Graded Mesh Refinement and Error Estimates of Higher Order for DGFE-solutions of Elliptic Boundary Value Problems in Polygons

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1 Graded Mesh Refinement and Error Estimates of Higher Order for DGFE-solutions of Elliptic Boundary Value Problems in Polygons Anna-Margarete Sändig, Miloslav Feistauer University Stuttgart, IANS Journées Singulières Augmentes en l honneur de Martin Costabel University of Rennes, August 26 to 30, 2013 Anna-Margarete Sändig Graded Mesh Refinement 1/19

2 1 Continuous problem 2 Discontinuous Galerkin Discretization 3 Error Estimates Anna-Margarete Sändig Graded Mesh Refinement 2/19

3 Preliminaries Finite volume method used in computational fluid dynamics, piecewise constant (and hence, discontinuous) approximations. Anna-Margarete Sändig Graded Mesh Refinement 3/19

4 Preliminaries Finite volume method used in computational fluid dynamics, piecewise constant (and hence, discontinuous) approximations. Conforming Galerkin finite element techniques used in structure mechanics, continuous piecewise polynomial approximations on a finite element mesh. Anna-Margarete Sändig Graded Mesh Refinement 3/19

5 Preliminaries Finite volume method used in computational fluid dynamics, piecewise constant (and hence, discontinuous) approximations. Conforming Galerkin finite element techniques used in structure mechanics, continuous piecewise polynomial approximations on a finite element mesh. Discontinuous Galerkin finite element method (DGFEM), piecewise polynomial approximations on a finite element mesh without any requirement on the continuity between neighbouring elements. Anna-Margarete Sändig Graded Mesh Refinement 3/19

6 Preliminaries Finite volume method used in computational fluid dynamics, piecewise constant (and hence, discontinuous) approximations. Conforming Galerkin finite element techniques used in structure mechanics, continuous piecewise polynomial approximations on a finite element mesh. Discontinuous Galerkin finite element method (DGFEM), piecewise polynomial approximations on a finite element mesh without any requirement on the continuity between neighbouring elements. Very flexible schemes. Anna-Margarete Sändig Graded Mesh Refinement 3/19

7 Preliminaries Finite volume method used in computational fluid dynamics, piecewise constant (and hence, discontinuous) approximations. Conforming Galerkin finite element techniques used in structure mechanics, continuous piecewise polynomial approximations on a finite element mesh. Discontinuous Galerkin finite element method (DGFEM), piecewise polynomial approximations on a finite element mesh without any requirement on the continuity between neighbouring elements. Very flexible schemes. Error estimates for DGFE-solutions are well investigated if one assumes that the exact solution is sufficiently regular. Some papers (Wihler 02, Brenner 09) about less regular solutions. Here, polygons, mixed boundary conditions, weighted spaces, Sobolev-Slobodetskii spaces. Anna-Margarete Sändig Graded Mesh Refinement 3/19

8 01 Linear elliptic boundary value problem P 3 Ω R 2 polygon with boundary Ω = Γ D Γ N, where Γ D Γ N =, meas{γ D } > 0, n = (n 1, n 2 ) unit outward normal to Ω. Γ N Γ 01 P 2 P 1 r D Ω ω Γ N P 4 Γ D P 5 Anna-Margarete Sändig Graded Mesh Refinement 4/19

9 01 Linear elliptic boundary value problem P 3 Ω R 2 polygon with boundary Ω = Γ D Γ N, where Γ D Γ N =, meas{γ D } > 0, n = (n 1, n 2 ) unit outward normal to Ω. Γ N Γ 01 P 2 P 1 r D Ω ω Γ N P 4 Γ D P 5 Lu = L(x, D x )u = 2 i,j=1 x i B 2 (x, D x )u = ( a ij (x) u ) + c(x)u = f in Ω, x j B 1 (x, D x )u = u = 0 on Γ D, 2 a ij (x) u n i = q x j on Γ N. i,j=1 Anna-Margarete Sändig Graded Mesh Refinement 4/19

10 Weak formulation Find u V = {v H 1 (Ω) : v ΓD = 0} such that Anna-Margarete Sändig Graded Mesh Refinement 5/19

11 Weak formulation Find u V = {v H 1 (Ω) : v ΓD = 0} such that a(u, v) = (f, v) + (q, v) ΓN v V, Anna-Margarete Sändig Graded Mesh Refinement 5/19

12 Weak formulation Find u V = {v H 1 (Ω) : v ΓD = 0} such that a(u, v) = (f, v) + (q, v) ΓN v V, a(u, v) = (f, v) = Ω Ω 2 u v a ij + cuv dx, x j x i i,j=1 fv dx, (q, v) ΓN = qv dv, Γ N Anna-Margarete Sändig Graded Mesh Refinement 5/19

13 Weak formulation Find u V = {v H 1 (Ω) : v ΓD = 0} such that a(u, v) = (f, v) + (q, v) ΓN v V, a(u, v) = (f, v) = Ω Ω 2 u v a ij + cuv dx, x j x i i,j=1 fv dx, (q, v) ΓN = qv dv, Γ N a ij, c, f, q sufficiently smooth. Assume, Lax-Milgram lemma holds and therefore, an unique weak solution u V exists. Anna-Margarete Sändig Graded Mesh Refinement 5/19

14 Regularity To which Sobolev-Slobodetskii spaces or weighted Sobolevs spaces belongs the weak solution? Anna-Margarete Sändig Graded Mesh Refinement 6/19

15 Regularity To which Sobolev-Slobodetskii spaces or weighted Sobolevs spaces belongs the weak solution? Sobolev-Slobodetskii spaces H k+γ (Ω),γ (0, 1), is defined as the subspace of H k (Ω) formed by all functions v for which the seminorm is finite, that means Anna-Margarete Sändig Graded Mesh Refinement 6/19

16 Regularity To which Sobolev-Slobodetskii spaces or weighted Sobolevs spaces belongs the weak solution? Sobolev-Slobodetskii spaces H k+γ (Ω),γ (0, 1), is defined as the subspace of H k (Ω) formed by all functions v for which the seminorm is finite, that means v H k+γ (Ω) = α =k Ω Ω D α v(x) D α v(y) 2 x y 2+2γ dxdy 1/2 < +. Anna-Margarete Sändig Graded Mesh Refinement 6/19

17 Regularity To which Sobolev-Slobodetskii spaces or weighted Sobolevs spaces belongs the weak solution? Sobolev-Slobodetskii spaces H k+γ (Ω),γ (0, 1), is defined as the subspace of H k (Ω) formed by all functions v for which the seminorm is finite, that means v H k+γ (Ω) = α =k The norm is defined as Ω Ω v H k+γ (Ω) = D α v(x) D α v(y) 2 x y 2+2γ dxdy ( v 2 H k (Ω) + v 2 H k+γ (Ω)) 1/2. 1/2 < +. Anna-Margarete Sändig Graded Mesh Refinement 6/19

18 Weighted Sobolev spaces Let be M = {P i } i=1,...,i the set of irregular points, β = (β 1, β 2,..., β I ), r = r(x) = dist(x, M), r i = x P i and C M (Ω) := {u C (Ω) : supp u M = }. Anna-Margarete Sändig Graded Mesh Refinement 7/19

19 Weighted Sobolev spaces Let be M = {P i } i=1,...,i the set of irregular points, β = (β 1, β 2,..., β I ), r = r(x) = dist(x, M), r i = x P i and C M (Ω) := {u C (Ω) : supp u M = }. The space V k,p β (Ω), 1 p <, is the closure of the set C M (Ω) with respect to the norm 1/p u V k,p β (Ω) = Ω i=1,...,i r pβ i i α k r p( k+ α ) D α u p dx. Anna-Margarete Sändig Graded Mesh Refinement 7/19

20 Weighted Sobolev spaces Let be M = {P i } i=1,...,i the set of irregular points, β = (β 1, β 2,..., β I ), r = r(x) = dist(x, M), r i = x P i and C M (Ω) := {u C (Ω) : supp u M = }. The space V k,p β (Ω), 1 p <, is the closure of the set C M (Ω) with respect to the norm 1/p u V k,p β (Ω) = Ω i=1,...,i r pβ i i α k r p( k+ α ) D α u p dx. In U(P i ) the norm is equivalent to ( D α u p dx U(Pi ) α k r p(β i k+ α ) i ) 1/p Anna-Margarete Sändig Graded Mesh Refinement 7/19

21 Weighted Sobolev spaces Let be M = {P i } i=1,...,i the set of irregular points, β = (β 1, β 2,..., β I ), r = r(x) = dist(x, M), r i = x P i and C M (Ω) := {u C (Ω) : supp u M = }. The space V k,p β (Ω), 1 p <, is the closure of the set C M (Ω) with respect to the norm 1/p u V k,p β (Ω) = Ω i=1,...,i r pβ i i α k r p( k+ α ) D α u p dx. In U(P i ) the norm is equivalent to ( D α u p dx U(Pi ) α k r p(β i k+ α ) i ) 1/p If β 1 = β 2 =... = β I = β, then we write shortly V k,p β (Ω) with ( u k,p = 1/p V β (Ω) Ω α k r p(β k+ α ) D α u dx) p. Anna-Margarete Sändig Graded Mesh Refinement 7/19

22 Weighted Sobolev spaces Let be M = {P i } i=1,...,i the set of irregular points, β = (β 1, β 2,..., β I ), r = r(x) = dist(x, M), r i = x P i and C M (Ω) := {u C (Ω) : supp u M = }. The space V k,p β (Ω), 1 p <, is the closure of the set C M (Ω) with respect to the norm 1/p u V k,p β (Ω) = Ω i=1,...,i r pβ i i α k r p( k+ α ) D α u p dx. In U(P i ) the norm is equivalent to ( D α u p dx U(Pi ) α k r p(β i k+ α ) i ) 1/p If β 1 = β 2 =... = β I = β, then we write shortly V k,p β (Ω) with ( u k,p = 1/p V β (Ω) Ω α k r p(β k+ α ) D α u dx) p. Babushka/Guo, Wihler have used weights for higher derivatives only. Anna-Margarete Sändig Graded Mesh Refinement 7/19

23 It is well known, that the distribution of the eigenvalues of a corresponding generalized eigenvalue problem (principal parts with frozen coefficients, localization, Mellin transform (r i r i α)) characterizes the regularity in each irregular boundary point. Anna-Margarete Sändig Graded Mesh Refinement 8/19

24 It is well known, that the distribution of the eigenvalues of a corresponding generalized eigenvalue problem (principal parts with frozen coefficients, localization, Mellin transform (r i r i α)) characterizes the regularity in each irregular boundary point. For an irregular point P i we consider an eigenvalue α0 i of the Mellin transformed eigenvalue problem with maximal real part, such that the strip 0 < R(α) < R(α0 i ) Anna-Margarete Sändig Graded Mesh Refinement 8/19

25 It is well known, that the distribution of the eigenvalues of a corresponding generalized eigenvalue problem (principal parts with frozen coefficients, localization, Mellin transform (r i r i α)) characterizes the regularity in each irregular boundary point. For an irregular point P i we consider an eigenvalue α0 i of the Mellin transformed eigenvalue problem with maximal real part, such that the strip 0 < R(α) < R(α0 i )is free of eigenvalues. We set H0 i = R(αi 0 ), H0 = {H0 1,..., HI 0 }. Anna-Margarete Sändig Graded Mesh Refinement 8/19

26 It is well known, that the distribution of the eigenvalues of a corresponding generalized eigenvalue problem (principal parts with frozen coefficients, localization, Mellin transform (r i r i α)) characterizes the regularity in each irregular boundary point. For an irregular point P i we consider an eigenvalue α0 i of the Mellin transformed eigenvalue problem with maximal real part, such that the strip 0 < R(α) < R(α0 i )is free of eigenvalues. We set H0 i = R(αi 0 ), H0 = {H0 1,..., HI 0 }. By some calculations (linear principal axes transformation, anisotropic to isotropic Laplacian, back transformation) we get H0 i > 1 4 for 0 < ωi 0 < 2π, Hi 0 > 1 2 for ωi 0 < π, Hi 0 = 1 2 for ω0 i = π. More precisely, H0 i = π for D-D conditions. λ i 1 tan ω0 i ) arctan( λ i 2 Anna-Margarete Sändig Graded Mesh Refinement 8/19

27 ondratjev 1967 Theorem Assume, f V k,2 1 k+ 1 H 2 (Ω) and q V,2 0 +ε+k 1 H (Γ 0 N), where ε > 0 +ε+k is an arbitrarily small real number, k is an integer. Then the weak solution u V is contained in V 2+k,2 1 H (Ω). 0 +ε+k Anna-Margarete Sändig Graded Mesh Refinement 9/19

28 ondratjev 1967 Theorem Assume, f V k,2 1 k+ 1 H 2 (Ω) and q V,2 0 +ε+k 1 H (Γ 0 N), where ε > 0 +ε+k is an arbitrarily small real number, k is an integer. Then the weak solution u V is contained in V 2+k,2 1 H (Ω). 0 +ε+k Two possibilities for simplification with respect to the number of irregular boundary points: We use a localization and consider different graded mesh refinements separately for individual irregular points. We set H 0 = min{h0 i, i = 1,, I } and consider the same (finest) graded mesh refinement for every singular point. Then, u V 2+k,2 1 H 0 +ε+k (Ω) and, furthermore u H1+H 0 ε (Ω). Anna-Margarete Sändig Graded Mesh Refinement 9/19

29 Discontinuous Galerkin Discretization {T h } h (0,h0 ) family of partitions with standard properties, h mesh parameter, F h = F D h F I h subset of faces of all elements T h. Anna-Margarete Sändig Graded Mesh Refinement 10/19

30 Discontinuous Galerkin Discretization {T h } h (0,h0 ) family of partitions with standard properties, h mesh parameter, F h = F D h F I h subset of faces of all elements T h. Consider broken Sobolev-Slobodetskii spaces H s (Ω, T h ) = {v; v H s () T h } equipped with the seminorm v H s (Ω,T h ) = ( T h v 2 H s ()) 1/2, and broken weighted spaces V k,2 β (Ω, T h). Anna-Margarete Sändig Graded Mesh Refinement 10/19

31 Discontinuous Galerkin Discretization {T h } h (0,h0 ) family of partitions with standard properties, h mesh parameter, F h = F D h F I h subset of faces of all elements T h. Consider broken Sobolev-Slobodetskii spaces H s (Ω, T h ) = {v; v H s () T h } equipped with the seminorm v H s (Ω,T h ) = ( T h v 2 H s ()) 1/2, and broken weighted spaces V k,2 β (Ω, T h). Using Greens formula we get A h (u, v) = l h (v) defined on X X, Anna-Margarete Sändig Graded Mesh Refinement 10/19

32 Discontinuous Galerkin Discretization {T h } h (0,h0 ) family of partitions with standard properties, h mesh parameter, F h = F D h F I h subset of faces of all elements T h. Consider broken Sobolev-Slobodetskii spaces H s (Ω, T h ) = {v; v H s () T h } equipped with the seminorm v H s (Ω,T h ) = ( T h v 2 H s ()) 1/2, and broken weighted spaces V k,2 β (Ω, T h). Using Greens formula we get A h (u, v) = l h (v) defined on X X, where X = H 1+γ (Ω, T h ) with γ = H 0 ε ( 1, 1) or 2 Anna-Margarete Sändig Graded Mesh Refinement 10/19

33 Discontinuous Galerkin Discretization {T h } h (0,h0 ) family of partitions with standard properties, h mesh parameter, F h = F D h F I h subset of faces of all elements T h. Consider broken Sobolev-Slobodetskii spaces H s (Ω, T h ) = {v; v H s () T h } equipped with the seminorm v H s (Ω,T h ) = ( T h v 2 H s ()) 1/2, and broken weighted spaces V k,2 β (Ω, T h). Using Greens formula we get A h (u, v) = l h (v) defined on X X, where X = H 1+γ (Ω, T h ) with γ = H 0 ε ( 1, 1) or 2 X = V 2,2 β+1 (Ω, T h) with β = H 0 + ε ( 1, 0). Anna-Margarete Sändig Graded Mesh Refinement 10/19

34 Discontinuous Galerkin Discretization {T h } h (0,h0 ) family of partitions with standard properties, h mesh parameter, F h = F D h F I h subset of faces of all elements T h. Consider broken Sobolev-Slobodetskii spaces H s (Ω, T h ) = {v; v H s () T h } equipped with the seminorm v H s (Ω,T h ) = ( T h v 2 H s ()) 1/2, and broken weighted spaces V k,2 β (Ω, T h). Using Greens formula we get A h (u, v) = l h (v) defined on X X, where X = H 1+γ (Ω, T h ) with γ = H 0 ε ( 1, 1) or 2 X = V 2,2 β+1 (Ω, T h) with β = H 0 + ε ( 1, 0). A h (u, v) = a h (u, v) + µj h (u, v) l h (v) = fv dx + qv ds. Ω Γ N Anna-Margarete Sändig Graded Mesh Refinement 10/19

35 a h (w, v) = 2 w v a ij + cwv dx T x h i,j=1 j x i 2 w 2 a ij n Γ v i Γ F h I Γ x [v] + θ a ij n Γ i i,j=1 j x [w] ds i,j=1 j 2 w 2 a ij n Γ i Γ F h D Γ x v + θ v a ij n Γ i i,j=1 j x w ds, i,j=1 j Anna-Margarete Sändig Graded Mesh Refinement 11/19

36 a h (w, v) = 2 w v a ij + cwv dx T x h i,j=1 j x i 2 w 2 a ij n Γ v i Γ F h I Γ x [v] + θ a ij n Γ i i,j=1 j x [w] ds i,j=1 j 2 w 2 a ij n Γ i Γ F h D Γ x v + θ v a ij n Γ i i,j=1 j x w ds, i,j=1 j J h (w, v) = κ(γ)[w] [v] ds + κ(γ)wv ds, Γ F h I Γ Γ F h D Γ Anna-Margarete Sändig Graded Mesh Refinement 11/19

37 a h (w, v) = 2 w v a ij + cwv dx T x h i,j=1 j x i 2 w 2 a ij n Γ v i Γ F h I Γ x [v] + θ a ij n Γ i i,j=1 j x [w] ds i,j=1 j 2 w 2 a ij n Γ i Γ F h D Γ x v + θ v a ij n Γ i i,j=1 j x w ds, i,j=1 j J h (w, v) = κ(γ)[w] [v] ds + κ(γ)wv ds, Γ F h I Γ Γ F h D Γ ( J h are penalty terms, θ = 1(NIPG), θ = 0(IIPG), θ = 1(SIPG), v = 1 v (L) + v (R)), [v] = v (L) v (R). 2 Anna-Margarete Sändig Graded Mesh Refinement 11/19

38 a h (w, v) = 2 w v a ij + cwv dx T x h i,j=1 j x i 2 w 2 a ij n Γ v i Γ F h I Γ x [v] + θ a ij n Γ i i,j=1 j x [w] ds i,j=1 j 2 w 2 a ij n Γ i Γ F h D Γ x v + θ v a ij n Γ i i,j=1 j x w ds, i,j=1 j J h (w, v) = κ(γ)[w] [v] ds + κ(γ)wv ds, Γ F h I Γ Γ F h D Γ ( J h are penalty terms, θ = 1(NIPG), θ = 0(IIPG), θ = 1(SIPG), v = 1 v (L) + v (R)), [v] = v (L) v (R). 2 The approximate solution u h will be sought in the space of discontinuous piecewise polynomial functions of fixed degree p, S hp = {v L 2 (Ω); v P p (), T h }, such that A h (u h, v h ) = l h (v h ) v h S hp. Anna-Margarete Sändig Graded Mesh Refinement 11/19

39 Error estimates on regular meshes The error e h = u h u of the DG method satisfies the Galerkin orthogonality condition A h (e h, v h ) = 0 v h S hp. Anna-Margarete Sändig Graded Mesh Refinement 12/19

40 Error estimates on regular meshes The error e h = u h u of the DG method satisfies the Galerkin orthogonality condition A h (e h, v h ) = 0 v h S hp. We introduce the DG-norm in H 1 (Ω, T h ) ( w DG = w 2 H 1 (Ω,T h ) + J h(w, w) ) 1/2 Anna-Margarete Sändig Graded Mesh Refinement 12/19

41 Error estimates on regular meshes The error e h = u h u of the DG method satisfies the Galerkin orthogonality condition A h (e h, v h ) = 0 v h S hp. We introduce the DG-norm in H 1 (Ω, T h ) ( w DG = w 2 H 1 (Ω,T h ) + J h(w, w) ) 1/2 Theorem Let be the exact solution u H 1+γ (Ω), γ ( 1 2, 1) and u H 1+ν ( Ω), where Ω Ω with dist( Ω, M) > 0, ν = min(p, κ), κ is an integer, which depends on the smoothness of the loads. Then e h DG C ( h 2ν u 2 H ν+1 ( Ω) + h2γ u 2 H 1+γ (Ω)) 1/2. Anna-Margarete Sändig Graded Mesh Refinement 12/19

42 Error estimates on graded meshes Graded partitions Ω polygon with the set of irregular points M, h mesh parameter, {T h } h (0,h0 ) family of partitions. Anna-Margarete Sändig Graded Mesh Refinement 13/19

43 Error estimates on graded meshes Graded partitions Ω polygon with the set of irregular points M, h mesh parameter, {T h } h (0,h0 ) family of partitions. For T h we set h = diam(), h = max Th h, ρ is the radius of the largest circle inscribed into, r = dist(, M). Anna-Margarete Sändig Graded Mesh Refinement 13/19

44 Error estimates on graded meshes Graded partitions Ω polygon with the set of irregular points M, h mesh parameter, {T h } h (0,h0 ) family of partitions. For T h we set h = diam(), h = max Th h, ρ is the radius of the largest circle inscribed into, r = dist(, M). Furthermore, T h = Th M Th 0 : T h M, if M. Assume (a) {T h } h (0,h0 ) is regular: h / ρ C T h, h Anna-Margarete Sändig Graded Mesh Refinement 13/19

45 Error estimates on graded meshes Graded partitions Ω polygon with the set of irregular points M, h mesh parameter, {T h } h (0,h0 ) family of partitions. For T h we set h = diam(), h = max Th h, ρ is the radius of the largest circle inscribed into, r = dist(, M). Furthermore, T h = Th M Th 0 : T h M, if M. Assume (a) {T h } h (0,h0 ) is regular: h / ρ C T h is graded with respect to M : T h, h Anna-Margarete Sändig Graded Mesh Refinement 13/19

46 Error estimates on graded meshes Graded partitions Ω polygon with the set of irregular points M, h mesh parameter, {T h } h (0,h0 ) family of partitions. For T h we set h = diam(), h = max Th h, ρ is the radius of the largest circle inscribed into, r = dist(, M). Furthermore, T h = Th M Th 0 : T h M, if M. Assume (a) {T h } h (0,h0 ) is regular: h / ρ C T h is graded with respect to M : T h, h (b 1 ) if T M h, then C 1h 1/µ h C 1 h 1/µ, (b 2 ) if T 0 h, then C 2hr 1 µ h C 2 hr 1 µ, where µ (0, 1] is the grading parameter. Anna-Margarete Sändig Graded Mesh Refinement 13/19

47 Examples for graded partitions Radial partitions using layers (Oganesyan/Rukhovets 1979) Anna-Margarete Sändig Graded Mesh Refinement 14/19

48 Examples for graded partitions Radial partitions using layers (Oganesyan/Rukhovets 1979) N = 3, µ = 0.4, Apel 1996 Anna-Margarete Sändig Graded Mesh Refinement 14/19

49 Radial partitions using an initial triangulation (Raugel 1978) Anna-Margarete Sändig Graded Mesh Refinement 15/19

50 Radial partitions using an initial triangulation (Raugel 1978) N = 3, µ = 0.4, Apel 1996 Anna-Margarete Sändig Graded Mesh Refinement 15/19

51 Feistauer/Sändig 2012 Theorem Let Ω be a polygon, β = H 0 + ε, γ = H 0 ε ( 1 2, 1). Consider the exact solution u V 2+k,2 β+1+k (Ω) H1+γ (Ω) of the mixed problem and set p = k + 1. Anna-Margarete Sändig Graded Mesh Refinement 16/19

52 Feistauer/Sändig 2012 Theorem Let Ω be a polygon, β = H 0 + ε, γ = H 0 ε ( 1 2, 1). Consider the exact solution u V 2+k,2 β+1+k (Ω) H1+γ (Ω) of the mixed problem and set p = k + 1.Let {T h } h (0,h0 ) be a regular system of triangulations graded towards the set M. Then there exists a constant C > 0 such that Anna-Margarete Sändig Graded Mesh Refinement 16/19

53 Feistauer/Sändig 2012 Theorem Let Ω be a polygon, β = H 0 + ε, γ = H 0 ε ( 1 2, 1). Consider the exact solution u V 2+k,2 β+1+k (Ω) H1+γ (Ω) of the mixed problem and set p = k + 1.Let {T h } h (0,h0 ) be a regular system of triangulations graded towards the set M. Then there exists a constant C > 0 such that u u h DG Ch α ( u 2 V 2+k,2 β+1+k (Ω) + u 2 H 1+γ (Ω) α = with ε > 0 arbitrarily small. { k + 1 for µ < H 0 k+1, H 0 ε µ for µ H 0 k+1, ) 1/2, Anna-Margarete Sändig Graded Mesh Refinement 16/19

54 Main ideas of the proof For η = u I h u it holds ( ( e h DG C η 2 H 1 () + h 2 η 2 H 2 () + h 2 T 0 h + T M h η 2 L 2 () ( η 2 H 1 () + h 2γ η 2 H 1+γ () + h 2 η 2 L 2 ())) 1/2, h (0, h0). ) Anna-Margarete Sändig Graded Mesh Refinement 17/19

55 Main ideas of the proof For η = u I h u it holds ( ( e h DG C η 2 H 1 () + h 2 η 2 H 2 () + h 2 For T 0 h T 0 h + T M h η 2 L 2 () ( η 2 H 1 () + h 2γ η 2 H 1+γ () + h 2 η 2 L 2 ())) 1/2, h (0, h0). we have u H 2+k () and e.g. ) Anna-Margarete Sändig Graded Mesh Refinement 17/19

56 Main ideas of the proof For η = u I h u it holds ( ( e h DG C η 2 H 1 () + h 2 η 2 H 2 () + h 2 For T 0 h T 0 h + T M h η 2 L 2 () ( η 2 H 1 () + h 2γ η 2 H 1+γ () + h 2 η 2 L 2 ())) 1/2, h (0, h0). we have u H 2+k () and e.g. η 2 H 1 () Ch 2(k+1) = Ch 2(k+1) Ch 2(k+1) 2+k u 2 L 2 () r 2(β+1+k) r 2(β+1+k) u 2 V 2+k,2 (). β+1+k r 2(β+1+k) 2+k u 2 dx negative exponent ) Anna-Margarete Sändig Graded Mesh Refinement 17/19

57 Main ideas of the proof For η = u I h u it holds ( ( e h DG C η 2 H 1 () + h 2 η 2 H 2 () + h 2 For T 0 h T 0 h + T M h η 2 L 2 () ( η 2 H 1 () + h 2γ η 2 H 1+γ () + h 2 η 2 L 2 ())) 1/2, h (0, h0). we have u H 2+k () and e.g. η 2 H 1 () Ch 2(k+1) = Ch 2(k+1) Ch 2(k+1) 2+k u 2 L 2 () r 2(β+1+k) r 2(β+1+k) u 2 V 2+k,2 (). β+1+k Exploiting the grading h C 2hr 1 µ r 2(β+1+k) 2+k u 2 dx negative exponent we get h 2(k+1) r 2(β+1+k) Ch 2(k+1) r 2(1 µ)(k+1) 2(β+1+k) Ch 2(k+1) for µ β k+1 = H 0 ε k+1. Anna-Margarete Sändig Graded Mesh Refinement 17/19 )

58 For Th M we have u V 2,2 β+1 () H1+γ () for β = H 0 + ε and e.g. Anna-Margarete Sändig Graded Mesh Refinement 18/19

59 For Th M we have u V 2,2 β+1 () H1+γ () for β = H 0 + ε and e.g. u 2 H 1 () = r 2( β) r 2β u 2 dx r 2β u 2 dx positive exponent h 2( β) h 2( β) u 2 = V 2,2 h2γ β+1 () u 2 V 2,2 (). β+1 Anna-Margarete Sändig Graded Mesh Refinement 18/19

60 For Th M we have u V 2,2 β+1 () H1+γ () for β = H 0 + ε and e.g. u 2 H 1 () = r 2( β) r 2β u 2 dx r 2β u 2 dx positive exponent h 2( β) h 2( β) u 2 = V 2,2 h2γ β+1 () u 2 V 2,2 (). β+1 Using the inverse inequality we get I h u 2 H 1 () Ch 2 π hu L 2 () 2 h 2 u 2 L 2 () h 2( β) u 2 = V 2,2 h2γ β+1 () u 2 V 2,2 (). β+1 Anna-Margarete Sändig Graded Mesh Refinement 18/19

61 Happy birthday, dear Martin! Oberwolfach 2002 Anna-Margarete Sändig Graded Mesh Refinement 19/19