Hybridized Discontinuous Galerkin Methods

Transcription

1 Hybridized Discontinuous Galerkin Methods Theory and Christian Waluga Joint work with Herbert Egger (Uni Graz) 1st DUNE User Meeting, Stuttgart Christian Waluga (AICES) HDG Methods October 6-8, / 26

2 Outline Outline 1 Hybridization: From DG to HDG Preliminaries A simple DG method Hybridization: HDG Example: Oseen problem 2 Hybridized Mortar Methods Relation between HDG and Hybridized Mortar Example: Stokes problem 3 Conclusion Christian Waluga (AICES) HDG Methods October 6-8, / 26

3 Outline Hybridization: From DG to HDG 1 Hybridization: From DG to HDG Preliminaries A simple DG method Hybridization: HDG Example: Oseen problem 2 Hybridized Mortar Methods Relation between HDG and Hybridized Mortar Example: Stokes problem 3 Conclusion Christian Waluga (AICES) HDG Methods October 6-8, / 26

4 Model problem Hybridization: From DG to HDG Preliminaries Ω Model problem: Poisson s equation Given f L 2 (Ω), find u H 1 (Ω), such that Ω u = f, u = 0, on Ω on Ω Assume Ω R d is bounded, sufficiently regular domain. Notation: (u, v) D := D u v dx D Ω v 0,D := (v, v) D, v 1,D := ( v, v) D We will abbreviate H 1 (Ω) := [ H 1 (Ω) ] d etc. Christian Waluga (AICES) HDG Methods October 6-8, / 26

5 Hybridization: From DG to HDG Meshes and basic notation Preliminaries Meshes: (hanging nodes also possible) T i T e ij j T h := { T i }, where Ω = T i T h := { T i } E h := { e ij : e ij = T i T j, i > j } Broken Sobolev spaces: H s (T h ) := { v L 2 (Ω) : v T H s (T ) for all T T h } (u, v) Th := (u, v) L 2 (T ), T T h v Th := (v, v) 1/2 T h, etc. Jump and average: v := v Ti v Tj, {v} := 1 2 (v T i + v Tj ) Note: functions defined on E h can be interpreted as functions on T h. Christian Waluga (AICES) HDG Methods October 6-8, / 26

6 A simple DG method Hybridization: From DG to HDG A simple DG method Example: Symmetric Interior Penalty Galerkin (SIPG 1 ) Find u H s (T h ) H 1 0 (Ω), s > 3/2, such that a SIP G h (u, v) = (f, v) Th, v H s (T h ), where we define a symmetric bilinear form a SIP h G (u, v) :=( u, v) Th ({ n u}, v ) Eh ({ n v}, u ) Eh + τ( u, v ) Eh. Here, τ := α p2 h is a penalty parameter, with α R+. Natural energy norm for the discrete analysis: v 1,h := ( v 2 T h + τ 1/2 v 2 E h ) 1/2. 1 e.g. [Arnold 1982], [Rivière 2008] Christian Waluga (AICES) HDG Methods October 6-8, / 26

7 Hybridization: From DG to HDG Hybridized DG method Hybridization: HDG Example: Hybridized Symmetric Interior Penalty Galerkin Find (u, û) (H s (T h ) H0 1 (Ω)) L 2 (E h ), s > 3/2, such that a h (u, û; v, ˆv) = (f, v) Th, (v, ˆv) H s (T h ) L 2 (E h ), where û := {u} and ˆv := {v}. We define a symmetric bilinear form a h (u, û; v, ˆv) :=( u, v) Th ( n u, v ˆv) Th ( n v, u û) Th + τ(u û, v ˆv) Th. Here, τ := α p2 h is a penalty parameter, with α R+. Natural energy norm for the discrete analysis: (v, ˆv) 1,h := ( v 2 T h + τ 1/2 (v ˆv) 2 T h ) 1/2. Christian Waluga (AICES) HDG Methods October 6-8, / 26

8 Hybridization: From DG to HDG Hybridization: HDG Hybridization: From DG to HDG (4/4) Given a triangulation T h, define finite dimensional spaces V h and V h : V h := { v H 1 (T h ) : v T P p (T ), T T h } ˆV h := { ˆv L 2 (E h ) : v E P p (E), E E h, ˆv = 0 on Ω } Discrete problem: Find (u h, û h ) V h V h such that a h (u h, û h ; v h, ˆv h ) = (f, v h ) Ω, (v h, ˆv h ) V h V h. Stability analysis: For any α > α 0, there holds 2 a h (v h, ˆv h ; v h, ˆv h ) 1 2 (v h, ˆv h ) 2 1,h, (v h, ˆv h ) V h V h, where α 0 depends on the element shape. One can choose α large enough (e.g. α = 10) or explicitly compute α 0 T on d-simplices and d-hypercubes. 2 e.g. [Egger 2008] Christian Waluga (AICES) HDG Methods October 6-8, / 26

9 Some remarks Hybridization: From DG to HDG Hybridization: HDG The hybridized method is consistent by construction! Standard error analysis yields optimal error estimates. Stability is independent of the particular choice of V h. More general boundary conditions are possible. Other problems were also investigated, e.g. Convection-diffusion [Egger and Schöberl 2009] Stokes problem [Cockburn et. al. 2010], [Egger and W. 2010b] Oseen problem (in preparation) Assembly in an element-wise fashion. Static condensation on element level. Upwinding can be easily incorporated. Locally varying polynomial degrees and nonconforming h-refinements possible. Christian Waluga (AICES) HDG Methods October 6-8, / 26

10 Hybridization: From DG to HDG HDG methods: Implementation (1/2) Implementation uses dune-pdelab and the core modules. Approximations in the interior of elements by monomials (using MonomLocalFiniteElementMap) Approximations on the element borders require some extra work: We use the IntersectionIndexSet provided in PDELab and a self-written FaceMonomLocalFiniteElementMap to define a grid function space on the faces of the mesh. Problem: The orientation of intersections may differ in two intersecting elements. This causes problems when mapping from intersection coordinates to coordinates inside the element. Christian Waluga (AICES) HDG Methods October 6-8, / 26

11 HDG methods (2/2) Hybridization: From DG to HDG The only solution we know so far is a little helper that finds the corresponding intersection in the outside element if it has a lower index or a higher level than the current intersection. We then map the coordinates as follows: if(wrongintersection) return rightintersection->geometryinoutside().global(x); else return rightintersection->geometryininside().global(x); The big disadvantage here is that we need to instantiate a quadratic number of intersections (performance issues?!) Christian Waluga (AICES) HDG Methods October 6-8, / 26

12 Hybridization: From DG to HDG Example: Oseen problem Numerical example: Kovaznay (1/3) Example: Oseen problem; Ω = ( 0.5, 2) ( 0.5, 1.5) [Kovasznay 1947] } u + w u + p = 0 on Ω, u = u div u = 0 exact on Ω Exact solution: ( 1 exp(λx) cos(2πy) u(x, y) = λ 2π exp(λx) sin(2πy) p(x, y) = 1 2 exp(2λx) + p, with parameters λ and p given as 8π 2 λ := ν 1 + ν π 2 ), and p = exp(2λx) dx. 2 Figure: Velocity vectors (ν = 1/10) Christian Waluga (AICES) HDG Methods October 6-8, / 26

13 Hybridization: From DG to HDG Example: Oseen problem Numerical example: Kovaznay (2/3) level P 1 P P 2 P P 3 P dimension L 2 error energy error K S error rate error rate K S error rate error rate K S error rate error rate Table: Kovsznay flow: Errors of the numerical solution for different inf-sup stable finite element approximations and a sequence of uniformly refined meshes. Christian Waluga (AICES) HDG Methods October 6-8, / 26

14 Hybridization: From DG to HDG Example: Oseen problem Numerical example: Kovaznay (3/3) y x Figure: Kovasznay flow (ν = 1): Streamlines and nonconforming mesh after 4 subsequent h-refinements. Christian Waluga (AICES) HDG Methods October 6-8, / 26

15 Outline Hybridized Mortar Methods 1 Hybridization: From DG to HDG Preliminaries A simple DG method Hybridization: HDG Example: Oseen problem 2 Hybridized Mortar Methods Relation between HDG and Hybridized Mortar Example: Stokes problem 3 Conclusion Christian Waluga (AICES) HDG Methods October 6-8, / 26

16 Hybridized Mortar Methods Hybridized mortar method (1/2) Relation between HDG and Hybridized Mortar Similarly as for the HDG method, we can define a hybridized mortar method 3 : V h := { v H 1 0 (Ω h ) : v T P p (T ), T T h } ˆV h := { ˆv L 2 (Γ h ) : v E P p (E), E T h (Γ h ) } Γ 13 Ω 1 Ω 2 Ω 3 Ω 4 Γ 34 partition: Ω h := {Ω 1, Ω 2,..., Ω N } interfaces: Γ ij := Ω i Ω j Γ h := {Γ ij } skeleton: Γ := Γ ij 3 cf. [Egger 2008] Christian Waluga (AICES) HDG Methods October 6-8, / 26

17 Hybridized Mortar Methods Hybridized mortar method (2/2) Relation between HDG and Hybridized Mortar Discrete problem: Find (u h, û h ) V h V h such that a h (u h, û h ; v h, ˆv h ) = (f, v h ) Ω, (v h, ˆv h ) V h V h. where we define Important results: a h (u, û, v, ˆv) :=( u, v) Ωh ( n u, v ˆv) Ωh ( n v, u û) Ωh + τ(u û, v ˆv) Ωh. No direct coupling between subdomain solutions. Elimination on subdomains leads to a Schur complement system for the hybrid variables only ( domain decomposition methods). Space for the hybrid variable can be chosen with great flexibility (no inf-sup-condition necessary for multiplier). Hybridized mortar methods for other problems were also analyzed, e.g. Maxwell [Hollaus et. al. 2010], Stokes [Egger and W. 2010a]. For the finest partition Ω h = T h, we recover the hybridized DG method. Christian Waluga (AICES) HDG Methods October 6-8, / 26

18 Hybridized Mortar Methods Hybrid mortar: Implementation In many applications, there exists a parametrization for the interface (e.g. planar, cylindrical, spherical,...) Idea: Lagrange multipliers live on d 1 dimensional structured meshes that are transformed to the physical space. Overlaps between multiplier mesh and subdomain meshes are computed using dune-grid-glue by C. Engwer and O. Sander. Christian Waluga (AICES) HDG Methods October 6-8, / 26

19 Hybridized Mortar Methods Hybrid mortar: Implementation In many applications, there exists a parametrization for the interface (e.g. planar, cylindrical, spherical,...) Idea: Lagrange multipliers live on d 1 dimensional structured meshes that are transformed to the physical space. Overlaps between multiplier mesh and subdomain meshes are computed using dune-grid-glue by C. Engwer and O. Sander. Christian Waluga (AICES) HDG Methods October 6-8, / 26

20 Hybridized Mortar Methods Hybrid mortar: Implementation In many applications, there exists a parametrization for the interface (e.g. planar, cylindrical, spherical,...) Idea: Lagrange multipliers live on d 1 dimensional structured meshes that are transformed to the physical space. Overlaps between multiplier mesh and subdomain meshes are computed using dune-grid-glue by C. Engwer and O. Sander. Christian Waluga (AICES) HDG Methods October 6-8, / 26

21 Hybridized Mortar Methods Hybrid mortar: Stokes problem Example: Stokes problem Stokes problem [Egger and W. 2010a]: Partitioning and triangulations: The subdomain interfaces are extracted using a Codim1Extractor The entire OneDGrid is extracted with a Codim0Extractor A transformation is given to place the interface grids on the coupling boundaries. We use the PSurface backend to generate remote intersections. Christian Waluga (AICES) HDG Methods October 6-8, / 26

22 Numerical example Hybridized Mortar Methods Example: Stokes problem Example: Stokes problem: Colliding flow, Ω = ( 1, 1) 2 u + p = 0 div u = 0 } on Ω, u = u exact on Ω u exact = ( 20xy 3, 5x 4 5y 4), p exact = 60x 2 y 20y 3 Plot of analytic solution; velocity vectors and pressure field Christian Waluga (AICES) HDG Methods October 6-8, / 26

23 Hybridized Mortar Methods Mortar: Numerical results (1/2) Example: Stokes problem Figure: Numerical results (p=2): h: 1.00; L 2 -error: 1.418; energy error: Christian Waluga (AICES) HDG Methods October 6-8, / 26

24 Hybridized Mortar Methods Mortar: Numerical results (1/2) Example: Stokes problem Figure: Numerical results (p=2): h: 0.50; L 2 -error: 0.171; energy error: Christian Waluga (AICES) HDG Methods October 6-8, / 26

25 Hybridized Mortar Methods Mortar: Numerical results (1/2) Example: Stokes problem Figure: Numerical results (p=2): h: 0.25; L 2 -error: 0.021; energy error: Christian Waluga (AICES) HDG Methods October 6-8, / 26

26 Hybridized Mortar Methods Mortar: Numerical results (2/2) Example: Stokes problem Figure: Simple example for a diffusion problem with non-matching meshes in 3D Christian Waluga (AICES) HDG Methods October 6-8, / 26

27 Outline Conclusion 1 Hybridization: From DG to HDG Preliminaries A simple DG method Hybridization: HDG Example: Oseen problem 2 Hybridized Mortar Methods Relation between HDG and Hybridized Mortar Example: Stokes problem 3 Conclusion Christian Waluga (AICES) HDG Methods October 6-8, / 26

28 Conclusion Conclusion Summary Presentation of hybridized DG methods. Implementation of HDG and hybridized mortar methods in DUNE. Possible applications include different interface problems (e.g. propellers). Known issues HDG methods in DUNE not (yet) naturally implementable? Future work Implementation of parallel codes for high performance computing. Analysis and implementation for time-dependent problems. Use of efficient domain decomposition solvers. Christian Waluga (AICES) HDG Methods October 6-8, / 26

29 Conclusion Conclusion Summary Presentation of hybridized DG methods. Implementation of HDG and hybridized mortar methods in DUNE. Possible applications include different interface problems (e.g. propellers). Known issues HDG methods in DUNE not (yet) naturally implementable? Future work Implementation of parallel codes for high performance computing. Analysis and implementation for time-dependent problems. Use of efficient domain decomposition solvers. Thanks for your attention! Christian Waluga (AICES) HDG Methods October 6-8, / 26

30 Acknowledgement Acknowledgement Financial support from the German Research Association (DFG) through grant GSC 111 is gratefully acknowledged. Christian Waluga (AICES) HDG Methods October 6-8, / 26

31 Literature Literature D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. SIAM Journal for Numerical Analysis, 39: , R. Becker, P. Hansbo, and R. Stenberg. A finite element method for domain decomposition with non-matching grids. ESAIM: Mathematical Modeling and Numerical Analysis, 37(2): , B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM Journal for Numerical Analysis, 47(2): , H. Egger. A class of hybrid mortar finite element methods for interface problems with non-matching meshes. Technical report, AICES , RWTH-Aachen University, H. Egger, and C. Waluga. A hybridized discontinuous Galerkin method for incompressible flow. Part I: Stokes problem. in preparation, H. Egger, and C. Waluga. A hybrid mortar finite element method for the Stokes problem. submitted, Christian Waluga (AICES) HDG Methods October 6-8, / 26