Discontinuous Galerkin Methods: Theory, Computation and Applications

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1 Discontinuous Galerkin Methods: Theory, Computation and Applications Paola. Antonietti MOX, Dipartimento di Matematica Politecnico di Milano.MO. X MODELLISTICA E CALCOLO SCIENTIICO. MODELING AND SCIENTIIC COMPUTING International Doctoral School Gran Sasso Science Institute (GSSI), L Aquila 2-5 May 2017 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 1 / 1

2 Lecture 3 The flux formulation of DG methods for elliptic problems Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 2 / 1

3 Model Problem R d (d = 2, 3) bounded convex (polygonal) domain { u = f in u = 0 on σ = u σ = u div(σ) = f u = 0 in in on [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 3 / 1

4 Model Problem R d (d = 2, 3) bounded convex (polygonal) domain { u = f in u = 0 on σ = u σ = u div(σ) = f u = 0 in in on [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 3 / 1

5 Model Problem R d (d = 2, 3) bounded convex (polygonal) domain { u = f in u = 0 on σ = u σ = u div(σ) = f u = 0 in in on [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 3 / 1

6 Starting point Starting Point: σ = u in, div(σ) = f σ τ = u div(τ ) + u τ n σ v = f v + σ n v in, u = 0 on τ smooth enough v smooth enough Define: V p h = { v h L 2 () : v h T P p (T ) T T h }, Σ p h = [V p h ]d Discretize: σ σ h Σ p h u u h V p h Sum over all the elements of T h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 4 / 1

7 Starting point Starting Point: σ = u in, div(σ) = f σ τ = u div(τ ) + u τ n σ v = f v + σ n v in, u = 0 on τ smooth enough v smooth enough Define: V p h = { v h L 2 () : v h T P p (T ) T T h }, Σ p h = [V p h ]d Discretize: σ σ h Σ p h u u h V p h Sum over all the elements of T h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 4 / 1

8 Starting point Starting Point: σ = u in, div(σ) = f σ τ = u div(τ ) + u τ n σ v = f v + σ n v in, u = 0 on τ smooth enough v smooth enough Define: V p h = { v h L 2 () : v h T P p (T ) T T h }, Σ p h = [V p h ]d Discretize: σ σ h Σ p h u u h V p h Sum over all the elements of T h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 4 / 1

9 Starting point Starting Point: σ = u in, div(σ) = f σ τ = u div(τ ) + u τ n σ v = f v + σ n v in, u = 0 on τ smooth enough v smooth enough Define: V p h = { v h L 2 () : v h T P p (T ) T T h }, Σ p h = [V p h ]d Discretize: σ σ h Σ p h u u h V p h Sum over all the elements of T h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 4 / 1

10 Starting point Starting Point: σ = u in, div(σ) = f σ τ = u div(τ ) + u τ n σ v = f v + σ n v in, u = 0 on τ smooth enough v smooth enough Define: V p h = { v h L 2 () : v h T P p (T ) T T h }, Σ p h = [V p h ]d Discretize: σ σ h Σ p h u u h V p h Sum over all the elements of T h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 4 / 1

11 Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] ind (σ h, u h ) Σ p h V p h s.t. σ h τ = u h div(τ ) + û τ n τ Σ p h, σ h v = f v + σ n v, v V p h, Numerical luxes û u σ σ = u Consistent : û(v) = v and σ(v, v) = v Conservative: if fluxes are single-valued on each e v smooth Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 5 / 1

12 Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] ind (σ h, u h ) Σ p h V p h s.t. σ h τ = u h div(τ ) + û τ n τ Σ p h, σ h v = f v + σ n v, v V p h, Numerical luxes û u σ σ = u Consistent : û(v) = v and σ(v, v) = v Conservative: if fluxes are single-valued on each e v smooth Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 5 / 1

13 Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] ind (σ h, u h ) Σ p h V p h s.t. σ h τ = u h div(τ ) + û τ n τ Σ p h, σ h v = f v + σ n v, v V p h, Numerical luxes û u σ σ = u Consistent : û(v) = v and σ(v, v) = v Conservative: if fluxes are single-valued on each e v smooth Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 5 / 1

14 Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] Magic ormula [[v]] {τ } + T h vτ n = I {v } [[τ ]] ind (σ h, u h ) Σ p h V p h : σ h τ = u h h τ + [[û]] {τ } + {û } [[τ ]] I σ h h v { σ } [[v]] [[ σ]] {v } = fv I τ Σ p h v V p h Integration by parts and the identity σ h τ = h u h τ + [[û u h ]] {τ } + {û u h } [[τ ]] I Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 6 / 1

15 Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] Magic ormula [[v]] {τ } + T h vτ n = I {v } [[τ ]] ind (σ h, u h ) Σ p h V p h : σ h τ = u h h τ + [[û]] {τ } + {û } [[τ ]] I σ h h v { σ } [[v]] [[ σ]] {v } = fv I τ Σ p h v V p h Integration by parts and the identity σ h τ = h u h τ + [[û u h ]] {τ } + {û u h } [[τ ]] I Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 6 / 1

16 Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001/2002] Magic ormula [[v]] {τ } + T h vτ n = I {v } [[τ ]] ind (σ h, u h ) Σ p h V p h : σ h τ = u h h τ + [[û]] {τ } + {û } [[τ ]] I σ h h v { σ } [[v]] [[ σ]] {v } = fv I τ Σ p h v V p h Integration by parts and the identity σ h τ = h u h τ + [[û u h ]] {τ } + {û u h } [[τ ]] I Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 6 / 1

17 Eliminate σ h Lifting operators: τ Σ p h R([[v h ]]) τ = [[v h ]] {τ } L( {v h }) τ = I {v h } [[τ ]] The first equation can be rewritten as [σ h h u h + R([[û u h ]]) + L( {û u h })] τ = 0 τ Σ p h û = û(u h ) = eliminate σ h element-by-element σ h = h u h R([[û u h ]]) L( {û u h }) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 7 / 1

18 Eliminate σ h Lifting operators: τ Σ p h R([[v h ]]) τ = [[v h ]] {τ } L( {v h }) τ = I {v h } [[τ ]] The first equation can be rewritten as [σ h h u h + R([[û u h ]]) + L( {û u h })] τ = 0 τ Σ p h û = û(u h ) = eliminate σ h element-by-element σ h = h u h R([[û u h ]]) L( {û u h }) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 7 / 1

19 Eliminate σ h Lifting operators: τ Σ p h R([[v h ]]) τ = [[v h ]] {τ } L( {v h }) τ = I {v h } [[τ ]] The first equation can be rewritten as [σ h h u h + R([[û u h ]]) + L( {û u h })] τ = 0 τ Σ p h û = û(u h ) = eliminate σ h element-by-element σ h = h u h R([[û u h ]]) L( {û u h }) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 7 / 1

20 Primal ormulation σ h = h u h R([[û u h ]]) L( {û u h }) Primal ormulation ind u h V p h s.t. A h (u h, v) = fv v V p h A h (u h, v) = h u h h v + [[û u h ]] { h v } + {û u h } [[ h v]] { σ } [[v]] [[ σ]] {v } I I Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 8 / 1

21 Primal ormulation σ h = h u h R([[û u h ]]) L( {û u h }) Substitute into 2 nd equation: σ h h v Apply the definition of lifting operators { σ } [[v]] [[ σ]] {v } = fv I Primal ormulation ind u h V p h s.t. A h (u h, v) = fv v V p h A h (u h, v) = h u h h v + [[û u h ]] { h v } + {û u h } [[ h v]] { σ } [[v]] [[ σ]] {v } I I Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 8 / 1

22 Primal ormulation Primal ormulation ind u h V p h s.t. A h (u h, v) = fv v V p h A h (u h, v) = h u h h v + [[û u h ]] { h v } + {û u h } [[ h v]] { σ } [[v]] [[ σ]] {v } I I Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 8 / 1

23 Primal ormulation Primal ormulation ind u h V p h s.t. A h (u h, v) = fv v V p h A h (u h, v) = h u h h v + [[û u h ]] { h v } + {û u h } [[ h v]] { σ } [[v]] [[ σ]] {v } I I Remark Different choices of û and σ determine different DG methods. Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 8 / 1

24 Example: Interior Penalty Methods {u h } + 1 θ [[u h ]] n I û = 2 σ = { h u h } γ [[u h ]] 0 B Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 9 / 1

25 Example: Interior Penalty Methods {u h } + 1 θ [[u h ]] n I û = 2 σ = { h u h } γ [[u h ]] 0 B {û u h } = 0 { σ } = { h u h } γ [[u h ]] [[û u h ]] = θ [[u h ]], [[ σ]] = 0, Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 9 / 1

26 Example: Interior Penalty Methods {u h } + 1 θ [[u h ]] n I û = 2 σ = { h u h } γ [[u h ]] 0 B A h (u h, v h ) = h u h h v h { h u h } [[v h ]] θ [[u h ]] { h v h } γ [[u h ]] [[v h ]] Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 9 / 1

27 Example: Interior Penalty Methods {u h } + 1 θ [[u h ]] n I û = 2 σ = { h u h } γ [[u h ]] 0 B A h (u h, v h ) = h u h h v h { h u h } [[v h ]] θ [[u h ]] { h v h } γ [[u h ]] [[v h ]] θ = 1 SIP [Arnold, SINUM, 1982] θ = 1 NIP [Rivière, Wheeler & Girault, Comp. Geosc.,1999] θ = 0 IIP [Wheeler, Dawson & Sun, CMAME, 2004] Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 9 / 1

28 Different Ways to Enforce Stability α j (w, v) = α r (w, v) = γ j [[w]] [[v]] γ r r ([[w]]) r ([[v]]) γ j αp 2 h 1 γ r α r : [L 1 ( )] d Σ p h r ([[u h ]]) τ = [[u h ]] {τ } τ Σ p h Global and Local Lifting Operators R(ϕ) = r (ϕ) ϕ [L 1 ()] d Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 10 / 1

29 Numerical luxes: Summary Method û σ SIP {u h } { h u h } γ j [[u h ]] BRMPS {u h } { h u h } + γ r {r ([[u h ]]) } SIP(δ) {u h } (1 δ) { h u h } δ γ j [[u h ]] [[u h ]] BO {u h } + [[u h ]] n { h u h } NIP {u h } + [[u h ]] n { h u h } γ j [[u h ]] IIP {u h } + 1/2 [[u h ]] n { h u h } γ j [[u h ]] BR {u h } {σ h } BMMPR {u h } {σ h } + γ r {r ([[u h ]]) } LDG {u h } β [[u h ]] {σ h } + β [[σ h ]] γ j [[u h ]] BZ (u h ) αh (2p+1) [[u h ]] BMMPR 2 (u h ) αh 2p {r ([[u h ]]) } Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 11 / 1

30 Abstract error estimate u u h DG u Π p h u DG + Π p h u u h DG triangle inequality Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 12 / 1

31 Abstract error estimate u u h DG u Π p h u DG + Π p h u u h DG Π p h u u h 2 DG A(Πp h u u h, Π p h u u h) A(Π p h u u, Πp h u u h) Π p h u u DG Π p h u u h DG Π p h u u DG Π p h u u h DG triangle inequality Coercivity on V p h Galerkin orthogonality Continuity on Ṽ p h Norms equivalence on V p h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 12 / 1

32 Abstract error estimate u u h DG u Π p h u DG + Π p h u u h DG Π p h u u h 2 DG A(Πp h u u h, Π p h u u h) A(Π p h u u, Πp h u u h) Π p h u u DG Π p h u u h DG Π p h u u DG Π p h u u h DG triangle inequality Coercivity on V p h Galerkin orthogonality Continuity on Ṽ p h Norms equivalence on V p h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 12 / 1

33 Abstract error estimate u u h DG u Π p h u DG + Π p h u u h DG Π p h u u h 2 DG A(Πp h u u h, Π p h u u h) A(Π p h u u, Πp h u u h) Π p h u u DG Π p h u u h DG Π p h u u DG Π p h u u h DG triangle inequality Coercivity on V p h Galerkin orthogonality Continuity on Ṽ p h Norms equivalence on V p h u u h DG u Π p h u DG Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 12 / 1

34 Summary Method G.O. A.C. Stab. Type Cond. DG L2 () SIP α j α > α h p h p+1 BRMPS α r α > α h p h p+1 SIP(δ) α j α > α h p h p+1 BO(p 2) X X - - h p h p NIP X α j α > 0 h p h p IIP X α j α > α h p h p BR X - - [h p ] [h p ] BMMPR 1 α r α > 0 h p h p+1 LDG α j α > 0 h p h p+1 BZ X X α j α h 2p h p h p+1 BMMPR2 X X α r α h 2p h p h p+1 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 13 / 1

35 Numerical Results = (0, π) (0, π), u(x, y) = sin(x) sin(y) Triangulations Cartesian Grids made of 4 N elements (N = 2, 3, 4, 5, 6) Unstructured triangular grids made of 2 4 N elements (N = 2, 3, 4, 5) DG finite elements Q p - P p (p = 1, 2, 3) O(h p ) in the energy norm O(h p+1 ) in the L 2 -norm Matlab code Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 14 / 1

36 Methods in Primal orm: Cartesian Grids, p = 1, α = Log(error) SIPG L2 BRMPS L2 SIPG(δ=1) L2 NIPG L2 IIPG L2 SIPG H1 BRMPS H1 SIPG(δ=1) H1 NIPG H1 IIPG H Log(1/h) 2.00 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 15 / 1

37 Methods in Mixed orm: Cartesian Grids, p = 1, α = 1, β = [1, 1] Log(error) LDG L2 BMMPR1 L2 LDG H1 BMMPR1 H Log(1/h) ne u u h L2 () u u h H1 () LDG BMMPR LDG BMMPR Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 16 / 1

38 Super Penalty Mehotds: Cartesian Grids, p = 1, α = BZ L2 BMMPR2 L2 BZ H1 BMMPR2 H1 Log(error) Log(1/h) ne u u h L2 () u u h H1 () BZ BMMPR2 BZ BMMPR Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 17 / 1

39 LDG Method: Non-matching grids, p = 1, α = 1, β = [0.5, 0.5] Log(error) u u h L 2 u u h H Log(1/h) ne u u h L2 () u u h H1 () Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 18 / 1

40 A Super Convergence Result: LDG Method, p = 1, [Cockburn, anshat, Perugia & Shötzau, SINUM, 2001] u(x, y) = exp(xy) α j (u h, v) = γ j [[u h ]] [[v]] β = [0.5, 0.5] γ j = 1 u σ h 0, Log(error) u σ h L 2 () Log(1/h) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 19 / 1

41 SIP and NIP Methods: Unstructured Triangular Grids, p = 1, 2, 3, α = 10 SIP Method NIP Method Log(error) P1 u u h 0, P2 u u h 0, Log(error) P1 u u h 0, P2 u u h 0, P3 u u h 0, P1 u u h 1,h P2 u u h 1,h P3 u u h 1,h Log(1/h) P3 u u h 0, P1 u u h 1,h P2 u u h 1,h P3 u u h 1,h Log(1/h) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 20 / 1

42 Ö Ð Ø º º Æ ÙÑ ÒÒ º º The Darcy low with a Discontinuous κ: Cartesian Grids, p = 2, α = 10 Ö Ð Ø º º div(κ u) = f in ( 1, 1) 2 κ = 10 κ = 100 We choose f and the BC s.t u(x, y) = cos(πx) cos(πy) κ = 1000 κ = 1 Ö Ð Ø º º Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 21 / 1

43 Load The Darcy low with a Discontinuous κ: Cartesian Grids, p = 2, α = 10 div(κ u) = f in ( 1, 1) 2 x We choose f and the BC s.t u(x, y) = cos(πx) cos(πy) Y axis X axis Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 21 / 1

44 The Darcy low with a Discontinuous κ: Cartesian Grids, p = 2, α = 10 SIP Method NIP Method Log(error) σ σ h 0, u u h 1,h u u h 0, Log(1/h) 1.99 Log(error) σ σ h 0, u u h 1,h u u h 0, Log(1/h) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 21 / 1

Discontinuous Galerkin Methods: Theory, Computation and Applications

Discontinuous Galerkin Methods: Theory, Computation and Applications Paola. Antonietti MOX, Dipartimento di Matematica Politecnico di Milano.MO. X MODELLISTICA E CALCOLO SCIENTIICO. MODELING AND SCIENTIIC