Discontinuous Galerkin Methods: Theory, Computation and Applications

Transcription

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 / 48

2 Lecture 2 An introduction to DG methods for elliptic problems Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 2 / 48

3 What are Discontinuous Galerkin methods? Discontinuous Galerkin (DG) methods are a family of finite element methods for the approximation of partial differential equations The idea The discrete solution is seek in a discrete space made of polynomials that are completely discontinuous across mesh elements V h V Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 3 / 48

4 eatures of DG Methods Wide range of PDE s treated within the same unified framework Weak approximation of boundary conditions lexibility in mesh design lexibility in polynomial degree distribution T i T j X Higher number of degrees of freedom X Larger algebraic linear systems to be solved (need of fast solvers) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 4 / 48

5 Historical roots Introduced in the 70 s for purely hyperbolic problems (Reed-Hill, Lesaint-Raviart) Extended in the mid 70 s to second order elliptic PDEs (Douglas-Dupont) and to fourth order problems (Baker) Abandoned in 80 s-90 s due to a much larger number of degrees of freedom compared to their conforming cousins Great revival since the late 90 s, application to a wide range of problems (linear/non-linear PDEs, time-dependent/stationary problems, spectrum approximation, multi-physics problems) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 5 / 48

6 Essential Bibliography: Books B. Rivière. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, J.S. Hesthaven, T. Warburton, Discontinuous Galerkin Methods. Algorithms, Analysis, and Applications, Springer, H. ahs, High-order discontinuous Galerkin methods for the Maxwell equations, 2010, Editions Universitaires Europèennes, D. A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Springer, 2012 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 6 / 48

7 Essential Bibliography Elliptic/parabolic PDEs Babuška & Zlámal (1973) Douglas & Dupont (1976) Baker (1977) Wheeler (1978), Riviére & Wheeler (1999 ) Arnold (1979, 1982) Cockburn, Perugia & Schötzau (2000 ) Arnold, Brezzi, Cockburn & Marini (SINUM, 2001/2002 ) Hyperbolic PDEs Reed & Hill (1973, Los Alamos Technical report) Lesaint & Raviart (1974, 1978) Johnson, Nävert & Pitkäranta (1984), Johnson & Pitkäranta (1986) Baumann (1997), Baumann & Oden (1997 ) Cockburn & Shu (1989 ) Houston & Süli (1999 ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 7 / 48

8 A first look at DG finite element methods Model problem { u = f in Ω u = 0 on Ω Take the equation u = f, multiply it by a (elementwise smooth) test function v and integrate over an element K T h uv = fv K Integrate by parts and sum over all the elements K T h u v u n K v = fv K T h K T h K To deal with the boxed term we need some further notation K K Ω Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 8 / 48

9 A first look at DG finite element methods Model problem { u = f in Ω u = 0 on Ω Take the equation u = f, multiply it by a (elementwise smooth) test function v and integrate over an element K T h uv = fv K Integrate by parts and sum over all the elements K T h K T h K u v K T h K K u n K v = To deal with the boxed term we need some further notation Ω fv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 8 / 48

10 A first look at DG finite element methods Model problem { u = f in Ω u = 0 on Ω Take the equation u = f, multiply it by a (elementwise smooth) test function v and integrate over an element K T h uv = fv K Integrate by parts and sum over all the elements K T h K T h K u v K T h K K u n K v = To deal with the boxed term we need some further notation Ω fv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 8 / 48

11 Trace Operators for any I (= set of interior faces) shared by K ± {v } = (v + + v )/2 [[v]] = v + n + + v n {τ } = (τ + + τ )/2 [[τ ]] = τ + n + + τ n for any B (= set of boundary faces) {v } = v [[v]] = vn {τ } = τ [[τ ]] = τ n Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 9 / 48

12 Trace Operators for any I (= set of interior faces) shared by K ± {v } = (v + + v )/2 [[v]] = v + n + + v n {τ } = (τ + + τ )/2 [[τ ]] = τ + n + + τ n for any B (= set of boundary faces) {v } = v [[v]] = vn {τ } = τ [[τ ]] = τ n Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 9 / 48

13 Magic ormula (Arnold, 82) K T h K τ n K v = {τ } [[v]] + I [[τ ]] {v } Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 10 / 48

14 Magic ormula (Arnold, 82) Proof K T h K } {{ } (A) Observe that (A) = I (B) = I τ n K v = {τ } [[v]] + I [[τ ]] {v } } {{ } (B) (τ + n + v + + τ n v ) + ( {τ } [[v]] + [[τ ]] {v }) + B B τ nv τ nv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 10 / 48

15 Magic ormula (Arnold, 82) Proof K T h K } {{ } (A) Observe that (A) = I (B) = I τ n K v = {τ } [[v]] + I [[τ ]] {v } } {{ } (B) (τ + n + v + + τ n v ) + ( {τ } [[v]] + [[τ ]] {v }) + B B τ nv τ nv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 10 / 48

16 Magic ormula: proof (cont d) (Arnold, 82) Proof Therefore, it is enough to show that on each internal face I (τ + n + v + + τ n v ) = ( {τ } [[v]] + [[τ ]] {v }) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 11 / 48

17 Magic ormula: proof (cont d) (Arnold, 82) Proof Therefore, it is enough to show that on each internal face I (τ + n + v + + τ n v ) = ( {τ } [[v]] + [[τ ]] {v }) } {{ } (C) Using the definition of the jump and average operators and that n + = n we have (C) = 1 (τ + + τ )(v + v ) n + + (v + + v )(τ + τ ) n + 2 = 1 (2τ + v + + τ v + τ + v 2τ v + v τ + v + τ ) n + 2 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 11 / 48

18 Magic ormula: proof (cont d) (Arnold, 82) Proof Therefore, it is enough to show that on each internal face I (τ + n + v + + τ n v ) = ( {τ } [[v]] + [[τ ]] {v }) } {{ } (C) (C) = 1 (τ + + τ )(v + v ) n + + (v + + v )(τ + τ ) n + 2 = 1 (2τ + v + +τ v + τ + v 2τ v +v τ + v + τ ) n + 2 = (τ + n + v + + τ n v ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 11 / 48

19 A first look at DG finite element methods (cont d) Magic formula K T h K τ n K v = {τ } [[v]] + I [[τ ]] {v } u v K T h K T h K K u n K v = Ω fv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 12 / 48

20 A first look at DG finite element methods (cont d) Magic formula K T h K τ n K v = {τ } [[v]] + I [[τ ]] {v } K T h K u v { u } [[v]] I [[ u]] {v } = Ω fv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 12 / 48

21 A first look at DG finite element methods (cont d) Magic formula K T h K τ n K v = {τ } [[v]] + I [[τ ]] {v } K T h K u v { u } [[v]] + [[ u]] {v } = fv I Ω Since u H 2 (Ω), then [[ u]] = 0 I. Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 12 / 48

22 A first look at DG finite element methods (cont d) Use that [[u]] = 0 (since u H0 1 (Ω)) to add a symmetry term u v { u } [[v]] { h v } [[u]] = fv K Ω K T h where h is the elementwise gradient (v is only piecewise smooth). We also add a stabilization term that controls the jumps K T h K u v { u } [[v]] + [[u]] { h v } γ [[u]] [[v]] = where γ is a stabilization function (that might depend on the discretization parameters) [Douglas-Dupont, Wheeler, Arnold], see later... Ω fv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 13 / 48

23 A first look at DG finite element methods (cont d) Use that [[u]] = 0 (since u H0 1 (Ω)) to add a symmetry term u v { u } [[v]] { h v } [[u]] = fv K Ω K T h where h is the elementwise gradient (v is only piecewise smooth). We also add a stabilization term that controls the jumps K T h K u v { u } [[v]] + [[u]] { h v } γ [[u]] [[v]] = where γ is a stabilization function (that might depend on the discretization parameters) [Douglas-Dupont, Wheeler, Arnold], see later... Ω fv Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 13 / 48

24 A first look at DG finite element methods (cont d) or p 1, define the DG discrete space V p h = { v h L 2 (Ω) : v h T P p (T ) T T h } H 1 0 (Ω) Discretize u u h, v v h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 14 / 48

25 A first look at DG finite element methods (cont d) or p 1, define the DG discrete space V p h = { v h L 2 (Ω) : v h T P p (T ) T T h } H 1 0 (Ω) Discretize u u h, v v h ind u h V p h s.t. A(u h, v h ) = Ω fv h v h V p h A(w, v) = w v { h w } [[v]] K T K h [[w]] { h v } + γ [[w]] [[v]] Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 14 / 48

26 The class of Interior Penalty DG methods ind u h V p h s.t. A(u h, v h ) = Ω fv h v h V p h A(w, v) = w v { h w } [[v]] K T K h θ [[w]] { h v } + γ [[w]] [[v]] θ = 1: Symmetric Interior Penalty (SIP). [Wheeler, 78],[Arnold, 82] θ = 1: Non-symmetric Interior Penalty (NIP). [Riviére, Wheeler & Girault, 99] θ = 0: Incomplete Interior Penalty (IIP). [Dawson, Sun, Wheeler, 04] Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 15 / 48

27 The stabilization function γ γ [[w]] [[v]] γ = α p2 h p = p K { max{pk, p K } if I h if B h h = h K { min{hk +, h K } if I h if B h Assumptions p K + p K h K + h K Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 16 / 48

28 A little bit of theory: notation or an interger s 1, define the broken Sobolev space H s (T h ) = { v L 2 (Ω) : v K H s (K) K T h } v 2 H s (T h ) = K T h v 2 H s (K) Define also v 2 L 2 ( h ) = h v 2 L 2 ( ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 17 / 48

29 A little bit of theory: notation (cont d) Define Ṽ p h = V p h + H2 (T h ) and v h 2 DG = hv h 2 γ L 2 (Ω) + 1/2 [[v h ]] v 2 γ DG = v 2 DG + 1/2 { h v } 2 L 2 ( h ) 2 L 2 ( h ) v h V p h v Ṽ p h It can be shown v h DG v h DG v h DG v h V p h. (trivial) (see later) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 18 / 48

30 Key ingredients 1 Continuity on Ṽ p h : A(v, w) v DG w DG v, w Ṽ p h 2 Coercivity on V p h : A(v h, v h ) v h 2 DG v h V p h 3 Consisentcy A(u, v h ) = Ω fv h v h V p h 4 Approximation. Let Π p h u V p h a suitable approximation of u. Then u Π p h u DG?? Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 19 / 48

31 Abstract error estimate or the analysis, suppose, for simplicity, that the exact solution u is (at least) u H 2 (Ω) the mesh T h is quasi-uniform (h=mesh-size) p K = p for any K T h 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 20 / 48

32 Abstract error estimate or the analysis, suppose, for simplicity, that the exact solution u is (at least) u H 2 (Ω) the mesh T h is quasi-uniform (h=mesh-size) p K = p for any K T h u u h DG u Π p h u DG + Π p h u u h DG (triangle inequality) Π p h u u h DG A(Πp h u u h, Π p h u u h) (Coercivity on V p h ) A(Π p h u u, Πp h u u h) (Consistency) Π p h u u DG Π p h u u h DG (Continuity on Ṽ p h ) Π p h u u DG Π p h u u h DG (Norms equivalence on V p h ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 20 / 48

33 Abstract error estimate or the analysis, suppose, for simplicity, that the exact solution u is (at least) u H 2 (Ω) the mesh T h is quasi-uniform (h=mesh-size) p K = p for any K T h u u h DG u Π p h u DG + Π p h u u h DG (triangle inequality) Π p h u u h DG A(Πp h u u h, Π p h u u h) (Coercivity on V p h ) A(Π p h u u, Πp h u u h) (Consistency) Π p h u u DG Π p h u u h DG (Continuity on Ṽ p h ) Π p h u u DG Π p h u u h DG (Norms equivalence on V p h ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 20 / 48

34 Abstract error estimate or the analysis, suppose, for simplicity, that the exact solution u is (at least) u H 2 (Ω) the mesh T h is quasi-uniform (h=mesh-size) p K = p for any K T h u u h DG u Π p h u DG + Π p h u u h DG (triangle inequality) Π p h u u h DG A(Πp h u u h, Π p h u u h) (Coercivity on V p h ) A(Π p h u u, Πp h u u h) (Consistency) Π p h u u DG Π p h u u h DG (Continuity on Ṽ p h ) Π p h u u DG Π p h u u h DG (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 20 / 48

35 Preliminary (key) estimate Let v, w Ṽ p h, we want to estimate a term of the form { h w } [[v]] Via Cauchy-Schwarz inequality ( a i b i ( ai 2 ) 1/2 ( bi 2 ) 1/2 ) we get { h w } [[v]] = γ 1/2 { h w } γ 1/2 [[v]] ( γ 1/2 ) 1/2 ( { h w } γ 1/2 [[v]] = γ 1/2 { h w } L 2 ( h ) 2 L 2 ( ) γ 1/2 [[v]] L 2 ( h ) 2 L 2 ( ) ) 1/2 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 21 / 48

36 Preliminary (key) estimate (cont d) γ { h w } [[v]] 1/2 { h w } L 2 ( h ) γ 1/2 [[v]] L 2 ( h ) w, v Ṽ p h If w is piecewise polynomial, the term γ 1/2 { h w } L can be 2 ( h ) further manipulated using the following trace/inverse estimate η L 2 ( ) p h 1/2 η L 2 (K) K η P p (K) Indeed, if w V p h we have γ 1/2 { h w } 2 h L 2 ( h ) αp 2 { h w } 2 L 2 ( ) h h p 2 αp 2 h K T h h w 2 L 2 (K + K ) 1 α hw 2 L 2 (Ω) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 22 / 48

37 Preliminary (key) estimate (cont d) In summary γ { h w } [[v]] 1/2 { h w } γ 1/2 [[v]] L 2 ( h ) { h w } [[v]] 1 h w α L2 (Ω) γ 1/2 [[v]] L2 ( h ) L 2 ( h ) w, v Ṽ p h w, v V p h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 23 / 48

38 Key ingredients 1 Continuity on Ṽ p h : A(v, w) v DG w DG v, w Ṽ p h 2 Coercivity on V p h : A(v h, v h ) v h 2 DG v h V p h 3 Consistency A(u, v h ) = Ω fv h v h V p h 4 Approximation. Let Π p h u V p h a suitable approximation of u. Then u Π p h u DG?? Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 24 / 48

39 Continuity on Ṽ p h A(v, w) v DG w DG v, w Ṽ p h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 25 / 48

40 Continuity on Ṽ p h Let v, w Ṽ p h, then A(v, w) w v + { K T K h h w } [[v]] }{{}}{{} (I ) (II ) + [[w]] { h v } + γ [[w]] [[v]] }{{}}{{} (III ) (IV ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 26 / 48

41 Continuity on Ṽ p h (cont d) Via Cauchy-Schwarz inequality ( a i b i ( a 2 i )1/2 ( b 2 i )1/2 ) we get 1/2 (I ) h w 2 L 2 (K) h v 2 L 2 (K) K Th K Th 1/2 = h w L 2 (Ω) hv L 2 (Ω) w DG v DG Analogously (IV ) γ 1/2 [[w]] h = γ 1/2 [[w]] L 2 ( h ) 2 L 2 ( ) 1/2 h γ 1/2 [[w]] 2 L 2 ( ) γ 1/2 [[v]] w L 2 DG v DG ( h ) 1/2 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 27 / 48

42 Continuity on Ṽ p h (cont d) (II ) γ 1/2 { h w } h = γ 1/2 { h w } L 2 ( h ) 2 L 2 ( ) 1/2 h γ 1/2 [[v]] 2 L 2 ( ) γ 1/2 [[v]] w L 2 DG v DG ( h ) 1/2 (II ) γ 1/2 { h v } h = γ 1/2 { h v } L 2 ( h ) 2 L 2 ( ) 1/2 h γ 1/2 [[w]] 2 L 2 ( ) γ 1/2 [[w]] v L 2 DG w DG ( h ) 1/2 Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 28 / 48

43 Coercivity on V p h A(v h, v h ) v h 2 DG v h V p h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 29 / 48

44 Coercivity on V p h Recalling the definition of A(, ), we have A(v h, v h ) = h v h 2 L 2 (Ω) (1 + θ) { h v h } [[v h ]] + γ 1/2 [[v h ]] h 2 0, If θ = 1 (i.e. NIP method), the result is trivial as A(v h, v h ) = h v h 2 γ L 2 (Ω) + 1/2 [[v h ]] 2 = v h 2 DG 0, If θ = 0, 1 (i.e. SIP/IIP methods), we have to estimate the term (I ) = { h v h } [[v h ]] { h v h } [[v h ]] = γ 1/2 { h v h } γ 1/2 [[v h ]] L 2 ( h ) L 2 ( h ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 30 / 48

45 Coercivity on V p h (cont d) rom the arithmetic-geometric inequality we have, for any ε > 0, (I ) ε γ 1/2 { h v h } γ 1/2 [[v h ]] L 2 ( h ) 2ε rom which it follows 2 L 2 ( h ) A(v h, v h ) = h v h 2 L 2 (Ω) (1 + θ) { h v h } [[v h ]] + γ 1/2 [[v h ]] 2 L 2 ( h ) h v h 2 L 2 (Ω) (1 + θ) { h v h } [[v h ]] + γ 1/2 [[v h ]] 2 L 2 ( h ) h v h 2 L 2 (Ω) (1 + θ) ε ( γ 1/2 { h v h } θ ) γ 1/2 [[v h ]] L 2 ( h ) 2ε 2 L 2 ( h ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 31 / 48

46 Coercivity on V p h (cont d) rom the arithmetic-geometric inequality we have, for any ε > 0, (I ) ε γ 1/2 { h v h } γ 1/2 [[v h ]] L 2 ( h ) 2ε rom which it follows 2 L 2 ( h ) A(v h, v h ) = h v h 2 L 2 (Ω) (1 + θ) { h v h } [[v h ]] + γ 1/2 [[v h ]] 2 L 2 ( h ) h v h 2 L 2 (Ω) (1 + θ) { h v h } [[v h ]] + γ 1/2 [[v h ]] 2 L 2 ( h ) h v h 2 L 2 (Ω) (1 + θ) ε ( γ 1/2 { h v h } θ ) γ 1/2 [[v h ]] 2 L 2 ( h ) 2ε We now bound the term in the box 2 L 2 ( h ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 31 / 48

47 Coercivity on V p h (cont d) Trace/inverse estimate η L 2 ( ) p h 1/2 η L 2 (K) K η P p (K) Using the above trace-inverse estimate γ 1/2 { h v h } 2 L 2 ( h ) h p 2 α α h p 2 αp 2 h { h v h } 2 L 2 ( ) h 1 α C I h v h L 2 (Ω) K T h h v h L 2 (K + K ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 32 / 48

48 Coercivity on V p h Therefore we get (cont d) ( A(v h, v h ) 1 C ) ( I(1 + θ)ε h v h 2 L 2α 2 (Ω) θ ) γ 1/2 [[v h ]] 2ε 2 L 2 ( h ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 33 / 48

49 Coercivity on V p h Therefore we get (cont d) ( A(v h, v h ) 1 C ) ( I(1 + θ)ε h v h 2 L 2α 2 (Ω) θ ) γ 1/2 [[v h ]] 2ε Now choose (for example) ε = 1 + θ to get A(v h, v h ) (1 C I(1 + θ) 2 ) h v h 2 L 2α 2 (Ω) + 1 γ 1/2 [[v h ]] 2 2 L 2 ( h ) 2 L 2 ( h ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 33 / 48

50 Coercivity on V p h Therefore we get (cont d) ( A(v h, v h ) 1 C ) ( I(1 + θ)ε h v h 2 L 2α 2 (Ω) θ ) γ 1/2 [[v h ]] 2ε Now choose (for example) ε = 1 + θ to get A(v h, v h ) (1 C I(1 + θ) 2 ) h v h 2 L 2α 2 (Ω) + 1 γ 1/2 [[v h ]] 2 2 L 2 ( h ) 2 L 2 ( h ) Therefore we obtain A(v h, v h ) 1 2 hv h 2 L 2 (Ω) + 1 γ 1/2 [[v h ]] 2 2 = 1 L 2 ( h ) 2 v h 2 DG provided 1 C I(1 + θ) 2 2α 1 2 α C I (1 + θ) 2 α Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 33 / 48

51 Consistency A(u, v h ) = Ω fv h v h V p h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 34 / 48

52 Consistency Recalling the Magic ormula, we have the following Integration by parts formula τ v = K T h K T h K K τ v {τ } [[v]] I [[τ ]] {v } for all τ [H 1 (T h )] d, v H 1 (T h ) Now, let u solves the boundary value problem u = f in Ω u = 0 on Ω Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 35 / 48

53 Consistency (cont d) rom the integration by parts formula we get u v = { u } [[v]]+ K T h K T h K K uv + I [[ u]] {v } for any v H 1 (T h ). Since [[ u]] = 0 on each interior face and u = f, we obtain u v { u } [[v]] = fv v H 1 (T h ). K Ω K T h Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 36 / 48

54 Consistency (cont d) Since [[u]] = 0 on each interior face and [[u]] = u = 0 on each boundary face, we can also write u v { u } [[v]] K T K h + γ [[u]] [[v]] = Ω [[u]] { h v } fv v H 1 (T h ). That is A(u, v) = Ω fv v H 1 (T h ). Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 37 / 48

55 Consistency (cont d) A(u, v h ) = Ω fv h v V p h. By linearity we than obtain Galerkin orthogonality A(u u h, v h ) = 0 v V p h. Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 38 / 48

56 Approximation bound for u Π p h u DG Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 39 / 48

57 Approximation bound for u Π p h u DG Set e π = u Π p hu and recall that e π 2 DG = he π 2 L 2 (Ω) + γ 1/2 [[e π ]] 2 }{{} 0, h (I ) }{{} (II ) + h We make use of the following hp approximation result γ 1/2 { h e π } 2 0, } {{ } (III ) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 40 / 48

58 An hp approximation result [Babuska, Suri, 1987] Let K T h and let v H s (K), s 1. Then, there exists a sequence of operators Π p h v : Hs (K) P p (K), p = 1, 2,... such that v Π p h v hmin(p+1,s) q H q (K) p s q v H s (K) 0 q s The hidden constant is independent of v, h and p, but depends on the shape-regularity of K an on s. or the proof, see [Babuska, Suri, 1987], for d = 2. The case d = 3 is analogous. Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 41 / 48

59 Approximation bound for u Π p h u DG (cont d) Then, for the term (I) we get (I ) = h (u Π p h u) 2 L 2 (Ω) h2 min(p+1,s) 2 p 2s 2 v 2 H s (T h ) To estimate (II), we make use of the multiplicative trace inequality Multiplicative trace inequality η 2 L 2 ( K) η L 2 (K) η L 2 (K) + h 1 η 2 L 2 (K) η H 1 (K) where the hidden constant depends on the shape regularity of K and on the dimension d. Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 42 / 48

60 Approximation bound for u Π p h u DG (cont d) (II ) = h p2 h p2 h Analogously, we have (III ) = h p2 h γ 1/2 [[e π ]] 2 0, p2 h K T h e π 2 L 2 ( K) K T h e π L 2 (K) e π L 2 (K) + h 1 e π 2 L 2 (K) h 2 min(p+1,s) 1 p 2s 1 u 2 H s (T h ) γ 1/2 { h e π } h 2 min(p+1,s) 3 2 0, p 2s 3 u 2 H s (T h ) p2 h K T h e π 2 L 2 ( K) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 43 / 48

61 Approximation bound for u Π p h u DG (cont d) If the exact solution u H s (T h ), s 2, then u Π p h u DG hmin(p+1,s) 1 p s 1/2 u H s (T h ) In particular, if p s 1, the estimate becomes u Π p h u DG ( ) h s 1 p 1/2 u p. Hs(Th) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 44 / 48

62 Abstract error estimate If u H s (T h ), s 2, then u u h DG hmin(p+1,s) 1 p s 1/2 u H s (T h ) or the SIP and IIP methods the above estimate holds provided that the penalty constant α is chosen sufficiently large. The bound is optimal in h and suboptimal in p by a factor p 1/2. See, for example, [Houston, Schwab, Suli, 2001], [Riviere, Wheeler, Girault, 1999], [Perugia, Schotzau, 2001]. Optimal error estimates with respect to p can be shown using the projector of [Georgoulis & Suli, 2005] provided the solution belongs to a suitable augmented space, or whenever a continuous interpolant can be built; cf. [Stamm & Wihler, 2010]. Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 45 / 48

63 L 2 -norm error estimates by duality argument An estimate for the L 2 -error can be obtained by using a duality argument. Elliptic regularity Assume that Ω is such that the following ellitpic regularity result holds: for any g L 2 (Ω), the solution z of the problem satisfies z H 2 (Ω) and z = g in Ω z = 0 on Ω z H 2 (Ω) g L 2 (Ω) Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 46 / 48

64 L 2 norm error estimates by duality argument (cont d) If the exact solution u H s (T h ), s 2 and if u h is the solution obtained with the SIP method (θ = 1), it holds u u h L 2 (Ω) hmin(p+1,s) p s+1/2 u H s (T h ) The essential ingredient in the duality argument for proving L 2 estimates is the following adjoint consistency property A(v h, z) = fv h v h V p h Ω Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 47 / 48

65 L 2 norm error estimates by duality argument (cont d) or NIP and IIP formulations it only holds u u h L 2 (Ω) hmin(p+1,s) 1 p s 1/2 u H s (T h ) since the corresponding bilinear form is non-symmetric and does not satisfy the adjoint consistency property Paola. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 48 / 48