Discontinuous Galerkin Finite Element Methodsperugia/TEACHING/DOTT2005/intro.pdf · 2005. 2....

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Discontinuous Galerkin Finite Element Methods Ilaria Perugia Dipartimento di Matematica Universit ` a di Pavia http://www-dimat.unipv.it/perugia Ciao Ciao Ciao Milano, February 16, 2004 Discontinuous Galerkin Finite Element Methods – p.1/10

Transcript of Discontinuous Galerkin Finite Element Methodsperugia/TEACHING/DOTT2005/intro.pdf · 2005. 2....

Page 1: Discontinuous Galerkin Finite Element Methodsperugia/TEACHING/DOTT2005/intro.pdf · 2005. 2. 18. · get a TVDM and TVB method, under fixed CFL n. f0( t= x) that can be chosen 1=2

Discontinuous Galerkin Finite Element Methods

Ilaria PerugiaDipartimento di Matematica

Universita di Paviahttp://www-dimat.unipv.it/perugia

Ciao

Ciao

Ciao

Milano, February 16, 2004

Discontinuous Galerkin Finite Element Methods – p.1/10

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Introduction

Th = K partition of Ω

Pk(Th) discontinuous finite element spaces

Local variational formulation (element-by-element)

Interelement continuity conditions: numerical fluxes(integral terms in the formulation, no special boundary d.o.f.,no Lagrange multipliers)

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Features of DG Methods

Wide range of PDE’s treated within the same unified framework

Flexibility in the mesh design

K i

K j non-matching grids (hanging nodes)

non-uniform approximation degrees

Freedom in the choice of basis functions

Orthogonal bases can be easily constructed

Parallelization

Drawback: high number of degrees of freedom

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The Original DG Method (I)

[Reed & Hill, Los Alamos Tech. Rep., 1973]

Neutron transport equation:

σ u + ∇ · (a u) = f in Ω

Construct a triangulation Th = K of Ω, and define Vh = Pk(Th)

Multiply by a test function v and integrate by parts over any K

σ(u, v)K − (u, a · ∇v)K + 〈a · nKu, v〉∂K = (f, v)K

Take the approximate solution uh ∈ Vh

Substitute a · nKuh in the integral over ∂K by a numerical flux h

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The Original DG Method (I)

[Reed & Hill, Los Alamos Tech. Rep., 1973]

Neutron transport equation:

σ u + ∇ · (a u) = f in Ω

Construct a triangulation Th = K of Ω, and define Vh = Pk(Th)

Multiply by a test function v and integrate by parts over any K

σ(u, v)K − (u, a · ∇v)K + 〈a · nKu, v〉∂K = (f, v)K

Take the approximate solution uh ∈ Vh

Substitute a · nKuh in the integral over ∂K by a numerical flux h

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The Original DG Method (II)

DG method: find uh ∈ Vh s.t., for any K ∈ Th,

σ(uh, v)K − (uh, a · ∇v)K + 〈h, v〉∂K = (f, v)K ∀v ∈ Pk(K)

with upstream flux h(x) = a · nK(x) lims↓0

uh(x − s a)

Mathematical analysis:[LeSaint & Raviart, in Math. Aspects of FE in PDE, Acad. Press, 1974]See also [Johnson & Pitkaränta, Math. Comp, 1986]and [Lin & Zhou, Acta Math. Sci. 1993]

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The Original DG Method (II)

DG method: find uh ∈ Vh s.t., for any K ∈ Th,

σ(uh, v)K − (uh, a · ∇v)K + 〈h, v〉∂K = (f, v)K ∀v ∈ Pk(K)

with upstream flux h(x) = a · nK(x) lims↓0

uh(x − s a)

Mathematical analysis:[LeSaint & Raviart, in Math. Aspects of FE in PDE, Acad. Press, 1974]See also [Johnson & Pitkaränta, Math. Comp, 1986]and [Lin & Zhou, Acta Math. Sci. 1993]

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DG for Conservation Laws: RKDG (I)

Survey: [Cockburn & Shu, JSC, 2001]

One-dimensional model problem:

ut + f(u)x = 0 in (0, 1) × (0, T )

u(x, 0) = u0(x) ∀x ∈ (0, 1)

with periodic b.c.

[Chavent & Salzano, JCP, 1982]: P1-DG in space, forward Euler intime (unconditionally unstable for constant ∆t/∆x)

[Chavent & Cockburn, M2AN, 1989]: introd. of a slope limiter toget a TVDM and TVB method, under fixed CFL n. f ′(∆t/∆x) thatcan be chosen ≤ 1/2 (only first order accurate in time)

[Cockburn & Shu, Math. Comp., 1989]: Pk-DG in space, explicit(k + 1)-th order RK in time, generalized slope limiter (high-orderRKDG)

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DG for Conservation Laws: RKDG (I)

Survey: [Cockburn & Shu, JSC, 2001]

One-dimensional model problem:

ut + f(u)x = 0 in (0, 1) × (0, T )

u(x, 0) = u0(x) ∀x ∈ (0, 1)

with periodic b.c.

[Chavent & Salzano, JCP, 1982]: P1-DG in space, forward Euler intime (unconditionally unstable for constant ∆t/∆x)

[Chavent & Cockburn, M2AN, 1989]: introd. of a slope limiter toget a TVDM and TVB method, under fixed CFL n. f ′(∆t/∆x) thatcan be chosen ≤ 1/2 (only first order accurate in time)

[Cockburn & Shu, Math. Comp., 1989]: Pk-DG in space, explicit(k + 1)-th order RK in time, generalized slope limiter (high-orderRKDG)

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DG for Conservation Laws: RKDG (I)

Survey: [Cockburn & Shu, JSC, 2001]

One-dimensional model problem:

ut + f(u)x = 0 in (0, 1) × (0, T )

u(x, 0) = u0(x) ∀x ∈ (0, 1)

with periodic b.c.

[Chavent & Salzano, JCP, 1982]: P1-DG in space, forward Euler intime (unconditionally unstable for constant ∆t/∆x)

[Chavent & Cockburn, M2AN, 1989]: introd. of a slope limiter toget a TVDM and TVB method, under fixed CFL n. f ′(∆t/∆x) thatcan be chosen ≤ 1/2 (only first order accurate in time)

[Cockburn & Shu, Math. Comp., 1989]: Pk-DG in space, explicit(k + 1)-th order RK in time, generalized slope limiter (high-orderRKDG)

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DG for Conservation Laws: RKDG (I)

Survey: [Cockburn & Shu, JSC, 2001]

One-dimensional model problem:

ut + f(u)x = 0 in (0, 1) × (0, T )

u(x, 0) = u0(x) ∀x ∈ (0, 1)

with periodic b.c.

[Chavent & Salzano, JCP, 1982]: P1-DG in space, forward Euler intime (unconditionally unstable for constant ∆t/∆x)

[Chavent & Cockburn, M2AN, 1989]: introd. of a slope limiter toget a TVDM and TVB method, under fixed CFL n. f ′(∆t/∆x) thatcan be chosen ≤ 1/2 (only first order accurate in time)

[Cockburn & Shu, Math. Comp., 1989]: Pk-DG in space, explicit(k + 1)-th order RK in time, generalized slope limiter (high-orderRKDG)

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DG for Conservation Laws: RKDG (II)

One-dimensional, linear model problem:

ut + (c u)x = 0 in (0, 1) × (0, T )

u(x, 0) = u0(x) ∀x ∈ (0, 1)

with periodic b.c.

Partition Th = Kj1≤j≤J of (0, 1), with Kj = (xj−1/2, xj+1/2)

DG space Vh = Pk(Th)

Multiply by a test function v(x) and integrate by parts:

(∂tu(x, t), v(x))Kj− (c u(x, t), ∂xv(x))Kj

+ c u(xj+1/2, t) v(x−j+1/2) − c u(xj−1/2, t) v(x+

j−1/2) = 0

(u(x, 0), v(x))Kj= (u0(x), v(x))Kj

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DG for Conservation Laws: RKDG (II)

One-dimensional, linear model problem:

ut + (c u)x = 0 in (0, 1) × (0, T )

u(x, 0) = u0(x) ∀x ∈ (0, 1)

with periodic b.c.

Partition Th = Kj1≤j≤J of (0, 1), with Kj = (xj−1/2, xj+1/2)

DG space Vh = Pk(Th)

Multiply by a test function v(x) and integrate by parts:

(∂tu(x, t), v(x))Kj− (c u(x, t), ∂xv(x))Kj

+ c u(xj+1/2, t) v(x−j+1/2) − c u(xj−1/2, t) v(x+

j−1/2) = 0

(u(x, 0), v(x))Kj= (u0(x), v(x))Kj

Discontinuous Galerkin Finite Element Methods – p.7/10

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DG for Conservation Laws: RKDG (II)

One-dimensional, linear model problem:

ut + (c u)x = 0 in (0, 1) × (0, T )

u(x, 0) = u0(x) ∀x ∈ (0, 1)

with periodic b.c.

Partition Th = Kj1≤j≤J of (0, 1), with Kj = (xj−1/2, xj+1/2)

DG space Vh = Pk(Th)

Multiply by a test function v(x) and integrate by parts:

(∂tu(x, t), v(x))Kj− (c u(x, t), ∂xv(x))Kj

+ c u(xj+1/2, t) v(x−j+1/2) − c u(xj−1/2, t) v(x+

j−1/2) = 0

(u(x, 0), v(x))Kj= (u0(x), v(x))Kj

Discontinuous Galerkin Finite Element Methods – p.7/10

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DG for Conservation Laws: RKDG (III)

Replace smooth v by v ∈ Vh and approximate u by uh ∈ Vh

At points xj+1/2, replace c u(xj+1/2, t) by a numerical flux

h(u)j+1/2(t) = h(u(x−j+1/2, t), u(x+

j+1/2, t))

Here: h(a, b) =

c a if c ≥ 0

c b if c < 0(upwind)

Nonlinear pb.: Lipschitz, consistent, monotone flux

Diagonalizing the mass matrix: Legendre polynomials

d

dtuh = Lh(uh) in (0, T ), uh(t = 0) = u0h

L2-stability; order of convergence: k + 1/2, if u0 ∈ Hk+1

L2-stability; order of convergence: (k + 1, if u0 ∈ Hk+2)

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DG for Conservation Laws: RKDG (III)

Replace smooth v by v ∈ Vh and approximate u by uh ∈ Vh

At points xj+1/2, replace c u(xj+1/2, t) by a numerical flux

h(u)j+1/2(t) = h(u(x−j+1/2, t), u(x+

j+1/2, t))

Here: h(a, b) =

c a if c ≥ 0

c b if c < 0(upwind)

Nonlinear pb.: Lipschitz, consistent, monotone flux

Diagonalizing the mass matrix: Legendre polynomials

d

dtuh = Lh(uh) in (0, T ), uh(t = 0) = u0h

L2-stability; order of convergence: k + 1/2, if u0 ∈ Hk+1

L2-stability; order of convergence: (k + 1, if u0 ∈ Hk+2)

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DG for Conservation Laws: RKDG (III)

Replace smooth v by v ∈ Vh and approximate u by uh ∈ Vh

At points xj+1/2, replace c u(xj+1/2, t) by a numerical flux

h(u)j+1/2(t) = h(u(x−j+1/2, t), u(x+

j+1/2, t))

Here: h(a, b) =

c a if c ≥ 0

c b if c < 0(upwind)

Nonlinear pb.: Lipschitz, consistent, monotone flux

Diagonalizing the mass matrix: Legendre polynomials

d

dtuh = Lh(uh) in (0, T ), uh(t = 0) = u0h

L2-stability; order of convergence: k + 1/2, if u0 ∈ Hk+1

L2-stability; order of convergence: (k + 1, if u0 ∈ Hk+2)Discontinuous Galerkin Finite Element Methods – p.8/10

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DG for Conservation Laws: RKDG (IV)

Explicit RK time stepping: partition tn0≤n≤N of [0, T ]

Set u0h = u0h. For 0 ≤ n ≤ N , un

h → un+1h as

• set u(0)h = un

h

• for 1 ≤ i ≤ K, compute

u(i)h =

i−1∑

l=0

αil wilh wil

h = u(l)h +

βil

αil∆tnLh(u

(l)h )

(RK parameters s.t. stability follows from stab. of u(l)h 7→ wil

h )

• set un+1h = uK

h

Pk-DG in space → (k + 1)-stage RK in time

Stability condition: |c| (∆t/∆x) ≤ CFLL2 ≈ 1/(2k + 1)

Introduction of slope limiters for nonlinear problems

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DG for Conservation Laws: RKDG (IV)

Explicit RK time stepping: partition tn0≤n≤N of [0, T ]

Set u0h = u0h. For 0 ≤ n ≤ N , un

h → un+1h as

• set u(0)h = un

h

• for 1 ≤ i ≤ K, compute

u(i)h =

i−1∑

l=0

αil wilh wil

h = u(l)h +

βil

αil∆tnLh(u

(l)h )

(RK parameters s.t. stability follows from stab. of u(l)h 7→ wil

h )

• set un+1h = uK

h

Pk-DG in space → (k + 1)-stage RK in time

Stability condition: |c| (∆t/∆x) ≤ CFLL2 ≈ 1/(2k + 1)

Introduction of slope limiters for nonlinear problemsDiscontinuous Galerkin Finite Element Methods – p.9/10

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DG for Elliptic Problems

DG Methods:

Interior Penalty: [Douglas & Dupont, LN in Physics, 1976],Interior Penalty: [Wheeler, SINUM, 1978], [Arnold, SINUM, 1982]Bassi-Rebay: [Bassi & Rebay, JCP, 1997]Local Discontinuous Galerkin: [Cockburn & Shu, SINUM, 1998]Baumann-Oden: [Baumann & Oden, CMAME, 1999]Non-symmetric Interior Penalty: [Rivière, Wheeler & Girault,Non-symmetric Interior Penalty: Comp. Geosc., 1999]

Theoretical Analysis:

[Oden, Babuška & Baumann, JCP, 1998],[Rivière, Wheeler & Girault, Comp. Geosc., 1999],[Castillo, Cockburn, Perugia & Schötzau, SINUM, 2000],[Arnold, Brezzi, Cockburn & Marini, SINUM, 2001],[Houston, Schwab & Süli, SINUM, 2002]

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DG for Elliptic Problems

DG Methods:

Interior Penalty: [Douglas & Dupont, LN in Physics, 1976],Interior Penalty: [Wheeler, SINUM, 1978], [Arnold, SINUM, 1982]Bassi-Rebay: [Bassi & Rebay, JCP, 1997]Local Discontinuous Galerkin: [Cockburn & Shu, SINUM, 1998]Baumann-Oden: [Baumann & Oden, CMAME, 1999]Non-symmetric Interior Penalty: [Rivière, Wheeler & Girault,Non-symmetric Interior Penalty: Comp. Geosc., 1999]

Theoretical Analysis:

[Oden, Babuška & Baumann, JCP, 1998],[Rivière, Wheeler & Girault, Comp. Geosc., 1999],[Castillo, Cockburn, Perugia & Schötzau, SINUM, 2000],[Arnold, Brezzi, Cockburn & Marini, SINUM, 2001],[Houston, Schwab & Süli, SINUM, 2002]

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