Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems

Transcription

1 Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems Christian Waluga 1 advised by Prof. Herbert Egger 2 Prof. Wolfgang Dahmen 3 1 Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University 2 M2 Center of Mathematics, Technische Universität München 3 Institut für Geometrie und Praktische Mathematik, RWTH Aachen University February 3, 2012 Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

2 Outline Outline 1 Introduction Motivation Governing equations Discretization Overview of the thesis 2 A Hybrid Discontinuous Galerkin Method Preliminaries Poisson problem Stokes problem Navier-Stokes problem 3 A posteriori error estimators and adaptivity Error estimation An adaptive algorithm Numerical results 4 Conclusions Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

3 Outline Introduction 1 Introduction Motivation Governing equations Discretization Overview of the thesis 2 A Hybrid Discontinuous Galerkin Method 3 A posteriori error estimators and adaptivity 4 Conclusions Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

4 Introduction Motivation Simulation aims to predict physical phenomena which are difficult, expensive, or even impossible to observe in conventional experiments. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

5 Introduction Motivation Source: Wikimedia Commons Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

6 Introduction Motivation Real-world problems Mathematical language Computable problems Approximate solutions Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

7 Introduction Governing equations Motion of fluids Incompressible Navier-Stokes equations: ν u + u u + p = f div u = 0 } on Ω R d velocity: u = [u 1,..., u d ] pressure: p Boundary conditions: u = g D on Ω D (Dirichlet) ν nu pn = g N on Ω N (Neumann) R 2 Ω Ω N Ω D For simplicity, let us suppose that u = 0 on Ω. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

8 Introduction Governing equations Motion of fluids Incompressible Navier-Stokes equations: ν u + u u + p = f div u = 0 } on Ω R d velocity: u = [u 1,..., u d ] pressure: p Boundary conditions: u = g D on Ω D (Dirichlet) ν nu pn = g N on Ω N (Neumann) R 2 Ω Ω N Ω D For simplicity, let us suppose that u = 0 on Ω. Difficulties? incompressibility, convective terms, nonlinearity Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

9 Introduction Governing equations Motion of fluids Incompressible Navier-Stokes equations: ν u + u u + p = f div u = 0 } on Ω R d velocity: u = [u 1,..., u d ] pressure: p Boundary conditions: u = g D on Ω D (Dirichlet) ν nu pn = g N on Ω N (Neumann) R 2 Ω Ω N Ω D For simplicity, let us suppose that u = 0 on Ω. Difficulties? incompressibility, convective terms, nonlinearity Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

10 Introduction Discretization Popular discretization methods Finite Volume (FV) locally conservative suitable for convection dominated flow extension to higher orders is complicated FE Finite Element (FE) straightforward extension to higher orders not locally conservative unstable for dominant convection on coarse meshes DG Discontinuous Galerkin (DG) combines advantages of FV and FE methods very suitable for adaptivity increased number of degrees of freedom reduced sparsity in the discrete system Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

11 Introduction Discretization Circumventing the drawbacks of DG... add additional unknowns at the interfaces (hybridization*). relax the coupling across interfaces. eliminate element unknowns (static condensation). DG HDG* HDG Literature: Cockburn, Gopalakrishnan and Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

12 Introduction Discretization Hybrid Discontinuous Galerkin (HDG) Methods DG methods with some (algorithmic) advantages better sparsity structure (for higher orders) static condensation element-based assembly but: Implementation more involved! Overview: Cockburn, Gopalakrishnan, Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

13 Introduction Overview of the thesis Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte D i s s e r t a t i o n vorgelegt von Diplom-Ingenieur Christian Waluga aus Würselen Berichter: Univ.-Prof. Dr. Herbert Egger Univ.-Prof. Dr. Wolfgang Dahmen Tag der mündlichen Prüfung: 3. Februar 2012 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Updated and corrected version: May 23, 2013 Overview of the thesis A priori analysis for different flow model-problems. Technical results for the hp analysis. Suitable error estimators to drive adaptive algorithms. Hybrid mortar methods for domain decomposition. Numerical experiments. Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

14 Introduction Overview of the thesis Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte D i s s e r t a t i o n vorgelegt von Diplom-Ingenieur Christian Waluga aus Würselen Berichter: Univ.-Prof. Dr. Herbert Egger Univ.-Prof. Dr. Wolfgang Dahmen Tag der mündlichen Prüfung: 3. Februar 2012 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Updated and corrected version: May 23, 2013 Overview of the thesis A priori analysis for different flow model-problems. Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems Suitable error estimators to drive adaptive algorithms. Numerical experiments. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

15 Outline HDG for incompressible flow 1 Introduction 2 A Hybrid Discontinuous Galerkin Method Preliminaries Poisson problem Stokes problem Navier-Stokes problem 3 A posteriori error estimators and adaptivity 4 Conclusions Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

16 HDG for incompressible flow Preliminaries Triangulations of the domain Ω Hybrid meshes with a bounded level of nonconformity (shape regular, quasi-uniform) E 0,6 E 0,10 E 0,1 E 0,10 E 0,9 E 0,8 E 0,6 E 8,9 E T E 6 T T 6,7 9, E 7,8 T 10 E 9,11 E 8,12 T 7 E 5,6 E E 10,11 T 11 T 7,12 12 E E 4,5 11,12 E 4,7 T 4 T E 5 1,10 E 2,11 E 2,12 E 3,4 E 0,5 T 1 E 1,2 T 2 E 2,3 T 3 E 0,3 E 0,1 E 0,2 Collection of elements: T h = {T 1, T 2,... } Boundary + interior facets: E h := {E 0,1, E 0,2,..., E 1,2, E 2,3,...} Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

17 HDG for incompressible flow Preliminaries Hybrid Discontinuous Galerkin? Galerkin? seek a weak solution in a finite dimensional space solution can be computed by solving a linear system of equations AU = F A : stiffness matrix, U : unknowns, F : right hand side Discontinuous? find a discrete solution u h V h on T h space V h consists of piecewise discontinuous polynomial functions. V h = { v h L 2 (Ω) : v h T P k (T ), T T h } Hybrid? also approximate the trace û h V h on E h V h = { v h L 2 (E h ) : v h E P k (E), E E h, v h = 0 on Ω } Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

18 HDG for incompressible flow Poisson problem HDG for Poisson u = f in Ω and u = 0 on Ω. Discrete problem Find (u h, û h ) V h V h, such that where we define a h (u h, û h ; v h, v h ) = f h (v h, v h ) for all (v h, v h ) V h V h. a h (u h, û h ; v h, v h ) := ( u h v h dx nu h (v h v h ) ds T T T T h (u h û h ) nv h ds + T T f h (v h, v h ) := f v h dx T T T h Consistency For u H 1 0 (Ω) H2 (T h ), there holds k γ 2 ) T T ht (u h û h ) (v h v h ) ds a h (u, u Eh ; v h, v h ) = f h (v h, v h ) for all (v h, v h ) V h V h. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

19 HDG for incompressible flow Poisson problem A priori analysis for Poisson Theorem (Coercivity): For γ T sufficiently large, there holds a h (v h, v h ; v h, v h ) 1 2 (v h, v h ) 2 1,h, (v h, v h ) V h V h. Energy norm: (v h, v h ) 2 1,h := T T h ( v h 2 T + γ T k2 T ht v h v h 2 T ) 1/2. Remarks: Optimal γ T is explicitly given by sharp trace inverse estimates. Existence, uniqueness and stability bounds of discrete solution (Lax-Milgram). Standard arguments and approximation results yield order-optimal error estimates. Literature: Arnold et. al. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems Warburton, Hesthaven. On the constants in hp-finite element trace inverse inequalities, Burman, Ern. Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, Egger. A class of hybrid mortar finite element methods for interface problems with non-matching meshes, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

20 HDG for incompressible flow Stokes problem HDG for Stokes System of equations for u = [u 1,..., u d ] with incompressibility constraint. Finite dimensional spaces u + p = f and div u = 0 in Ω and u = 0 on Ω. discrete velocity (u h, û h ): V h := V d h (on T h ), Vh := V d h (on E h ) discrete pressure p h : Q h := { q h L 2 0(Ω) : q h T P k 1 (T ) } Discrete (saddle-point) problem Find (u h, û h ) V h V h and p h Q h, such that a h (u h, û h ; v h, v h ) + b h (v h, v h ; p h ) = f h (v h, v h ) for all (v h, v h ) V h V h, b h (u h, û h ; q h ) = 0 for all q h Q h. The bilinear form associated with the incompressibility constraint is defined as b h (u h, û h ; q h ) := ( ) div u h q h dx (u h û h ) n q h ds T T T T h Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

21 HDG for incompressible flow Stokes problem A priori analysis for Stokes The crucial part of the analysis is the following stability condition. Theorem (discrete inf-sup stability): There exists a constant β independent of the mesh and the polynomial degree k, such that b h (v h, v h ; q h ) sup βk 1/2 q h 0,h, q h Q h. (v h, v h ) V h V (v h h, v h ) 1,h Remarks: The proof is based on an argument due to Fortin. We employ new hp-estimates for the L 2 -orthogonal projections: Π T : H 1 (T ) P k (T ) ΠE : H 1 (T ) P k (E). The analysis applies to hybrid meshes (e.g. tri/quad or tet/hex) with hanging nodes. Error estimates of such mixed methods depend on the approximation properties of the finite element spaces and the discrete inf-sup estimate. Hence, the k-dependence is important for high order discretizations. Literature: Fortin. Analysis of the Convergence of Mixed Finite Element Methods, Brezzi, Fortin. Mixed and Hybrid Finite Element Methods. Springer, Boffi et. al. Mixed finite elements, compatibility conditions, and applications, Springer, Egger, W. hp-analysis of a Hybrid DG Method for Stokes Flow, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

22 HDG for incompressible flow Stokes problem Comparison with related work The discrete inf-sup constant is usually of the order k s (optimal method: s = 0). b h (v h, q h ) sup βk s q h Qh q h Q h, v h V h v h Vh Type Reference Suboptimality s Element types Balanced approx. Spectral [BM:99] 0 quad, hex yes CG [AC:02] 0 quad yes CG [SS:96] ɛ quad, hex yes CG [S:98] 1/2 quad no HDG [EW:11] 1/2 tri, quad, tet, hex yes d 1 DG [T:02] quad, hex no 2 DG [SST:03] 1 quad, hex yes CG [S:98] ( 3 ) tri no Literature: [SS:96] Stenberg, Suri. Mixed finite element methods for problems in elasticity and Stokes flow, [S:98] Schwab. p- and hp- finite element methods: theory and applications in solid and fluid mechanics, [BM:99] Bernardi, Maday. Uniform inf-sup conditions for the spectral discretization of the Stokes problem, [AC:02] Ainsworth, Coggins. A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow. [T:02] Toselli. hp discontinuous Galerkin approximations for the Stokes problem, [SST:03] Schötzau, Schwab, Toselli. Mixed hp-dgfem for incompressible flows, [EW:11] Egger, W. hp-analysis of a Hybrid DG Method for Stokes Flow, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

23 HDG for incompressible flow Stokes problem Convergence rates for Stokes We can prove order-optimal convergence rates; i.e, for (u, p) H m+1 (T h ) H m (T h ): (u u h, u Eh û h ) 1,h + 1 k p p h 0,h hm k m 1 u m+1,ω + hm k m 1/2 p m,ω k level velocity error rate pressure error rate Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

24 HDG for incompressible flow Navier-Stokes problem HDG for Navier-Stokes Add nonlinear convective terms... ν u + u u + p = f and div u = 0 in Ω and u = 0 on Ω. Discrete (nonlinear) problem Find (u h, û h ) V h V h and p h Q h, such that νa h (u h, û h ; v h, v h ) + c h (u h, û h ; u h, û h ; v h, v h ) + b h (v h, v h ; p h ) = f h (v h, v h ), for all (v h, v h ) V h V h and all q h Q h. The form associated with the convective terms is defined as b h (u h, û h ; q h ) = 0. c h (w h, ŵ h ; u h, û h ; v h, v h ) := ( u h (w h v h ) dx + (ŵ h n) {û h /u h } (v h v h ) ds T T T T h b h(w h, ŵ h ; u h v h ). where {û h /u h } = f(w h, ŵ h ; u h, û h ) denotes an upwind value. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

25 HDG for incompressible flow Navier-Stokes problem What is upwinding? Numerical scheme adapts to the direction of propagation of information in the flow field. {û h /u h } := { û h if ŵ h n 0, u h if ŵ h n > 0. ŵ h {û h /u h } = u h {û h /u h } = û h Literature: Reed and Hill. Triangular mesh methods for the neutron transport equation Egger and Schöberl. A mixed-hybrid-discontinuous Galerkin finite element method for convection-diffusion problems Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

26 HDG for incompressible flow Navier-Stokes problem Dealing with the nonlinearity... The discrete Navier-Stokes problem is equivalent to a fixed point problem (u h, û h ) = Φ h (u h, û h ) The operator Φ h : (w h, ŵ h ) (u h, û h ) is defined by the following discrete Oseen problem Remarks: Find (u h, û h ) V h V h and p h Q h, such that νa h (u h, û h ; v h, v h ) + c h (w h, ŵ h ; u h, û h ; v h, v h ) + b h (p h ; v h, v h ) = f h (v h, v h ), for all (v h, v h ) V h V h and all q h Q h. The fixed-point operator is well-defined. b h (q h ; u h, û h ) = 0. Existence of fixed points (thus discrete solutions) by Leray-Schauder principle. For small f Ω, we also obtain Uniqueness of a discrete solution (Banach) Order-optimal convergence rates. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

27 HDG for incompressible flow Navier-Stokes problem Lid-driven cavity flow Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

28 HDG for incompressible flow Navier-Stokes problem Lid-driven cavity flow (ν = 1/100) Comparison of a third order solution with reference data by Ghia et. al y u 1 (x = 0.5) Literature: Ghia et. al. High-Re Solutions for Incompressible Flow using the Navier-Stokes Equations and a Multigrid Method, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

29 HDG for incompressible flow Navier-Stokes problem Lid-driven cavity flow (ν = 1/1000) Comparison of a third order solution with reference data by Ghia et. al y u 1 (x = 0.5) Literature: Ghia et. al. High-Re Solutions for Incompressible Flow using the Navier-Stokes Equations and a Multigrid Method, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

30 Outline A posteriori error estimators and adaptivity 1 Introduction 2 A Hybrid Discontinuous Galerkin Method 3 A posteriori error estimators and adaptivity Error estimation An adaptive algorithm Numerical results 4 Conclusions Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

31 A posteriori error estimators and adaptivity Error estimation Error estimation Exact solution of relevant problems is usually unknown. The jump error estimator η J is given by a sum of local contributions η J := ( T T h η2 T ) 1/2 where η 2 T := γ T k2 T ht uh û h 2 L 2 ( T ) Estimator bounds the error from below (efficiency) and above (reliability) η J (u u h, u û h ) 1,h + p p h 0,h k η J + osc Reliability proved for HDG Methods for Poisson and Stokes. η T can be used as error indicator to drive adaptive refinement strategies. Literature: Egger, W. hp-analysis of a Hybrid DG Method for Stokes Flow, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

32 A posteriori error estimators and adaptivity An adaptive algorithm The adaptive algorithm Given an initial mesh T 0 h of Ω, we invoke the algorithm: For i = 0, 1,... SOLVE ESTIMATE MARK REFINE, (SOLVE) The discrete problem on Th i is solved by the HDG Method. (ESTIMATE) For each T T i h, we compute the local error indicator η T. (MARK) Dörfler: obtain minimal M(T h ) T h, such that T M(T h ) η2 T θ2 η T T T 2, here: θ = 0.5 h (REFINE) Refine T i h by subdividing all marked triangles in M(T h) into four similar ones. Ensure that the maximal difference of the refinement levels between two neighboring elements is one (1-irregular). Literature: Dörfler. A convergent adaptive algorithm for Poissons equation, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

33 A posteriori error estimators and adaptivity Numerical results Stokes in L-shape domain Exact solution due to Verfürth exhibits corner singularity. Initial mesh and adaptively refined meshes after 20 refinement steps. initial k = 1 k = 2 k = 3 Literature: Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner, Egger, W. hp-analysis of a Hybrid DG Method for Stokes Flow, Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

34 A posteriori error estimators and adaptivity Numerical results Convergence rates and effectivity index energy error η J estimate 10 0 energy error η J estimate 10 0 energy error η J estimate number of elements (k=1) number of elements (k=2) number of elements (k=3) 10 8 effectivity index 10 8 effectivity index 10 8 effectivity index number of elements (k=1) number of elements (k=2) number of elements (k=3) Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

35 Outline Conclusions 1 Introduction 2 A Hybrid Discontinuous Galerkin Method 3 A posteriori error estimators and adaptivity 4 Conclusions Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

36 Conclusions Conclusions We derived analyzed HDG methods for a class of incompressible flow problems. Some technical results may also be useful for related work. HDG methods are promising for high order simulations. Reliable, efficient and simple error estimators. Outlook: HDG methods for other interesting physical models. Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34

37 Financial support from the Deutsche Forschungsgemeinschaft through grant GSC 111 is gratefully acknowledged

38 Conclusions Selected references Discontinuous Galerkin (DG) Arnold et. al. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems Di Pietro, Ern. Mathematical Aspects of Discontinuous Galerkin Methods Cockburn et. al. A locally conservative LDG method for the incompressible Navier-Stokes equations Girault et. al. A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes problems Hybrid Discontinuous Galerkin (HDG) Cockburn et. al. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems Egger and Schöberl. A mixed-hybrid-discontinuous Galerkin finite element method for convection-diffusion problems Nguyen et. al. An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations Cockburn et. al. Analysis of HDG methods for Stokes flow Egger and W. hp-analysis of a hybrid DG method for Stokes flow Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, / 34