Discontinuous Galerkin Finite Element Methods

Discontinuous Galerkin Finite Element Methods Ilaria Perugia Dipartimento di Matematica Universita` di Pavia http://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)


Features of DG Methods Wide range of PDE’s treated within the same unified framework Flexibility in the mesh design

Ki

Kj

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


The Original DG Method (I) [Reed & Hill, Los Alamos Tech. Rep., 1973] Neutron transport equation: σ u + ∇ · (a u) = f

in Ω


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 + ha · nK u, vi∂K = (f, v)K Take the approximate solution uh ∈ Vh Substitute a · nK uh in the integral over ∂K by a numerical flux b h


The Original DG Method (II) DG method: find uh ∈ Vh s.t., for any K ∈ Th , h, vi∂K = (f, v)K σ(uh , v)K − (uh , a · ∇v)K + hb

with upstream flux

∀v ∈ Pk (K)

b h(x) = a · nK (x) lim uh (x − s a) s↓0


The Original DG Method (II) DG method: find uh ∈ Vh s.t., for any K ∈ Th , h, vi∂K = (f, v)K σ(uh , v)K − (uh , a · ∇v)K + hb

with upstream flux

∀v ∈ Pk (K)

b h(x) = a · nK (x) lim uh (x − s a) s↓0

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]


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.



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 in time (unconditionally unstable for constant ∆t/∆x)



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 in time (unconditionally unstable for constant ∆t/∆x) [Chavent & Cockburn, M2 AN, 1989]: introd. of a slope limiter to get a TVDM and TVB method, under fixed CFL n. f 0 (∆t/∆x) that can be chosen ≤ 1/2 (only first order accurate in time)



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 in time (unconditionally unstable for constant ∆t/∆x) [Chavent & Cockburn, M2 AN, 1989]: introd. of a slope limiter to get a TVDM and TVB method, under fixed CFL n. f 0 (∆t/∆x) that can 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-order RKDG)


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.



in (0, 1) × (0, T )

u(x, 0) = u0 (x)

∀x ∈ (0, 1)

with periodic b.c. Partition Th = {Kj }1≤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:



in (0, 1) × (0, T )

u(x, 0) = u0 (x)

∀x ∈ (0, 1)

with periodic b.c. Partition Th = {Kj }1≤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: (∂t u(x, t), v(x))Kj − (c u(x, t), ∂x v(x))Kj + + c u(xj+1/2 , t) v(x− ) − c u(x , t) v(x j−1/2 j+1/2 j−1/2 ) = 0

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


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− , t), u(x j+1/2 j+1/2 , t))

(

c a if c ≥ 0 (upwind) Here: h(a, b) = c b if c < 0 Nonlinear pb.: Lipschitz, consistent, monotone flux



(

c a if c ≥ 0 (upwind) Here: h(a, b) = c b if c < 0 Nonlinear pb.: Lipschitz, consistent, monotone flux Diagonalizing the mass matrix: Legendre polynomials d uh = Lh (uh ) in (0, T ), dt

uh (t = 0) = u0h



(

c a if c ≥ 0 (upwind) Here: h(a, b) = c b if c < 0 Nonlinear pb.: Lipschitz, consistent, monotone flux Diagonalizing the mass matrix: Legendre polynomials d uh = Lh (uh ) in (0, T ), dt

uh (t = 0) = u0h

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


DG for Conservation Laws: RKDG (IV) Explicit RK time stepping: partition {tn }0≤n≤N of [0, T ] Set u0h = u0h . For 0 ≤ n ≤ N , unh → un+1 as h (0)

• set uh = unh • for 1 ≤ i ≤ K, compute (i) uh

=

i−1 X l=0

αil whil

(l)

whil = uh +

βil n (l) ∆t Lh (uh ) αil (l)

(RK parameters s.t. stability follows from stab. of uh 7→ whil ) = uK • set un+1 h h


DG for Conservation Laws: RKDG (IV) Explicit RK time stepping: partition {tn }0≤n≤N of [0, T ] Set u0h = u0h . For 0 ≤ n ≤ N , unh → un+1 as h (0)

• set uh = unh • for 1 ≤ i ≤ K, compute (i) uh

=

i−1 X

αil whil

(l)

whil = uh +

l=0

βil n (l) ∆t Lh (uh ) αil (l)

(RK parameters s.t. stability follows from stab. of uh 7→ whil ) = uK • set un+1 h 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


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]


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]
