A Discontinuous Galerkin Finite Element Method ... - Amazon S3
Discontinuous Galerkin Finite Element Methodsperugia/TEACHING/DOTT2005/intro.pdf · 2005. 2....
Transcript of Discontinuous Galerkin Finite Element Methodsperugia/TEACHING/DOTT2005/intro.pdf · 2005. 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
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)
Discontinuous Galerkin Finite Element Methods – p.2/10
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
Discontinuous Galerkin Finite Element Methods – p.3/10
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
Discontinuous Galerkin Finite Element Methods – p.4/10
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
Discontinuous Galerkin Finite Element Methods – p.4/10
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]
Discontinuous Galerkin Finite Element Methods – p.5/10
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]
Discontinuous Galerkin Finite Element Methods – p.5/10
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)
Discontinuous Galerkin Finite Element Methods – p.6/10
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)
Discontinuous Galerkin Finite Element Methods – p.6/10
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)
Discontinuous Galerkin Finite Element Methods – p.6/10
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)
Discontinuous Galerkin Finite Element Methods – p.6/10
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
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
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
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
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
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
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
Discontinuous Galerkin Finite Element Methods – p.9/10
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
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]
Discontinuous Galerkin Finite Element Methods – p.10/10
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]
Discontinuous Galerkin Finite Element Methods – p.10/10