Adaptive Cell-centered Finite Volume Method

Transcription

1 ASC Report No. 0/2008 Adaptive Cell-centered Finite Volume Method Christoph rath, Stefan Funken, Dirk Praetorius Institute for Analysis and Scientific Computing Vienna University of echnology U Wien ISBN

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3 ADAPIV CLL-CNRD FINI VOLUM MHOD CHRISOPH RAH, SFAN FUNKN, AND DIRK PRAORIUS Abstract. In our talk, we propose an adaptive mesh-refining strategy for the cell-centered FVM based on some a posteriori error control for the quantity (u Iu h ) L 2. Here, u h P 0 ( ) denotes the FVM approximation of u and I is a certain interpolation operator. As model example serves the Laplace equation with mixed boundary conditions, where our contributions extend a result of [NIC0]. Moreover, this approach allows the coupling of finite volume schemes with the boundary element method, which is a new and fruitful combination of the FVM with ideas from [CAR99a, CAR99b].. Introduction and lliptic Model Problem Let Ω R 2 be a bounded and connected domain with Lipschitz boundary Γ := Ω, which is divided into a closed Dirichlet boundary Γ D Γ with positive surface measure and a Neumann boundary Γ N := Γ\Γ D. We consider the elliptic boundary value problem (.) u = f in Ω, u = u D on Γ D and u/ n = g on Γ N. Here, f L 2 (Ω), u D H (Γ D ), and g L 2 (Γ N ) are given data, and L 2 ( ) and H ( ) denote the standard Lebesgue- and Sobolev-spaces equipped with the usual norms L 2 ( ) and H ( ). We aim to approximate the unique weak solution u H (Ω) by a postprocessed finite volume scheme. hroughout, denotes a certain triangulation of Ω, where N and are the corresponding sets of nodes and edges, respectively. he set of nodes (edges) on the Dirichlet resp. Neumann boundary are N D resp. N N ( D resp. N ). For brevity, we assume that the elements are non-degenerate rectangles and refer to [RA07] for triangular elements. For, h := diam() denotes the uclidean diameter, and for an edge, we denote by h its length. We say that the triangulation is almost regular, if (i) the mixed boundary conditions are resolved, i.e. each edge with Γ satisfies either Γ D or Γ N. (ii) the intersection 2 of two elements, 2 with 2 is either empty or a node or an edge. (iii) the edge contains an interior (i.e. hanging) node, there are two edges, 2 with = 2, cf. Figure 2. for examples. A node a N \(N D N N ) is a hanging node provided that there are elements, 2 such that a 2 is a node of but not of 2. Let N H be the set of all hanging nodes. Date: January, Key words and phrases. finite volume method, cell-centered method, diamond path, a posteriori error estimate, adaptive algorithm.

4 Figure 2.. he weights ψ for an almost regular mesh of squares. 2. Cell-Centered Finite Volume Method We integrate the differential equation (.) over a control volume and use the Gauss divergence theorem to obtain, with the set of edges of, u f dx = ds = Φ, (u) ds for all. n Here, Φ, (u) = u/ n, ds is the diffusive flux and n, is the outer normal vector of on. Let u h P 0 ( ) be a piecewise constant approximation of u, namely u := u h u(x ), where x denotes the center of an element. For the cell-centered finite volume method, u h is computed by replacing the diffusion flux Φ, (u) by a discrete diffusion flux F, (u h ). First, Φ, (u) = g ds is known for a Neumann edge N. One therefore defines F, (u h ) := Φ, (u h ) = g ds for N. Second, for a non-elementary edge with = 2 and, 2, where denotes the set of elementary edges (i.e. neither nor 2 contain an interior node), there holds Φ, (u) = Φ, (u) + Φ,2 (u), which leads to the definition F, (u h ) := F, (u h ) + F,2 (u h ) for = 2 with, 2. Finally, it remains to define F, (u h ) for elementary and Dirichlet edges D. his is done by the diamond path method in the following way: For each node a N, we define ψ (a) u, for all a N \(N D N N ), eω a (2.) u a = u D (a), for all a N D, u a + g a, for all a N N, with certain weights { ψ (a), a N } for each element of the patch ωa := { a }. Here, N denotes the set of nodes of. For details on the computation of the weights, the reader is referred to [COU00]. Figure 2. gives the precise values for almost regular triangulations of squares. Details on the computation of u a and g a are found in [RA07]. For an elementary edge, let x a and x b be the starting and end point of 2

5 x b x d = (x x ) n h t, n, (x x ) t x x a Figure 2.2. Notation for the diamond path. D and, the unique elements with =, cf. Figure 2.2. hen, ( ) u u u b u a F, (u h ) := h α d h (2.2) with α = (x x ) t, (x x ) n,, d = (x x ) n,. Here, the additional unknowns u b and u a are located at the nodes x b and x a and are computed by (2.). Finally, the tangential vector t, is chosen orthogonal to n, in mathematically positive sense. For a Dirichlet edge D, we compute F, (u h ) by (2.2), where x is now replaced by the midpoint x m of and u becomes u D (x m ). Altogether, the discrete problem reads: Find u h P 0 ( ) such that F, (u h ) = f dx, for all. Note, that the conservativity of the continuous flux Φ, (u) = Φ,(u) also holds for the discrete flux F, (u h ) = F,(u h ). Remark. We stress, that even for an admissible triangular mesh in the sense of [YM00, Definition 9.], local mesh-refinement is nontrivial, since admissibility necessarily implies that all angles are strictly less than π/2. For rectangular meshes, local mesh-refinement cannot avoid hanging nodes and thus contradicts the admissibility condition.. A Posteriori rror stimate For a non-degenerate rectangular element = conv{a, a 2, a, a } R 2 with edges j = conv{a j, a j+ } and a := a, we define P = P 2 span{x xy 2, y yx 2 } and Σ = (S,...,S 8 ), where S j (p) = p(a j ), S j+ (p) = j p n,j ds for j =,..., (p P ), cf. [NIC0, Section.2]. hen, the Morley-type element (, P, Σ ) is a nonconforming finite element. he corresponding Morley interpolant Iu h is now defined -elementwise by (.) (.). he definition of which is an extension of the definition in [NIC0, Section

6 ] to the case of hanging nodes and mixed boundary conditions. For each free node a N ( N \(N D N N N H ) ), we enforce (.) (Iu h ) (a) = ψ a (a)u h a. a eω a For each boundary node, the value Iu h (a) is prescribed by { u D (a) for a N N D, (.2) (Iu h ) (a) = u a + g a for a N N N. For each hanging node a N N H, there holds (.) (Iu h ) (a) = (Iu h ) a (a), where a is the unique element with a int() for some (non-elementary) edge a. Finally, for each edge holds (Iu h ) (.) ds = F, (u h ). n, We stress, that the Morley interpolant Iu h is uniquely defined by (.) (.). Moreover, the definition ensures the following orthogonality properties, which are essential for the analysis of the proposed error estimator: First, the residual R := f + (Iu h ) is L 2 -orthogonal to P 0 ( ), i.e. ( f + (Iuh ) ) dx = 0 for all. In particular, R = f f, where f denotes the -piecewise integral mean, i.e. f := f dx. Second, for boundary edges hold (u Iu h ) (u Iu h ) ds = 0 ( D ) resp. ds = 0 ( N ). t, n, Finally, for interior edges hold [ (Iuh ) ] ds = 0 ( 0 ) n, resp. [ (Iuh ) ] ds = 0 ( ). t, Here, 0 denotes the set of all interior edges which are not part of a non-elementary edge. o introduce our error estimator, we define the normal jump over an edge by [ (Iu h )/ n, ] = (Iu h )/ n, + (Iu h )/ n,, where, and =. Note that n, = n, so that the sum in the definition is in fact a difference. For each non-elementary edge = 2 with, 2, we define the normal jump [ (Iu h )/ n, ] (x) := [ (Iu h )/ n, ] i (x) for all x i, i =, 2.

7 0 0 error and error estimator I (unif.) η (unif.) h (unif.) I (adap.) η (adap.) h (adap.) / / number of elements Figure.. rrors I and h as well as a posteriori error estimator η in Problem (.) for uniform and adaptive mesh-refinement. he tangential jump [ (Iu h )/ t, ] is defined analogously. For each element, we now define the refinement indicator η 2 := h2 f f 2 L 2 () + h [ (Iu h )] 2 L 2 () (.) + N h (u Iu h) 2 n, \( D N ) L 2 () + D h (u Iu h) 2 t, L 2 () With techniques known from the finite element method [AIN00], one proves reliability ( C rel /2. (.6) (u Iu h ) L 2 (Ω) η := η) 2 As for non-conforming FM the proof employs a Helmholtz decomposition. he Galerkin orthogonality is replaced by the orthogonality properties of Iu h stated before. he converse inequality holds even -elementwise, namely for all (.7) C loc η ( (u Iu h ) 2 L 2 (ω ) + h2 f f 2 L 2 (ω )) /2, where the involved patch reads ω := { }. his estimate follows by usual techniques and the use of appropriate bubble functions. In particular, we obtain efficiency of η up to oscillation terms, (.8) C eff η (u Iu h ) L 2 (Ω) + h(f f ) L 2 (Ω), where h L (Ω), h := h denotes the local mesh-width. he constants C rel, C loc, C eff > 0 only depend on the shape of the elements in. We refer to [RA07] for the detailed proofs of the estimates (.6) (.8)..

8 # = 69 # = 67 # = 629 Figure.2. Adaptively generated meshes in problem (.).. Numerical xperiments We consider the Laplace problem (.) on the L-shaped domain Ω = (, ) 2 \ ( [0, ] [, 0] ). he given exact solution is the harmonic function u(x, y) = I ( (x + iy) 2/) which reads in polar coordinates u(x, y) = r 2/ sin(2ϕ/) with (x, y) = r (cosϕ, sin ϕ). Note that u has a generic singularity at the reentrant corner (0, 0). For the numerical computation, we prescribe the exact Neumann and Dirichlet data, where Γ D = Γ\Γ N and Γ N := {0} (, 0) (0, ) {0}. Note that Γ N includes the reentrant corner, where the normal derivative u/ n is singular. We compute the energy error h := ( u u h 2 L 2 (Ω) + /2, u u h,h) 2 with u P 0 ( ) the -piecewise integral mean of u, and where the discrete H -seminorm is defined by ( 2 ) /2 v h,h = (v v )/d h d D for any v h P 0 ( ). Provided that u H 2 (Ω), the diamond path method leads to h = O(N /2 ) with respect to the number N = of elements [COU00]. Moreover, we compute the Morley error I = (u Iu h ) L 2 (Ω) and the corresponding error estimator η. All computations are done in Matlab in the spirit of [ALB99, RA08]. We employ an adaptive mesh-refining algorithm, which is steered by the local refinement indicators η from (.): An element j is marked for refinement provided η j θ max{η,...,η N }, where we use θ = 0 for uniform and θ = 0. for adaptive mesh-refinement, respectively. he numerical outcome is plotted in Figure. over the number N of elements. For uniform mesh-refinement, the energy error h converges with a suboptimal order which appears to be slightly better than O(N / ). he proposed adaptive strategy regains the optimal order of convergence O(N /2 ). As can be expected from the finite element method, the Morley error I decreases like O(N / ) for uniform mesh-refinement. he adaptive algorithm leads to an improved order of convergence O(N /2 ). As predicted by theory, the error estimator η is observed to be reliable and efficient.. Outlook he finite volume method is a well adapted method for the discretization of various convection dominated partial differential equations. It is very popular in the engineering community (fluid mechanics) because of its conservative properties. his method is contrary to 6

9 the boundary element method (BM), which can be applied to the most important linear and homogeneous partial differential equations with constant coefficients also in unbounded domains. he coupling of FVM and BM combines the advantages of both methods, e.g. in problems where stationary diffusive heat and convection in different media are coupled. While convection would be modelled by the finite volume method, diffusive heat (in a possibly unbounded domain) is solved using the boundary element method. First numerical examples show the efficiency of the symmetric coupling of FVM and BM, where we used the results given here for the FVM and results from [CAR99a, CAR99b] to develop a stable discretization. he resulting system of linear equations is now a block-system instead of a system for the FM NC -BM coupling. he coupling of FVM and BM involves two further continuous ansatz functions on the interface to link the discontinuous displacement field to necessarily continuous boundary ansatz functions on the boundary. Quasi-optimal a priori error estimates and sharp a posteriori error estimates are almost established which justify adaptive mesh-refining algorithms. Numerical experiments show the adaptive coupling as an efficient tool for the numerical treatment of transmission problems. References [AIN00] M. Ainsworth, J.. Oden: A posteriori error estimation in finite element analysis, John Wiley & Sons, [ALB99] J. Alberty, C. Carstensen, S.A. Funken: Remarks around 0 lines of Matlab: Short finite element implementation, Num. Alg., 20:7 7, 999. [CAR99a] C. Carstensen, S. A. Funken: Coupling of nonconforming finite elements and boundary elements I: A priori estimates, Computing, 62:229 2, 999. [CAR99b] C. Carstensen, S. A. Funken: Coupling of nonconforming finite elements and boundary elements II: A posteriori estimates and adaptive mesh-refinement, Computing, 62:2 29, 999. [CIA78] P.G. Ciarlet: he finite element method for elliptic problems, North-Holland, Amsterdam, 978. [COU00] Y. Coudiére, P. Villedieu: Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes, SAIM: M2AN, (6):2 9, [RA08] C. rath, S. Funken. D. Praetorius: Matlab implementation of the finite volume method, part II: he cell centered FVM, in preparation, [RA07] C. rath, D. Praetorius: A posteriori error estimate and adaptive mesh-refinement for the cell-centered finite volume method for elliptic boundary value problems, submitted [YM00] R. ymard,. Gallouët, R. Herbin: Finite volume methods, Handbook of Numerical Analysis, Volume 7. lsevier Science B.V., first edition, 999. [NIC0] S. Nicaise: A posteriori error estimations of some cell-centered finite volume methods, SIAM J. Numer. Anal., (0):8 0, 200. University of Ulm, Institute for Numerical Mathematics, Helmholtzstraße 8, D Ulm, Germany -mail address: {Christoph.rath,Stefan.Funken}@uni-ulm.de Vienna University of echnology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8-0, A-00 Vienna, Austria -mail address: Dirk.Praetorius@tuwien.ac.at 7