hp-version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

Transcription

1 hp-version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes Andrea Cangiani Department of Mathematics University of Leicester Joint work with: E. Georgoulis & P. Dong (Leicester), P. Houston (Nottingham) Recent advances in discontinuous Galerkin methods University of Reading, September 2014 A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

2 Outline 1 Physical frame DG spaces on polygonal/polyhedral meshes: allowing for degenerating faces 2 Tools for hp-analysis 3 Applications: Second order elliptic problems First order hyperbolic problems C., Georgoulis, & Houston, M3AS, C., Dong, Georgoulis, & Houston, in preparation. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

3 Flexibility of DGFEMs DGFEMs are not restricted to employing standard polynomial spaces mapped from a reference frame. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

4 Flexibility of DGFEMs DGFEMs are not restricted to employing standard polynomial spaces mapped from a reference frame. Within this talk, the FE space Always defined in the physical frame; On each element κ it is given by P p (κ), 1 1 P p(κ)= space of polynomials of total degree p on κ. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

5 Flexibility of DGFEMs DGFEMs are not restricted to employing standard polynomial spaces mapped from a reference frame. Within this talk, the FE space Always defined in the physical frame; On each element κ it is given by P p (κ), 1 The order of convergence is independent of the element shape. Bassi et al., J. Comput. Phys., Antonietti, Giani, & Houston, SIAM J. Sci. Comput., P p(κ)= space of polynomials of total degree p on κ. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

6 Flexibility of DGFEMs DGFEMs are not restricted to employing standard polynomial spaces mapped from a reference frame. Within this talk, the FE space Always defined in the physical frame; On each element κ it is given by P p (κ), 1 The order of convergence is independent of the element shape. Bassi et al., J. Comput. Phys., Antonietti, Giani, & Houston, SIAM J. Sci. Comput., In contrast to mapped FE spaces, see eg. Arnold, Boffi, & Falk, Math. Comput., P p(κ)= space of polynomials of total degree p on κ. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

7 Finite Element Spaces Let Ω R d, d = 2, 3, bounded & open polyhedral domain. mesh T : disjoint subdivision of Ω into polygons/polyhedra; element κ T ; diameter h κ := diam(κ) polynomial degree p := (p κ N : κ T ) The DG finite element space S p T with respect to T and p defined by S p T := {u L2 (Ω) : u κ P pκ (κ), κ T }, REMARK: the local elemental polynomial spaces generating S p T defined in the physical space. are A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

8 Finite Element Spaces Let Ω R d, d = 2, 3, bounded & open polyhedral domain. mesh T : disjoint subdivision of Ω into polygons/polyhedra; element κ T ; diameter h κ := diam(κ) polynomial degree p := (p κ N : κ T ) The DG finite element space S p T with respect to T and p defined by S p T := {u L2 (Ω) : u κ P pκ (κ), κ T }, REMARK: the local elemental polynomial spaces generating S p T defined in the physical space. are A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

9 Poisson problem, quadrilateral meshes: p-convergence For a fixed 16 d square mesh we compare under p-refinement CGFEM: conforming Q p based DGFEM(P): P p based IP-DG DGFEM(Q): Q p based IP-DG x16 CGFEM DGFEM(P) DGFEM(Q) x16x16 CGFEM DGFEM(P) DGFEM(Q) u u h L 2 (Ω) u u h L 2 (Ω) dof 1/ dof 1/3 A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

10 Poisson problem, quadrilateral meshes: adaptive refinement Consider the classical L-shaped domain ( 1, 1) 2 \ [0, 1) ( 1, 0] impose an appropriate boundary condition for exact solution u so that u = r 2/3 sin(2ϕ/3) DGFEM(Q) DGFEM(P) u u h dof 1/3 hp adaptive algorithm based on residual based a posteriori error indicators of [Houston, Schötzau, and Wihler, M3AS, 2007]. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

11 The main goal In an hp-setting, extend IP-DG to general polygonal and polyhedral meshes such that: mesh element faces may have arbitrarily small measure in two dimensions; both mesh element faces and edges may have arbitrarily small measure in three dimensions. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

12 The main goal In an hp-setting, extend IP-DG to general polygonal and polyhedral meshes such that: mesh element faces may have arbitrarily small measure in two dimensions; both mesh element faces and edges may have arbitrarily small measure in three dimensions. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

13 General polygonal and polyhedral meshes Interfaces of T : intersections of the (d 1) dimensional facets of neighbouring elements. Assumption 0 (d = 3 only): Each interface of each κ T may be subdivided into a set of co-planar triangles. Faces always refer to (d 1) dimensional simplexes (line interfaces when d = 2) Skeleton Γ: union of all open mesh faces. Assumption 1 The number of faces of each element κ T is uniformly bounded by a constant C F N. hp-analysis tools for such elements are missing. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

14 General polygonal and polyhedral meshes Interfaces of T : intersections of the (d 1) dimensional facets of neighbouring elements. Assumption 0 (d = 3 only): Each interface of each κ T may be subdivided into a set of co-planar triangles. Faces always refer to (d 1) dimensional simplexes (line interfaces when d = 2) Skeleton Γ: union of all open mesh faces. Assumption 1 The number of faces of each element κ T is uniformly bounded by a constant C F N. hp-analysis tools for such elements are missing. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

15 General polygonal and polyhedral meshes Interfaces of T : intersections of the (d 1) dimensional facets of neighbouring elements. Assumption 0 (d = 3 only): Each interface of each κ T may be subdivided into a set of co-planar triangles. Faces always refer to (d 1) dimensional simplexes (line interfaces when d = 2) Skeleton Γ: union of all open mesh faces. Assumption 1 The number of faces of each element κ T is uniformly bounded by a constant C F N. hp-analysis tools for such elements are missing. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

16 General polygonal and polyhedral meshes Interfaces of T : intersections of the (d 1) dimensional facets of neighbouring elements. Assumption 0 (d = 3 only): Each interface of each κ T may be subdivided into a set of co-planar triangles. Faces always refer to (d 1) dimensional simplexes (line interfaces when d = 2) Skeleton Γ: union of all open mesh faces. Assumption 1 The number of faces of each element κ T is uniformly bounded by a constant C F N. hp-analysis tools for such elements are missing. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

17 General polygonal and polyhedral meshes Interfaces of T : intersections of the (d 1) dimensional facets of neighbouring elements. Assumption 0 (d = 3 only): Each interface of each κ T may be subdivided into a set of co-planar triangles. Faces always refer to (d 1) dimensional simplexes (line interfaces when d = 2) Skeleton Γ: union of all open mesh faces. Assumption 1 The number of faces of each element κ T is uniformly bounded by a constant C F N. hp-analysis tools for such elements are missing. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

18 Trace estimates For each element κ T and each face F κ T, κ F represents any d dimensional simplex such that κ F κ F κ F. κ F F For v P p (κ), a classical inverse estimate on the face κ F v 2 L 2 (F ) p2 F κ F v 2 L 2 (κ F ) p2 F κ F v 2 L 2 (κ), v 2 L 2 (F ) p 2 F sup κ F κ κf v 2 L 2 (κ). gives This estimate is not sharp with respect to the measure F relative to the measure κ. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

19 Trace estimates For each element κ T and each face F κ T, κ F represents any d dimensional simplex such that κ F κ F κ F. κ F F For v P p (κ), a classical inverse estimate on the face κ F v 2 L 2 (F ) p2 F κ F v 2 L 2 (κ F ) p2 F κ F v 2 L 2 (κ), v 2 L 2 (F ) p 2 F sup κ F κ κf v 2 L 2 (κ). gives This estimate is not sharp with respect to the measure F relative to the measure κ. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

20 Trace estimates For each element κ T and each face F κ T, κ F represents any d dimensional simplex such that κ F κ F κ F. κ F F For v P p (κ), a classical inverse estimate on the face κ F v 2 L 2 (F ) p2 F κ F v 2 L 2 (κ F ) p2 F κ F v 2 L 2 (κ), v 2 L 2 (F ) p 2 F sup κ F κ κf v 2 L 2 (κ). gives This estimate is not sharp with respect to the measure F relative to the measure κ. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

21 Allowing for degenerating faces Define T as the subset of T such that κ T if it can be covered by at most m T shape-regular simplexes K i with K i c as κ, and such that dist(κ, K i ) diam(k i )/p 2, i = 1,..., m T, κ K 2 for some m T independent of κ and T. K 1 Lemma (Georgoulis, Math. Comput., 2008) Let K be a shape-regular simplex. Then, for each v P p(k), there exists a simplex ˆκ K, having the same shape as K and faces parallel to the faces of K, with dist( ˆκ, K) diam(k)/p 2, such that v L 2 (ˆκ) 1 2 v L 2 (K). A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

22 Allowing for degenerating faces Define T as the subset of T such that κ T if it can be covered by at most m T shape-regular simplexes K i with K i c as κ, and such that dist(κ, K i ) diam(k i )/p 2, i = 1,..., m T, κ K 2 for some m T independent of κ and T. K 1 Lemma (Georgoulis, Math. Comput., 2008) Let K be a shape-regular simplex. Then, for each v P p(k), there exists a simplex ˆκ K, having the same shape as K and faces parallel to the faces of K, with dist( ˆκ, K) diam(k)/p 2, such that v L 2 (ˆκ) 1 2 v L 2 (K). A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

23 Now, if κ T and F κ v 2 L 2 (F ) p2 F κ F v 2 L 2 (κ F ) p2 F v 2 L (κ F ) m T p 2 p 2d F K j v 2 L 2 (K j ) j=1 m T p 2 p 2d F K j (4 v 2 L 2 (ˆκ j ) ) p2d+2 F κ j=1 v 2 L 2 (κ) A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

24 Now, if κ T and F κ v 2 L 2 (F ) p2 F κ F v 2 L 2 (κ F ) p2 F v 2 L (κ F ) m T p 2 p 2d F K j v 2 L 2 (K j ) j=1 m T p 2 p 2d F K j (4 v 2 L 2 (ˆκ j ) ) p2d+2 F κ j=1 v 2 L 2 (κ) A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

25 Now, if κ T and F κ v 2 L 2 (F ) p2 F κ F v 2 L 2 (κ F ) p2 F v 2 L (κ F ) m T p 2 p 2d F K j v 2 L 2 (K j ) j=1 m T p 2 p 2d F K j (4 v 2 L 2 (ˆκ j ) ) p2d+2 F κ j=1 v 2 L 2 (κ) A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

26 The final trace estimate Lemma If κ T then for each v P p (κ) we have the inverse estimate v 2 L 2 (F ) C INV(p, κ, F ) p2 F v 2 L κ 2 (κ), { } κ with C INV (p, κ, F ) := min p2d. sup κ F κ κf, A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

27 Mesh covering A covering T = {K} is a set of shape-regular d simplexes K, such that for each κ T, there exists a K T such that κ K. The covering domain is given by Ω := ( K T K ). Note that Ω Ω. Given v H s (Ω), s N 0, there exists an extension Ev H s (R d ): Ev H s (R d ) C v H s (Ω), where C depends only on s and Ω [Stain 1970]. Used to extend the exact solution globally from Ω to Ω. Assumption 2 There exists a covering T of T and O Ω N such that each K T has non-empty intersection with less than O Ω elements κ T. diam(k) C diam h κ κ T ; K T with κ K A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

28 Mesh covering A covering T = {K} is a set of shape-regular d simplexes K, such that for each κ T, there exists a K T such that κ K. The covering domain is given by Ω := ( K T K ). Note that Ω Ω. Given v H s (Ω), s N 0, there exists an extension Ev H s (R d ): Ev H s (R d ) C v H s (Ω), where C depends only on s and Ω [Stain 1970]. Used to extend the exact solution globally from Ω to Ω. Assumption 2 There exists a covering T of T and O Ω N such that each K T has non-empty intersection with less than O Ω elements κ T. diam(k) C diam h κ κ T ; K T with κ K A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

29 Mesh covering A covering T = {K} is a set of shape-regular d simplexes K, such that for each κ T, there exists a K T such that κ K. The covering domain is given by Ω := ( K T K ). Note that Ω Ω. Given v H s (Ω), s N 0, there exists an extension Ev H s (R d ): Ev H s (R d ) C v H s (Ω), where C depends only on s and Ω [Stain 1970]. Used to extend the exact solution globally from Ω to Ω. Assumption 2 There exists a covering T of T and O Ω N such that each K T has non-empty intersection with less than O Ω elements κ T. diam(k) C diam h κ κ T ; K T with κ K A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

30 Mesh covering A covering T = {K} is a set of shape-regular d simplexes K, such that for each κ T, there exists a K T such that κ K. The covering domain is given by Ω := ( K T K ). Note that Ω Ω. Given v H s (Ω), s N 0, there exists an extension Ev H s (R d ): Ev H s (R d ) C v H s (Ω), where C depends only on s and Ω [Stain 1970]. Used to extend the exact solution globally from Ω to Ω. Assumption 2 There exists a covering T of T and O Ω N such that each K T has non-empty intersection with less than O Ω elements κ T. diam(k) C diam h κ κ T ; K T with κ K A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

31 hp-approximation For v L 2 (Ω) we define the polygonal hp-approximant Πv S p T Πv κ := Π pκ (Ev K ) κ κ T ; K T with κ K where Π p : L 2 (K) P p (K) standard optimal hp-approximant. by Lemma If v L 2 (Ω) is such that Ev K H kκ (K), for some k 0, then v Πv H q (κ) hsκ q κ pκ kκ q Ev H kκ(k), k κ 0, for 0 q k κ. Here, s κ = min{p κ + 1, k κ }. Further, v Πv L 2 (F ) F where 1/2 hsκ d/2 κ pκ kκ 1/2 C m (p κ, κ, F ) = min { C m (p κ, κ, F ) 1/2 Ev H kκ(k), hκ d 1 sup κ F κ κf, pκ 1 d }. k κ > d/2, A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

32 hp-approximation For v L 2 (Ω) we define the polygonal hp-approximant Πv S p T Πv κ := Π pκ (Ev K ) κ κ T ; K T with κ K where Π p : L 2 (K) P p (K) standard optimal hp-approximant. by Lemma If v L 2 (Ω) is such that Ev K H kκ (K), for some k 0, then v Πv H q (κ) hsκ q κ pκ kκ q Ev H kκ(k), k κ 0, for 0 q k κ. Here, s κ = min{p κ + 1, k κ }. Further, v Πv L 2 (F ) F where 1/2 hsκ d/2 κ pκ kκ 1/2 C m (p κ, κ, F ) = min { C m (p κ, κ, F ) 1/2 Ev H kκ(k), hκ d 1 sup κ F κ κf, pκ 1 d }. k κ > d/2, A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

33 hp-approximation For v L 2 (Ω) we define the polygonal hp-approximant Πv S p T Πv κ := Π pκ (Ev K ) κ κ T ; K T with κ K where Π p : L 2 (K) P p (K) standard optimal hp-approximant. by Lemma If v L 2 (Ω) is such that Ev K H kκ (K), for some k 0, then v Πv H q (κ) hsκ q κ pκ kκ q Ev H kκ(k), k κ 0, for 0 q k κ. Here, s κ = min{p κ + 1, k κ }. Further, v Πv L 2 (F ) F where 1/2 hsκ d/2 κ pκ kκ 1/2 C m (p κ, κ, F ) = min { C m (p κ, κ, F ) 1/2 Ev H kκ(k), hκ d 1 sup κ F κ κf, pκ 1 d }. k κ > d/2, A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

34 Poisson model Problem u = f in Ω R 2,3, u = g D on Γ D, n u = g N on Γ N, where Ω = Γ D Γ N and f L 2 (Ω). Ω Ω Greater generality in the model problem is by all means possible. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

35 Symmetric Interior Penalty dg method Find u h S p T such that B(u h, v h ) = l(v h ) for all v h S p T B(w, v)= w v dx ( { h w } [[v]] + { h v } [[w]] σ[[w]] [[v]]) ds κ T κ Γ\Γ N l(v) = fv dx g D ( h v n σv) ds + g N v ds Ω Γ D Γ N The jump penalisation parameter σ : Γ\Γ N R + is defined facewise by C σ C INV (p κ, κ, F ) p2 F, x F Γ D, F = κ Γ D, κ σ(x) := { C σ max C INV (p κ, κ, F ) p2 κ F }, x F Γ int, F = κ + κ κ {κ +,κ } κ with C σ > 0 large enough. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

36 Convergence of IP-DG on polygonal/polyhedral meshes with possibly arbitrarily small interfaces Continuity and coercivity can be proven with respect to the DG norm: ( ) 1/2, w := w 2 dx + σ [[w]] 2 ds κ Γ\Γ N κ T hence yielding existence, uniqueness, and the following a priori error bound. Theorem (C.-Georgoulis-Houston, M3AS, 2014) Suppose that for each κ T, Eu K H kκ (K), where K T with κ K. Then, u u h 2 C κ T hκ 2(sκ 1) pκ 2(kκ 1) (1 + G κ ( κ, c m, C INV, p κ )) Eu 2 H kκ (K), For uniform order p and h = max κt h κ, if the skeleton Γ is shape regular the theorem generalises known optimal bound O(h s 1 /p k 3/2 ). A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

37 Convergence of IP-DG on polygonal/polyhedral meshes with possibly arbitrarily small interfaces Continuity and coercivity can be proven with respect to the DG norm: ( ) 1/2, w := w 2 dx + σ [[w]] 2 ds κ Γ\Γ N κ T hence yielding existence, uniqueness, and the following a priori error bound. Theorem (C.-Georgoulis-Houston, M3AS, 2014) Suppose that for each κ T, Eu K H kκ (K), where K T with κ K. Then, u u h 2 C κ T hκ 2(sκ 1) pκ 2(kκ 1) (1 + G κ ( κ, c m, C INV, p κ )) Eu 2 H kκ (K), For uniform order p and h = max κt h κ, if the skeleton Γ is shape regular the theorem generalises known optimal bound O(h s 1 /p k 3/2 ). A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

38 Convergence of IP-DG on polygonal/polyhedral meshes with possibly arbitrarily small interfaces Continuity and coercivity can be proven with respect to the DG norm: ( ) 1/2, w := w 2 dx + σ [[w]] 2 ds κ Γ\Γ N κ T hence yielding existence, uniqueness, and the following a priori error bound. Theorem (C.-Georgoulis-Houston, M3AS, 2014) Suppose that for each κ T, Eu K H kκ (K), where K T with κ K. Then, u u h 2 C κ T hκ 2(sκ 1) pκ 2(kκ 1) (1 + G κ ( κ, c m, C INV, p κ )) Eu 2 H kκ (K), For uniform order p and h = max κt h κ, if the skeleton Γ is shape regular the theorem generalises known optimal bound O(h s 1 /p k 3/2 ). A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

39 Implementation of the DG space S p T on polygons/polyhedra On each κ T, the local space is defined as follows: Define the Cartesian bounding box B κ of each element κ; apple B apple construct P pκ (B κ ) spanned by, eg. tensor-product Legendre polynomials; restrict such space to κ. Thursday, 4 July 13 Quadratures over κ are performed constructing a simplicial non-overlapping sub-triangulation. (Cf. talk by S. Giani on Composite DG) Other approaches: construct the physical local basis by Gram-Schmidt: Bassi et al, J. Comput. Phys., A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

40 Implementation of the DG space S p T on polygons/polyhedra On each κ T, the local space is defined as follows: Define the Cartesian bounding box B κ of each element κ; apple B apple construct P pκ (B κ ) spanned by, eg. tensor-product Legendre polynomials; restrict such space to κ. Thursday, 4 July 13 Quadratures over κ are performed constructing a simplicial non-overlapping sub-triangulation. (Cf. talk by S. Giani on Composite DG) Other approaches: construct the physical local basis by Gram-Schmidt: Bassi et al, J. Comput. Phys., A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

41 Implementation of the DG space S p T on polygons/polyhedra On each κ T, the local space is defined as follows: Define the Cartesian bounding box B κ of each element κ; apple B apple construct P pκ (B κ ) spanned by, eg. tensor-product Legendre polynomials; restrict such space to κ. Thursday, 4 July 13 Quadratures over κ are performed constructing a simplicial non-overlapping sub-triangulation. (Cf. talk by S. Giani on Composite DG) Other approaches: construct the physical local basis by Gram-Schmidt: Bassi et al, J. Comput. Phys., A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

42 Numerical example u = sin(πx) sin(πy) Polygonal Elements I Polygonal Elements II 2 2 Generated by PolyMesher [Talischi-Paulino-Pereira, Struct. Multidiscip. Optim., 2012] A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

43 Convergence results h-convergence p-convergence 10 0 Square Elements Polygonal Elements I Polygonal Elements II p= Square Elements Polygonal Elements I Polygonal Elements II u u h p=2 p=3 u u h Elements 64 Elements 256 Elements Elements dof 1/2 p= Elements p A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

44 Linear transport We consider the problem L 0 u b u + cu = f in Ω, u = g on Ω. where Ω = {x Ω, b(x) n(x) < 0}. and n denotes the unit outward normal to Ω. We assume that b [W 1 (Ω)] d, c L (Ω), f L 2 (Ω), and g L 2 ( Ω ) and the existence of a positive constant γ 0 such that c 0 (x) 2 := c(x) 1 2 b(x) γ 0 for almost every x Ω. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

45 Interior Penalty dg method Find u h S p T such that B(u h, v h ) = l(v h ) for all v h S p T. B(u, v) l(v) = 0 u)v dx κ T κ(l (b n)uv + ds κ T κ\ Ω (b n)u + v + ds, κ T κ Ω = f v dx (b n)gv + ds, κ T κ κ T κ Ω A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

46 DG norms DG norm: v 2 DG := ( κ T κ c 0 v 2 dx Γ b n v 2 ds) A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

47 DG norms DG norm: v 2 DG := ( κ T κ c 0 v 2 dx For triangular and quadrilateral meshes we have Optimal hp-bounds if b h v h S p T ; p-suboptimal by p 3/2 a priori bounds in general. Houston, Schwab & Süli, SINUM, Γ b n v 2 ds) A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

48 DG norms DG norm: v 2 DG := ( κ T κ c 0 v 2 dx Γ b n v 2 ds) We shall follow the idea of proving (inf-sup) stability for a stronger norm including an SUPG-type term: Streamline DG norm: v 2 s := v 2 DG + κ T τ κ b h v 2 κ, Johnson & Pitkäranta, Math. Comp, Ayuso & Marini, SINUM, C., Chapman, Georgoulis & Jensen, JCP, A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

49 Inf-sup condition On S p T, define the linear operator T (ν) = ν + α τ κ (b h ν) κ T with α > 0 at out disposal and set W p T Then we can show the inf-sup condition. Lemma Let τ κ := h κ /(p 2 κ b L (κ)). Then, for α min( 2γ 0 1 c, L (Ω) inf ν S p T 2C κ, ( 1 sup µ W p T ) 1 2 C inv ) to be the graph space of T. B(ν, µ) ν s µ s Λ s A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

50 Inf-sup condition On S p T, define the linear operator T (ν) = ν + α τ κ (b h ν) κ T with α > 0 at out disposal and set W p T Then we can show the inf-sup condition. Lemma Let τ κ := h κ /(p 2 κ b L (κ)). Then, for α min( 2γ 0 1 c, L (Ω) inf ν S p T 2C κ, ( 1 sup µ W p T ) 1 2 C inv ) to be the graph space of T. B(ν, µ) ν s µ s Λ s A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

51 A priori error bound Theorem Suppose that for each κ T Eu K H kκ (K), where K T with κ K. Then, u u h 2 s κ T hκ 2sκ 1 ( pκ 2kκ 1 p κ b L (κ)+ F hκ d 1 for s κ = min{p κ + 1, k κ } and k κ > 1 + d/2. F κ ) C m (p κ, κ, F ) Eu 2 H kκ (K) For uniform order p and h = max κt h κ, if the skeleton Γ is shape regular the theorem gives p-suboptimal by only p 1/2 bound of O(h s 1/2 /p k 1 ). A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

52 A priori error bound Theorem Suppose that for each κ T Eu K H kκ (K), where K T with κ K. Then, u u h 2 s κ T hκ 2sκ 1 ( pκ 2kκ 1 p κ b L (κ)+ F hκ d 1 for s κ = min{p κ + 1, k κ } and k κ > 1 + d/2. F κ ) C m (p κ, κ, F ) Eu 2 H kκ (K) For uniform order p and h = max κt h κ, if the skeleton Γ is shape regular the theorem gives p-suboptimal by only p 1/2 bound of O(h s 1/2 /p k 1 ). A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

53 Example We consider the PDE problem: x 2 u yy + u x + u = 0 for 1 x 1, y > 0; u x + u = 0 for 1 x 1, y 0 which has analytic solution: {sin( 1 u(x, y) = 2 π(1 + y)) exp( (x + π2 x 3 12 )) for 1 x 1, y > 0; sin( 1 2π(1 + y)) exp( x) for 1 x 1, y 0, This problem is hyperbolic in the area y 0 and is parabolic in the area y > 0. A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

54 Polygonal mesh conforming with y = modified polygonal mesh A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28

55 Convergence results u u h DG , 64, 256 elements under p refinement 16 Polygon DGFEM(P) 16 Rectangle DGFEM(P) 16 Rectangle DGFEM(Q) 64 Polygon DGFEM(P) 64 Rectangle DGFEM(P) 64 Rectangle DGFEM(Q) 256 Polygon DGFEM(P) 256 Rectangle DGFEM(P) 256 Rectangle DGFEM(Q) Dof 1/2 A. Cangiani (University of Leicester) hp-dg on polytopes ReaDG / 28