Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems.

Transcription

1 Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems. Stefan Girke WWU Münster Institut für Numerische und Angewandte Mathematik 10th of January, 2011 Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

2 Inhalt 1 introduction (an example) Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

3 Inhalt 1 introduction (an example) 2 the framework Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

4 Inhalt 1 introduction (an example) 2 the framework 3 examples for hybridizable methods Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

5 Inhalt 1 introduction (an example) 2 the framework 3 examples for hybridizable methods 4 Other novel methods hybridizable methods well suited for adaptivity The RT-method on meshes with hanging nodes Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

6 Inhalt literature D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM journal on numerical analysis, 39(5): , B. Cockburn and J. Gopalakrishnan. A characterization of hybridized mixed methods for second order elliptic problems. SIAM Journal on Numerical Analysis, 42(1): , B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal, 47(2): , R.M. Kirby, S.J. Sherwin, and B. Cockburn. To CG or to HDG: A Comparative Study. M. Vohraĺık, J. Maryska, and O. Severýn. Mixed and nonconforming finite element methods on a system of polygons. Applied Numerical Mathematics, 57(2): , OC Zienkiewicz. Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages of Stress Analysis, Edited by OC Zienkiewicz and GS Holister, Published by John Wiley & Sons, International Journal for Numerical Methods in Engineering, 52(3): , Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

7 introduction (an example) 1 introduction (an example) 2 the framework 3 examples for hybridizable methods 4 Other novel methods hybridizable methods well suited for adaptivity The RT-method on meshes with hanging nodes Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

8 introduction (an example) Second order elliptic boundary value problem Search solution u for (a u)+du = f in Ω Ê n, u = g on Ω, where a(x) is a bounded symmetric positive definite matrix-valued function, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

9 introduction (an example) Second order elliptic boundary value problem Search solution u for (a u)+du = f in Ω Ê n, u = g on Ω, where a(x) is a bounded symmetric positive definite matrix-valued function, d(x) is a bounded nonnegative function. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

10 introduction (an example) Second order elliptic boundary value problem Search solution u for (a u)+du = f in Ω Ê n, u = g on Ω, where a(x) is a bounded symmetric positive definite matrix-valued function, d(x) is a bounded nonnegative function. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

11 introduction (an example) Second order elliptic boundary value problem Search solution u for where (a u)+du = f in Ω Ê n, u = g on Ω, a(x) is a bounded symmetric positive definite matrix-valued function, d(x) is a bounded nonnegative function. This can be rewritten mixed form Search solution (q, u) for q+a u = 0 in Ω, q+du = f in Ω, u = g on Ω. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

12 introduction (an example) We re choosing a triangulation of Ω, T h T h Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

13 introduction (an example) We re choosing a triangulation of Ω, T h and using Raviart-Thomas elements P k (K) n +xp k (K) T h RT 0 RT 1 RT 2 Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

14 introduction (an example) Notation E h interior edges, E h boundary faces of Ω, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

15 introduction (an example) Notation Eh interior edges, E h boundary faces of Ω, E h = Eh E h, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

16 introduction (an example) Notation Eh interior edges, E h boundary faces of Ω, E h = Eh E h, (u,v) D := D uvdx for u,v L2 (D), D Ê n, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

17 introduction (an example) Notation Eh interior edges, E h boundary faces of Ω, E h = Eh E h, (u,v) D := D uvdx for u,v L2 (D), D Ê n, u,v D := D uvdx for u,v L2 (D), D Ê n 1, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

18 introduction (an example) Notation Eh interior edges, E h boundary faces of Ω, E h = Eh E h, (u,v) D := D uvdx for u,v L2 (D), D Ê n, u,v D := D uvdx for u,v L2 (D), D Ê n 1, (u,v) Th = K T h (v,w) K for u,v on Ω, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

19 introduction (an example) Notation Eh interior edges, E h boundary faces of Ω, E h = Eh E h, (u,v) D := D uvdx for u,v L2 (D), D Ê n, u,v D := D uvdx for u,v L2 (D), D Ê n 1, (u,v) Th = K T h (v,w) K for u,v on Ω, µ,λ E = e E µ,λ e for µ,λ on E E h. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

20 introduction (an example) Notation Eh interior edges, E h boundary faces of Ω, E h = Eh E h, (u,v) D := D uvdx for u,v L2 (D), D Ê n, u,v D := D uvdx for u,v L2 (D), D Ê n 1, (u,v) Th = K T h (v,w) K for u,v on Ω, µ,λ E = e E µ,λ e for µ,λ on E E h. H(div,Ω) := {v L 2 (Ω) n v L 2 (Ω)} Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

21 introduction (an example) RT finite element approximation The approximation (q h,u h ) is sought in the finite element space V RT h Wh RT given by V RT h = {v H(div,Ω) v K P k (K) n +xp k (K) K T h }, W RT h = {w L 2 (Ω) w K P k (K) K T h }. and is defined by requiring that, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

22 introduction (an example) RT finite element approximation The approximation (q h,u h ) is sought in the finite element space V RT h Wh RT given by V RT h = {v H(div,Ω) v K P k (K) n +xp k (K) K T h }, W RT h = {w L 2 (Ω) w K P k (K) K T h }. and is defined by requiring that, (cq h,v) Ω (u h, v) Ω = g,v n Ω (w, q h ) Ω (du h,w) Ω = (f,w) Ω v V RT h, w W RT h, with c = a 1 for each element K T h Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

23 introduction (an example) This yields to a matrix equation of the form: ( t )( ) ( É = Í À ), where É and Í are vectors of coefficients of q h and u h with respect to their corresponding finite element basis. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

24 introduction (an example) problem System is not positive definite. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

25 introduction (an example) problem System is not positive definite. solutions using expensive solver, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

26 introduction (an example) problem System is not positive definite. solutions using expensive solver, using a positive definite system by elimination of É: solve equation ( 1 t + )Í = + 1 for Í. Required inversion 1 is difficult to compute and a full matrix. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

27 introduction (an example) problem System is not positive definite. solutions using expensive solver, using a positive definite system by elimination of É: solve equation ( 1 t + )Í = + 1 for Í. Required inversion 1 is difficult to compute and a full matrix. hybridization: Introduce new unkowns λ h (Lagrangian multipliers) and relax the continuity constraints between element interfaces for q h. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

28 introduction (an example) Definition (hybridized RT finite element approximation) The approximation (q h,u h,λ h ) is sought in the finite element space V h RT Ŵ h RT M h RT given by V h RT = {v L 2 (Ω) n v K P k (K) n +xp k (K) K T h }, Ŵ h RT = {w L 2 (Ω) w K P k (K) K T h }, M hrt = {µ L 2 (E h ) µ e P k (e) e E h } Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

29 introduction (an example) Definition (hybridized RT finite element approximation) The approximation (q h,u h,λ h ) is sought in the finite element space V h RT Ŵ h RT M h RT given by RT V h = {v L 2 (Ω) n v K P k (K) n +xp k (K) K T h }, RT Ŵ h = {w L 2 (Ω) w K P k (K) K T h }, M hrt = {µ L 2 (Eh ) µ e P k (e) e Eh } and is defined by requiring that, (cq h,v) Ω (u h, v) Ω + λ h, v E h = g,v n Ω v V RT h, (w, q h ) Ω (du h,w) Ω = (f,w) Ω w ŴhRT, µ, q h E h = 0 µ M hrt. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

30 introduction (an example) This yields to a matrix of the form A B t C t Q G B D 0 U = F, C 0 0 Λ 0 where Λ is the vector of dofs associated to the multiplier λ h. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

31 introduction (an example) This yields to a matrix of the form A B t C t Q G B D 0 U = F, C 0 0 Λ 0 where Λ is the vector of dofs associated to the multiplier λ h. New vectors of dofs Q and U define the same approximation (q h,u h ) as the original method. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

32 introduction (an example) This yields to a matrix of the form A B t C t Q G B D 0 U = F, C 0 0 Λ 0 where Λ is the vector of dofs associated to the multiplier λ h. New vectors of dofs Q and U define the same approximation (q h,u h ) as the original method. Both Q and U can be easily eliminated Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

33 introduction (an example) This yields to a matrix of the form A B t C t Q G B D 0 U = F, C 0 0 Λ 0 where Λ is the vector of dofs associated to the multiplier λ h. New vectors of dofs Q and U define the same approximation (q h,u h ) as the original method. Both Q and U can be easily eliminated Λ = À, =CA 1 (A B t (BA 1 B t +D) 1 B)A 1 C t, À = CA 1 (A B t (BA 1 B t +D) 1 B)A 1 G CA 1 B t (BA 1 B t +D) 1 F. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

34 introduction (an example) advantages of hybridization A is now a block diagonal matrix, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

35 introduction (an example) advantages of hybridization A is now a block diagonal matrix, is symmetric positive definite, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

36 introduction (an example) advantages of hybridization A is now a block diagonal matrix, is symmetric positive definite, number of dofs is remarkably smaller, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

37 introduction (an example) advantages of hybridization A is now a block diagonal matrix, is symmetric positive definite, number of dofs is remarkably smaller, once Λ has been obtained, both Q and U can be computed efficiently element by element, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

38 introduction (an example) advantages of hybridization A is now a block diagonal matrix, is symmetric positive definite, number of dofs is remarkably smaller, once Λ has been obtained, both Q and U can be computed efficiently element by element, the multiplier λ h can be used to improve the approximation to u by means of local post-processing. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

39 introduction (an example) history of hybridization year event 1965 first hybridization of finite elements for solving equations of linear elasticity (often: static condensation)/ implementation trick [6] 1985 proof: hybrid variable contains extra information about the exact solution ( local post-processing)[5] 2004 hybridized RT and BDM methods of arbitrary order[2] 2005 extended to finite element methods for stationary Stokes equation (DG, mixed) 2009 unifying framework[3] 2010 comparison CG-H vs. LDG-H[4] Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

40 the framework 1 introduction (an example) 2 the framework 3 examples for hybridizable methods 4 Other novel methods hybridizable methods well suited for adaptivity The RT-method on meshes with hanging nodes Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

41 the framework Unified framework provides approximations for 1 (q,u) in the interior of the elements K T h, (q h,u h ), 2 u on the interior border of the elements, λ h. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

42 the framework Definition (local solvers) For any single valued function m L 2 ( K), the functions (Qm,Um) are the solutions of cqm+ Um = 0 on K, Qm+dUm = 0 on K, Um = m on K Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

43 the framework Definition (local solvers) For any single valued function m L 2 ( K), the functions (Qm,Um) are the solutions of cqm+ Um = 0 on K, Qm+dUm = 0 on K, Um = m on K and for any single valued function f L 2 (K), the functions (Qf,Uf) are the solutions of cqf + Uf = 0 on K, Qf +duf = f on K, Uf = 0 on K. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

44 the framework With u = λ + g and the linearity of the problem we have that (q,u) = (Qλ+Qg +Qf,Uλ+Ug +Uf). Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

45 the framework With u = λ + g and the linearity of the problem we have that (q,u) = (Qλ+Qg +Qf,Uλ+Ug +Uf). The above property only holds iff Definition (transmission condition) This completely characterizes λ. Qλ+Qg +Qf = 0 Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

46 the framework Definition (discrete local solvers) (Qm,Um) and (Qf,Uf) are the discrete versions of the local solvers. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

47 the framework Definition (discrete local solvers) (Qm,Um) and (Qf,Uf) are the discrete versions of the local solvers. Then the discrete solution can be written as (q h,u h ) = (Qλ h +Qg h +Qf,Uλ h +Ug h +Uf), where λ h M h and g h M h are approximations to the values of u. u h λ h M h g h M h Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

48 the framework Definition (discrete version of transmission condition (weak)) a h (λ h,µ) = b h (µ) µ M h is the discrete version of the transmission condition. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

49 the framework Notation V(K) polynomial space in which q is approximated, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

50 the framework Notation V(K) polynomial space in which q is approximated, W(K) polynomial space in which u is approximated, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

51 the framework Notation V(K) polynomial space in which q is approximated, W(K) polynomial space in which u is approximated, V h = {v v K V(K)}, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

52 the framework Notation V(K) polynomial space in which q is approximated, W(K) polynomial space in which u is approximated, V h = {v v K V(K)}, W h = {w w K W(K)}. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

53 the framework Notation V(K) polynomial space in which q is approximated, W(K) polynomial space in which u is approximated, V h = {v v K V(K)}, W h = {w w K W(K)}. In general: A function h on K is double-valued, each branch is denoted by h K + or h K. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

54 the framework To define a hybridizable method, we had to define the discrete local solvers (Qm,Um) and (Qf,Uf), Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

55 the framework To define a hybridizable method, we had to define the discrete local solvers (Qm,Um) and (Qf,Uf), the transmission condition determining λ h, a h and b h, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

56 the framework To define a hybridizable method, we had to define the discrete local solvers (Qm,Um) and (Qf,Uf), the transmission condition determining λ h, a h and b h, the finite element spaces Mh and M h, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

57 the framework To define a hybridizable method, we had to define the discrete local solvers (Qm,Um) and (Qf,Uf), the transmission condition determining λ h, a h and b h, the finite element spaces Mh and M h, the finite element spaces V(K) and W(K), Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

58 the framework To define a hybridizable method, we had to define the discrete local solvers (Qm,Um) and (Qf,Uf), the transmission condition determining λ h, a h and b h, the finite element spaces Mh and M h, the finite element spaces V(K) and W(K), the trace spaces ˆQm and ˆQf (used in discrete local solvers). Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

59 the framework To define a hybridizable method, we had to define the discrete local solvers (Qm,Um) and (Qf,Uf), the transmission condition determining λ h, a h and b h, the finite element spaces Mh and M h, the finite element spaces V(K) and W(K), the trace spaces ˆQm and ˆQf (used in discrete local solvers). Finally, think about existence and uniqueness of λ h. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

60 the framework Definition (trace space M h ) M h := {µ M h µ = 0 on Ω}. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

61 the framework Definition (discrete local solver) 1 maps m M h to the function (Qm,Um): (cqm,v) K (Um, v) K = m,v n K ( w,qm) K + w, ˆQm n K +(dum,w) K = 0 v V(K), w W(K). Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

62 the framework Definition (discrete local solver) 1 maps m M h to the function (Qm,Um): (cqm,v) K (Um, v) K = m,v n K ( w,qm) K + w, ˆQm n K +(dum,w) K = 0 v V(K), w W(K). 2 maps f L 2 (Ω) to the pair (Qf,Uf): (cqf,v) K (Uf, v) K = 0 ( w,qf) K + w, ˆQf n K +(duf,w) K = (f,w) K v V(K), w W(K). Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

63 the framework Definition (characterization of λ h ) a h (η,µ) = (cqη,qµ) Th +(duη,uµ) Th + 1, (Uµ µ)(ˆqη Qη) Eh, b h (µ) = g h, ˆQµ Eh +(f,uµ) Th 1, (Uµ µ)(ˆqf Qf) Eh + 1, Uf(ˆQµ Qµ) Eh 1, (Uµ µ)(ˆqg h Qg h ) Eh + 1, (Ug h g)(ˆqµ Qµ) Eh. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

64 the framework Theorem (existence and uniqueness of λ h ) If the three assumptions are fulfilled, then there is a unique solution λ h of the weak formulation. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

65 the framework Theorem (existence and uniqueness of λ h ) If the three assumptions 1 existence and uniqueness of the local solvers, 2 3 are fulfilled, then there is a unique solution λ h of the weak formulation. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

66 the framework Theorem (existence and uniqueness of λ h ) If the three assumptions 1 existence and uniqueness of the local solvers, 2 positive semidefiniteness of the local solvers and 3 are fulfilled, then there is a unique solution λ h of the weak formulation. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

67 the framework Theorem (existence and uniqueness of λ h ) If the three assumptions 1 existence and uniqueness of the local solvers, 2 positive semidefiniteness of the local solvers and 3 the gluing condition (If µ Mh, then for every interior face there exists a branch K with P K µ = µ.) are fulfilled, then there is a unique solution λ h of the weak formulation. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

68 examples for hybridizable methods 1 introduction (an example) 2 the framework 3 examples for hybridizable methods 4 Other novel methods hybridizable methods well suited for adaptivity The RT-method on meshes with hanging nodes Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

69 examples for hybridizable methods general assumptions same local solvers in every element K, Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

70 examples for hybridizable methods general assumptions same local solvers in every element K, conforming simplicial triangulation. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

71 examples for hybridizable methods general assumptions same local solvers in every element K, conforming simplicial triangulation. M c h,k := {µ C(E h) µ e P k (e) for all faces e E h } M h,k := {µ L 2 (E h ) µ e P k (e) for all faces e E h } Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

72 examples for hybridizable methods RT-H (Raviart Thomas): V(K) W(K) P k (K) n +xp k (K) P k (K) M h M h,k ˆQm Qm ˆQf Qf a h (η,µ) (cqη,qµ) Th +(duη,uµ) Th b h (µ) g h,qµ n Ω +(f,uµ) Th Conservativity strong Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

73 examples for hybridizable methods BDM-H (Brezzi Douglas Marini): V(K) W(K) P k (K) n P k 1 (K) M h M h,k ˆQm Qm ˆQf Qf a h (η,µ) (cqη,qµ) Th +(duη,uµ) Th b h (µ) g h,qµ n Ω +(f,uµ) Th Conservativity strong Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

74 examples for hybridizable methods LDG-H (Local Discontinuous Galerkin): V(K) W(K) 1.) P k (K) n P k 1 (K) 2.) P k (K) n P k (K) 3.) P k 1 (K) n P k (K) M h M h,k ˆQm Qm+τ K (Um m)n ˆQf Qf +τ K (Uf)n a h (η,µ) (cqη,qµ) Th +(duη,uµ) Th + 1, (Uµ µ)(τ K (Uη η)n) Eh b h (µ) g h,qµ n+τ K Uµ Ω +(f,uµ) Th Conservativity strong The stabilization parameter τ K is nonnegative constant on each face in E h, double-valued on Eh. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

75 examples for hybridizable methods CG-H (Continuous Galerkin): V(K) W(K) P k 1 (K) n P k (K) M h M c h,k ˆQm a new unkown variable ˆQf a new unkown variable a h (η,µ) (a Uη, Uµ) Th +(duη,uµ) Th b h (µ) g h, ˆQµ Eh +(f,uµ) Th Conservativity weak Assume that a(x) is a constant on each element. CG-H is an LDG-H method with τ. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

76 examples for hybridizable methods Are all methods hybridizable? Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

77 examples for hybridizable methods Are all methods hybridizable? No. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

78 examples for hybridizable methods Are all methods hybridizable? No. Example (DG-methods) Remember the lecture Scientific Computing or see [1]. A hybrizable method has to be single valued for û h (= λ h ). Method û h Bassi-Rebay {u h } Brezzi et al. {u h } LDG {u h } β u h IP {u h } Bassi et al. {u h } Baumann-Oden {u h }+n K u h NIPG {u h }+n K u h Babuska-Zlamal (u h K ) K Brezzi et al. (u h K ) K Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

79 Other novel methods 1 introduction (an example) 2 the framework 3 examples for hybridizable methods 4 Other novel methods hybridizable methods well suited for adaptivity The RT-method on meshes with hanging nodes Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

80 Other novel methods hybridizable methods well suited for adaptivity 1 introduction (an example) 2 the framework 3 examples for hybridizable methods 4 Other novel methods hybridizable methods well suited for adaptivity The RT-method on meshes with hanging nodes Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

81 Other novel methods hybridizable methods well suited for adaptivity hybridizable methods well suited for adaptivity: V(K) W(K) 1.) P k(k) (K) n +xp k(k) P k(k) (K) 2.) P k(k) (K) n P k(k) 1 (K) 3.) P k(k) (K) n P k(k) (K) 4.) P k(k) 1 (K) n P k(k) (K) Mh {µ µ e M h,k(e) e Eh } {µ µ K C({x K τ K (x) = })} ˆQm Qm+τ K (Um m)n ˆQf Qf +τ K (Uf)n a h (η,µ) depending on V(K),W(K) b h (µ) depending on V(K),W(K) with τ K 0 for 3.) (on at least one face) and 4.) (for alle faces). Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

82 Other novel methods hybridizable methods well suited for adaptivity For e = K + K, we set max{k(k + ),k(k )}, if τ + < and τ < k(e) := k(k ± ), if τ ± = and τ < min{k(k + ),k(k )}, if τ + = and τ =. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

83 Other novel methods hybridizable methods well suited for adaptivity assumption K τ K (0, ) interior face E h [0, ] τ K face of T h corresponding to LDG-H depending on the selected method Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

84 Other novel methods hybridizable methods well suited for adaptivity Main features of this class of methods: 1 variable degree approximation spaces on conforming meshes (k(k)), Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

85 Other novel methods hybridizable methods well suited for adaptivity Main features of this class of methods: 1 variable degree approximation spaces on conforming meshes (k(k)), 2 automatical coupling of different methods on conforming meshes (τ K determines method), Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

86 Other novel methods hybridizable methods well suited for adaptivity Main features of this class of methods: 1 variable degree approximation spaces on conforming meshes (k(k)), 2 automatical coupling of different methods on conforming meshes (τ K determines method), 3 mortaring capabilities for nonconforming meshes (choice of τ for hanging nodes). Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

87 Other novel methods The RT-method on meshes with hanging nodes 1 introduction (an example) 2 the framework 3 examples for hybridizable methods 4 Other novel methods hybridizable methods well suited for adaptivity The RT-method on meshes with hanging nodes Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

88 Other novel methods The RT-method on meshes with hanging nodes Considere the case of variable degree RT-H method with τ 0 everywhere (assumption in last method is violated). Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

89 Other novel methods The RT-method on meshes with hanging nodes Considere the case of variable degree RT-H method with τ 0 everywhere (assumption in last method is violated). Mesh is locally refined into four congruent triangles. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

90 Other novel methods The RT-method on meshes with hanging nodes Considere the case of variable degree RT-H method with τ 0 everywhere (assumption in last method is violated). Mesh is locally refined into four congruent triangles. Idea: Impose special conditions on the meshes and link the definition of k(k) to the structure of the mesh. Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

91 Other novel methods The RT-method on meshes with hanging nodes k(k 2 ) " " max{k(k 2 ),k(k 4 )} k(k 4 ) arbitrarily max{k(k 3 ),k(k 4 )} k(k 3 ) " " k(k 1 ) Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43

92 Other novel methods The RT-method on meshes with hanging nodes Thanks for your attention! Stefan Girke (WWU Münster Institut für Numerische und Angewandte Hybridization Mathematik) 10th of January, / 43