω) is dense in L2(Ω) and thereforewe can choose a sequence of v converging to u −∇2u − f and weobtain ‖u −∇2u − f ‖L2(Ω) = 0.

Colin CotterFEM

Proof (Cont.)

On the other hand,

0 =

∫Ωuv +∇u · ∇v − fv dx =

∫Ωuv +∇u · ∇v − (u −∇2u)v dx

=

∫∂Ω

∂u

∂nv dS .

We can find arbitrary v ∈ L2(∂Ω), hence ‖∂u∂n‖L2(∂Ω) = 0.

Colin CotterFEM

Definition 18 (Galerkin approximation)

For a finite element space Vh ⊂ V = Hk(Ω), the Galerkinapproximation of the Hk variational problem above seeks to finduh ∈ Vh such that

b(uh, v) = F (v), ∀v ∈ Vh.

What is the size of the error u − uh?

Colin CotterFEM

Theorem 19 (Cea)

Let Vh ⊂ V , and let u solve a linear variational problem on V ,whilst uh solves the equivalent Galerkin approximation on Vh. Then

‖u − uh‖V ≤M

γminv∈Vh

‖u − v‖V ,

where M and γ are the continuity and coercivity constants ofb(u, v), respectively.

Colin CotterFEM

Proof

We have

b(u, v) = F (v) ∀v ∈ V ,

b(uh, v) = F (v) ∀v ∈ Vh.

Choosing v ∈ Vh ⊂ V means we can use it in both equations, andsubtraction and linearity lead to the “Galerkin orthogonality”condition

b(u − uh, v) = 0, ∀v ∈ Vh.

Colin CotterFEM

Proof (Cont.)

Then, for all v ∈ Vh,

γ‖u − uh‖2V ≤ b(u − uh, u − uh),

= b(u − uh, u − v) + b(u − uh, v − uh)︸︷︷︸=0

,

≤ M‖u − uh‖V ‖u − v‖V .

Minimising over all v completes the proof.

Colin CotterFEM

‖u − uh‖V ≤M

γminv∈Vh

‖u − v‖V .

I The interpretation is that the error is proportional to theminimal error in approximating u in Vh.

I We can estimate the error in approximating u in Vh ifu ∈ C 2(Ω), using Taylor series.

I If u ∈ H2(Ω), then u does not necessarily have a degree 2Taylor polynomial, since derivatives are not defined atarbitrary points. We need a more general definition.

Colin CotterFEM

Definition 20 (Averaged Taylor polynomial)

Let Ω ⊂ Rn be a domain with diameter d , that is star-shaped withrespect to a ball B with radius ε, contained within Ω. Forf ∈ Hk+1(Ω) the averaged Taylor polynomial Qk,B f ∈ Pk isdefined as

Qk,B f (x) =1

|B|

∫BT k f (y , x) dy ,

where T k f is the Taylor polynomial of degree k of f ,

T k f (y , x) =∑|α|≤k

Dαf (y)(x − y)α

α!,

evaluated using weak derivatives.

Colin CotterFEM

This definition makes sense since the Taylor polynomial coefficientsare in L1

loc(Ω) and thus their integrals over B are defined.

Proposition 21

Let Ω ⊂ Rn be a domain with diameter d , that is star-shaped withrespect to a ball B with radius ε, contained within Ω. There existsa constant C (k, n) such that for 0 ≤ |β| ≤ k + 1 and allf ∈ Hk+1(Ω),

‖Dβ(f − Qk,B f )‖L2 ≤ C|Ω|1/2

|B|1/2dk+1−|β|‖∇k+1f ‖L2(Ω).

Colin CotterFEM

Proof

Assume f ∈ C∞(Ω), pass to the limit. Remainder theorem gives

f (x)− Tk f (y , x) =

(k + 1)∑|α|=k+1

(x − y)α

α!

∫ 1

0Dαf (ty + (1− t)x)tk dt.

Integration over y in B and dividing by |B| gives

f (x)− Qk,B f (x) =k + 1

|B|∑|α|=k+1

∫B

(x − y)α

α!×

∫ 1

0Dαf (ty + (1− t)x)tk dt dy .

Colin CotterFEM

Proof (Cont.)

Then∫Ω|f (x)− Qk,B f (x)|2 dx

≤ Cd2(k+1)

|B|2∑|α|=k+1

∫Ω

(∫B

∫ 1

0|Dαf (ty + (1− t)x)|tk dt dy

)2

dx ,

≤ C0d2(k+1)

|B|2∑|α|=k+1

∫Ω

∫B

∫ 1

0|Dαf (ty + (1− t)x)|2 dt dy

×∫B

∫ 1

0t2k dt dy dx .

Colin CotterFEM

Proof (Cont.)

Then∫Ω|f (x)− Qk,B f (x)|2 dx

≤ C1d2(k+1)

|B|2∑|α|=k+1

∫Ω

∫B

∫ 1

0|Dαf (ty + (1− t)x)|2 dt dy dx .

For each α term we can split the t integral into [0, 1/2] and[1/2, 1]. Call these terms I and II.

Colin CotterFEM

Proof (Cont.)

We obtain the required bounds for |β| = 0. For higher derivativeswe use the fact that

DβQk,B f (x) = Qk−|β|,BDβf (x),

which immediately leads to the estimate for |β| > 0.

Colin CotterFEM

Corollary 22

Let K1 be a triangle with diameter 1. There exists a constantC (k, n) such that

‖f − Qk,B f ‖Hk (K1) ≤ C |∇k+1f |Hk+1(K1).

Proof.

Take the maximum over the constants for the derivativecontributions of the left-hand side with d = 1 and use the previousresult.

Colin CotterFEM

Now we will use the Taylor polynomial estimates to derive errorestimates for the local interpolation operator. First we need toobtain the following bound.

Proposition 23

Let (K1,P,N ) be a finite element such that K1 is a triangle withdiameter 1, and such that the nodal variables in N involve onlyevaluations of functions or evaluations of derivatives of degree ≤ l ,and ‖Ni‖W l

∞(K1)′ <∞, where

‖Ni‖W l∞(K1)′ = sup

‖u‖Wl∞(K1)

>0

|Ni (u)|‖u‖W l

∞(K1)

.

Let u ∈ Hk(K1) with k > l + n/2. Then‖IK1u‖Hk (K1) ≤ C‖u‖Hk (K1).

Colin CotterFEM

Proof (Cont.)

Let φini=1 be the nodal basis for P.

‖IK1u‖Hk (K1) ≤k∑

i=1

‖φi‖Hk (K1)|Ni (u)|

≤k∑

i=1

‖φi‖Hk (K1)‖Ni‖W l∞(K1)′︸︷︷︸

C0

‖u‖W l∞(K1),

≤ C‖u‖Hk (K1),

where the Sobolev inequality was used in the last line.

Colin CotterFEM

Proposition 24

Let (K1,P,N ) be a finite element such that K1 has diameter 1,and such that the nodal variables in N involve only evaluations offunctions or evaluations of derivatives of degree ≤ l , and P containall polynomials of degree k and below, with k > l + n/2. Letu ∈ Hk+1(K1). Then for i ≤ k, the local interpolation operatorsatisfies

|IK1u − u|H i (K1) ≤ C1|u|Hk+1(K1). (3)

Colin CotterFEM

Proof.

|IK1u − u|2H i (K1) ≤ ‖IK1u − u‖2Hk (K1)

= ‖IK1u − Qk,Bu + Qk,Bu − u‖2Hk (K1)

≤ ‖Qk,Bu − u‖2Hk (K1) + ‖I(u − Qk,Bu)‖2

Hk (K1),

≤ ‖Qk,Bu − u‖2Hk (K1) + C 2‖Qk,Bu − u‖2

Hk (K1),

≤ (1 + C 2)|u|2Hk+1(K1),

as required.

Colin CotterFEM

Proposition 25

Let (K ,P,N ) be a finite element such that K has diameter d , andsuch that the nodal variables in N involve only evaluations offunctions or evaluations of derivatives of degree ≤ l , and Pcontains all polynomials of degree k and below, with k > l + n/2.Let u ∈ Hk+1(K ). Then for i ≤ k , the local interpolation operatorsatisfies

|IKu − u|H i (K) ≤ CKdk+1−i |u|Hk+1(K). (4)

where CK is a constant that depends on the shape of K but notthe diameter.

Colin CotterFEM

Proof (Homogeneity argument.)

Consider the change of variables x → φ(x) = x/d . This map takesK to K1 with diameter 1. Then∫

K|Dβ(IKu − u)|2 dx = d−2|β|+1

∫K1

|Dβ(IK1u φ− u φ)|2 dx ,

≤ C 21 d−2|β+1

∑|α|=k+1

∫K1

|Dαu φ|2 dx ,

≤ C 21 d−2|β+2(k+1)

∑|α|=k+1

∫K|Dαu|2 dx ,

= C 21 d

2(−|β|+k+1)|u|2Hk+1(K),

and taking the square root gives the result.

Colin CotterFEM

Proposition 26

Let T be a triangulation with finite elements (Ki ,Pi ,Ni ), suchthat the minimum aspect ratio r of the triangles Ki satisfies r > 0,and such that the nodal variables in N involve only evaluations offunctions or evaluations of derivatives of degree ≤ l , and Pcontains all polynomials of degree k and below, with k > l + n/2.Let u ∈ Hk+1(Ω). Let h be the maximum over all of the trianglediameters, with 0 ≤ h < 1. Let V be the corresponding C r finiteelement space. Then for i ≤ k and i ≤ r + 1, the globalinterpolation operator satisfies

‖Ihu − u‖H i (Ω) ≤ Chk+1−i |u|Hk+1(Ω). (5)

Colin CotterFEM

Proof.

If V is a Cn finite element space, then the r + 1-th weakderivatives exist. Then we may write

‖Ihu − u‖2H i (Ω) =

∑K∈T‖IKu − u‖2

H i (K), (6)

≤∑K∈T

CKd2(k+1−i)K |u|2Hk+1(K), (7)

≤ Cmaxh2(k+1−i)

∑K∈T|u|2Hk+1(K), (8)

= Cmaxh2(k+1−i)|u|2Hk+1(Ω), (9)

where the existence of the Cmax = maxK CK <∞ is due to thelower bound in the aspect ratio.

Colin CotterFEM

Corollary 27

The degree m Lagrange finite element approximation uh to thesolution u of the variational Helmholtz problem satisfies

‖uh − u‖H1(Ω) ≤ Chm‖u‖H2(Ω).

Proof.

We combine Cea’s Theorem with the previous estimate, since

minv∈Vh

‖u − v‖H1(Ω) ≤ ‖u − Ihu‖H1(Ω).

Colin CotterFEM

We would also like estimates of the error in the L2 norm. To dothis we quote the following without proof.

Theorem 28 (Elliptic regularity)

Let w solve the equation

w −∇2w = f ,∂w

∂n= 0 on ∂Ω,

on a convex domain Ω, with f ∈ L2. Then there exists constantC > 0 such that

|w |H2(Ω) ≤ C‖f ‖L2(Ω).

Colin CotterFEM

Theorem 29

The degree m Lagrange finite element approximation uh to thesolution u of the variational Helmholtz problem satisfies

‖uh − u‖L2(Ω) ≤ Cdm+1‖u‖H2(Ω).

Proof

We use the Aubin-Nitsche duality argument. Let w be the solutionof

w −∇2w = u − uh.

Colin CotterFEM

Proof (Cont.)

Then equivalently,

b(w , v) = (u − uh, v)L2(Ω), ∀v ∈ H1(Ω).

‖u − uh‖2L2(Ω) = (u − uh, u − uh) = b(w , u − uh),

= b(w − Ihw , u − uh) (orthogonality) ,

≤ C‖u − uh‖H1(Ω)‖w − Ihw‖H1(Ω),

≤ Ch‖u − uh‖H1(Ω)|w |H2(Ω)

≤ C1hm+1‖u − uh‖L2(Ω)|u|H2(Ω)

and dividing both sides by ‖u − uh‖L2(Ω) gives the result.

Colin CotterFEM

Finite Elements · Finite Elements Colin Cotter March 12, 2019 Colin Cotter FEM. I Consider a...

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