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
Lecture 3 The flux formulation of DG methods for elliptic problems Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 2 / 1
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Methods in Primal orm: Cartesian Grids, p = 1, α = 10 10 1 1.00 Log(error) 10 2 10 3 SIPG L2 BRMPS L2 SIPG(δ=1) L2 NIPG L2 IIPG L2 SIPG H1 BRMPS H1 SIPG(δ=1) H1 NIPG H1 IIPG H1 10 0 10 1 Log(1/h) 2.00 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 15 / 1
Methods in Mixed orm: Cartesian Grids, p = 1, α = 1, β = [1, 1] 10 1 1.00 Log(error) 10 2 10 3 2.00 LDG L2 BMMPR1 L2 LDG H1 BMMPR1 H1 10 0 10 1 Log(1/h) ne u u h L2 () u u h H1 () LDG BMMPR LDG BMMPR 16 64 2.5150 1.8895 1.0797 1.0114 64 256 2.1405 1.9885 1.0110 1.0000 256 1204 2.0235 2.0000 1.0066 1.0000 1024 4096 2.0089 2.0000 1.0000 1.0000 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 16 / 1
Super Penalty Mehotds: Cartesian Grids, p = 1, α = 1 10 0 BZ L2 BMMPR2 L2 BZ H1 BMMPR2 H1 Log(error) 10 1 1.00 10 2 2.00 10 0 10 1 Log(1/h) ne u u h L2 () u u h H1 () BZ BMMPR2 BZ BMMPR2 16 64 2.3051 2.0510 1.9901 1.2890 64 256 2.1821 2.0161 1.4504 1.0824 256 1204 2.0958 2.0043 1.1255 1.0210 1024 4096 2.0486 2.0011 1.0297 1.0053 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 17 / 1
LDG Method: Non-matching grids, p = 1, α = 1, β = [0.5, 0.5] 3 2.5 2 1.5 1 0.5 Log(error) 10 1 10 2 1.00 0 0 1 2 3 2.00 10 3 u u h L 2 u u h H 1 10 0 Log(1/h) ne u u h L2 () u u h H1 () 10 40 1.9899 0.9667 40 160 2.0448 1.0032 160 640 2.0199 1.0015 640 2560 2.0080 1.0005 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 18 / 1
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) 10 1 10 2 1.50 u σ h L 2 () 16 64 0.8509 64 256 1.1605 256 1204 1.2931 1024 4096 1.3841 4096 16384 1.4389 10 1 Log(1/h) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 19 / 1
SIP and NIP Methods: Unstructured Triangular Grids, p = 1, 2, 3, α = 10 SIP Method NIP Method 0 0 1 1.14 1 1.19 2 2 2.12 2.02 Log(error) 3 4 5 P1 u u h 0, P2 u u h 0, 2.16 3.14 2.98 Log(error) 3 4 5 P1 u u h 0, P2 u u h 0, 2.14 3.02 6 P3 u u h 0, P1 u u h 1,h P2 u u h 1,h P3 u u h 1,h 7 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Log(1/h) 4.07 6 P3 u u h 0, P1 u u h 1,h P2 u u h 1,h P3 u u h 1,h 4.07 7 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Log(1/h) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 20 / 1
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
Load The Darcy low with a Discontinuous κ: Cartesian Grids, p = 2, α = 10 div(κ u) = f in ( 1, 1) 2 x 10 4 18000 16000 We choose f and the BC s.t u(x, y) = cos(πx) cos(πy) 2 1.5 1 14000 12000 10000 8000 0.5 1 6000 0.5 4000 0 1 0.5 0 Y axis 0.5 1 1 0.5 0 X axis 2000 0 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 21 / 1
The Darcy low with a Discontinuous κ: Cartesian Grids, p = 2, α = 10 SIP Method NIP Method 4 4 3 3 2 2 Log(error) 1 0 1 2 3 4 5 σ σ h 0, u u h 1,h u u h 0, 1.99 2.98 6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Log(1/h) 1.99 Log(error) 1 0 1 2 3 σ σ h 0, u u h 1,h u u h 0, 2.00 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Log(1/h) 1.99 2.00 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 21 / 1