Different Approaches to a Posteriori Error Analysis of the Discontinuous Galerkin Method

Transcription

1 WDS'10 Proceedings of Contributed Papers, Part I, , ISBN MATFYZPRESS Different Approaces to a Posteriori Error Analysis of te Discontinuous Galerkin Metod I. Šebestová Carles University, Faculty of Matematics and Pysics, Prague, Czec Republic. Abstract. We deal wit a posteriori error estimates of te discontinuous Galerkin (DG) metods applied to te Poisson equation. DG tecnique provides a ig order discontinuous approximation. We present DG formulation of te problem and discuss a use of tree types of a posteriori error analysis for te Poisson equation: estimates based on te Galerkin ortogonality, estimates based on te dual formulation and estimates based on te Helmoltz decomposition. Introduction Te subject of a posteriori error estimation forms an important part of numerical analysis. Typically, we face te problem of solving partial differential equations wic are obtained by simplification of a pysical situation of interest. Te equations ave to be someow discretized and finally resulting algebraic system as to be solved. Clearly, some information is lost by tis process. A natural question arises: How accurate approximation of te exact solution is obtained? Tere are various contributions to te error: errors of te matematical model, te discretization error and te algebraic error. In wat follows, we sall be dealing wit te discretization error. Te aim is to solve te partial differential equations wit prescribed tolerance in an effective way. For tis purpose, different adaptive strategies are used during computations. One of te finite element metods were te adaptation can be performed very easily is te discontinuous Galerkin (DG) metod. Tere is no requirement on interelement continuity of DG solution wic allows us to increase polynomial degree on elements were te solution is regular enoug (p-adaptivity). Moreover, it is not necessary to consider conforming meses (for definition see [Ciarlet, 1979]). As a result, local mes refinement can be easily performed (-adaptivity). To drive an adaptive process a tool for recognizing parts of te mes wit large error is needed. Te a posteriori error estimates are suc a tool. In general, te a posteriori error estimator η is defined as a quantity wic bounds or approximates te error e in some norm and can be computed from te discrete solution and given data. If η K denotes a local error estimator on element K, te global error estimator is usually defined by η = 1/2. η 2 K It sould possess several properties wit respect to its usage, namely, guaranteed upper and lower bounds on te error sould be provided: c 1 e η c 2 e, were c 1, c 2 are constants. Tese properties are in literature referred to as a reliability and efficiency of a posteriori error estimates. Te quality of an estimator can be measured by its global effectivity index I eff = η/ e if te exact solution is known. For overview of a posteriori error estimation te reader is referred to [Verfürt, 1996]. 151

2 Poisson s equation ŠEBESTOVÁ: A POSTERIORI ERROR ESTIMATES Let R d (d = 2 or 3) be a bounded polyedral domain wit a boundary = D N, D N =. Let us consider te problem: u = f in, u = g D on D, u n = g N on N, (1) were n denotes te outward unit normal vector to, g D H 1/2 ( D ) and g N H 1/2 ( N ). Let f L 2 (). Definition 1 We say tat function u is te weak solution of (1), if te following conditions are satisfied u u HD(), 1 u v dx = g N v ds + N fv dx for all v H 1 D (), were u H 1 () as te trace g D and H 1 D () {v H1 (); v = 0 on D }. Let T, > 0 be a family of partitions of into a finite number of closed triangles in 2D and tetraedra in 3D wit mutually disjoint interiors tat satisfies te following conditions: sape regularity: C s > 0 : K ρ K C s K T, (2) local quasi-uniformity: C H > 0 : K C H K K, K : K K, (3) were K = diam(k) for K T, ρ K denotes te radius of te largest d-dimensional ball inscribed into K and K denotes te boundary of element K. We assume tat eac face of T lying on is contained eiter in D or N. By F I, FD and FN we denote te set of all interior faces, faces on D, and faces on N, respectively. For a simpler notation, we put F DN F D FN, FID F I FD and F F I FD FN. Furter, we set = diam(). For eac F I tere exist two elements KL and KR of T suc tat K L KR. We define a unit normal vector n to eac F I so tat it points out of KL. Finally, we assume tat n, F DN, as te same orientation as te outward normal to. Over te triangulation T we define te so-called broken Sobolev space H s (, T ) = {v;v K H s (K) K T } equipped wit te norm v 2 H s (,T ) = v 2 H s (K). For v H1 (, T ) we define te broken gradient v of v by ( v) K = (v K ) for K T and use te following notation: v L te trace of v K L on, v R te trace of v K R on, v 1 2 (vl +vr ), [v] v L vr, FI, v L te trace of v K L on, FDN, v [v] v L, FDN. If [ ] and appear in an integral of te form... ds, we sall omit te subscript and write [ ] and instead. Finally, we define te space of discontinuous piecewise polynomial functions S p = {v;v L 2 (),v K P p (K) K T }, were P p (K) is te space of all polynomials on K of degree p {0,1,2,...}. Te problem (1) is discretized by te discontinuous Galerkin finite element metod (DGFEM) 152

3 ŠEBESTOVÁ: A POSTERIORI ERROR ESTIMATES as in [Dolejší et al., 2005]. For tat purpose, we define appropriate forms: a θ (u,v) = u v dx F θ (v) = F ID K fv dx + ( ) u n [v] θ v n [u] ds, F N g N v ds + θ F D ( v n)g D ds, were θ = 1 is connected wit te symetric form, θ = 1 te nonsymetric form and θ = 0 te incomplete form of DGFEM, J σ (u,v) = σ[u][v] ds, were σ = F ID J σ D (v) = F D C W max{ K L, K R } σg D v ds, for F I, (4) σ = C W K L for F D, (5) B θ,σ (u,v) = aθ (u,v) + Jσ (u,v), θ { 1,0,1}, l θ,σ (v) = F θ (v) + Jσ D(v), θ { 1,0,1}. Te constant C W > 0 in (4) and (5) is a suitable constant ensuring coercivity of B θ,σ. Now, we can define te discrete problem. Definition 2 Function u is called a discontinuous Galerkin approximation of te solution of te problem (1), if it is te solution of one of te following problems: Find u S p suc tat B θ,σ (u,v ) = l θ,σ (v ) v S p, were θ { 1,0,1}. Let u be te weak solution of (1) and u be its discontinuous Galerkin approximation. We set Galerkin ortogonality principle e = u u. Due to te nonconformity of te space S p, one sould study ow well functions from tat space can be approximated by continuous piecewise polynomial functions. Te answer is given by te following lemma (for details see [Karakasian et al., 2003, 2004]). Teorem 3 Let a triangulation T be conforming and satisfies (2) and (3). Let g D be te restriction to D of a function in S p H 1 (). For any v S p, i = 0,1 it olds: 153

4 ŠEBESTOVÁ: A POSTERIORI ERROR ESTIMATES (i) Tere exists χ 1 S p H 1 () satisfying χ 1 D = g D and v χ 1 2 i,k C 2 O1 1 2i [v ] 2 + F I F D (ii) Tere exists χ 2 S p H 1 () satisfying v χ 2 2 i,k C 2 O2 were constants C O1, C O2 are independent of and v. F I 1 2i 1 2i [v ] 2, v g D 2. Using te approximation result above and Galerkin ortogonality principle, te following error estimate as been proved in [Karakasian et al., 2007] for te symetric interior penalty discontinuous Galerkin metod (SIPDG). However, te same procedure can be applied to te nonsymetric and incomplete interior penalty discontinuous Galerkin metod (NIPDG and IIPDG). Te proof can be found in [Šebestová, 2009]. Teorem 4 Tere olds: e 2 K c 2 K f + u 2 K + [ u n] 2 F I + g N u n 2 + C2 W 1 [u ] 2 F N +C 2 W F D g D u 2, 1 were a constant c is independent of and C W. Duality principle Besides Galerkin ortogonality property, te Aubin-Nitsce duality argument, as described in, e.g., [Ciarlet, 1979], plays a crucial rule in te derivation of error estimates in L 2 norm. Terefore, tis approac requires te domain to be convex. In addition, we consider Neumann 1 boundary condition on te wole boundary. Let φ {v; v dx = 0} be te solution of te dual problem: and F I φ = e in, φ n = 0 on, C > 0 : φ 2, C e, (6) were C is independent of e. Now, it can be proved te following result (see [Šebestová, 2009]). Teorem 5 Tere olds: e c f + u 2 K 4 K + [ u n] F I g N u n σ 2 3 [u ] 2 F N + F I [u ] F I 1 F I [u ] 2 1/2, 154

5 were te constant c is independent of. ŠEBESTOVÁ: A POSTERIORI ERROR ESTIMATES A bit different equation involving a reaction term is studied in [Rivière et al., 2003] for NIPDG metod. Helmoltz decomposition Te last tecnique is based on te Helmoltz decomposition and properties of te so-called discrete normal flux. To overcome difficulties wit te nonconformity of S p, te gradient of te error is decomposed as follows: e = φ + curlχ, were φ HD 1 () {v H1 (); v = 0 on D } is te solution of te problem φ v dx = e v dx v HD(), 1 χ H(curl,) {v (L 2 ()) k ;curl v (L 2 ()) d } (k = 1 for d = 2 and k = 3 for d = 3) and n curlχ = 0 on N. Moreover, it olds: e 2 = φ 2 + curlχ 2. Te ortogonality of te splitting is crucial because it enables us to estimate eac part of te error independently. Introducing te discrete normal flux u n σ[u ], on K F I Σ n (u ) u n σ(u g D ), on K F D g N, on K F N, wic satisfies K f dx + K Σ n (u ) = 0 for K T, n uφds = Σ n (u )φds, K T K\ D K T K\ D we can state te following result, as proved in [Becker et al., 2003]: e 2 ( K c 2 K f + u 2 K were c is independent of. Conclusion + K Σ n (u ) u n 2 K F IN ) + 1 K [u ] 2, K F ID We ave presented tree approaces to a posteriori error estimation of DGFEM for Poisson s equation wit mixed boundary conditions. Tey allow us to estimate te error in L 2 norm and a mes-dependent DG-norm e 2 DG e edx + σ[e] 2 ds wit regard to te identity: σ[e] 2 ds = F ID F I K F ID σ[u ] 2 ds + F D σ(g D u ) 2 ds. In comparison to conforming finite element metods, te estimators, in addition, contain jumps of te discrete solution across element interfaces. In all of tem, unknown constants are involved. Terefore, a guaranteed upper bound is not provided. Neverteless, te estimators can be used as indicators of areas wit large error. 155

6 References ŠEBESTOVÁ: A POSTERIORI ERROR ESTIMATES Becker R., Hansbo P., Larson M., Energy norm a posteriori error estimation for discontinuous Galerkin metods, Comput. Metods Appl. Mec. Engrg. 192 (2003), Ciarlet P.-G., Te Finite Element Metod for Elliptic Problems, Nort-Holland, Amsterdam, New York, Oxford, Dolejší V., Feistauer M., Sobotíková V., Analysis of te discontinuous Galerkin metod for nonlinear convection-diffusion problems, Comput. Metods Appl. Mec. Engrg. 194 (2005), Karakasian O. A., Pascal F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), Karakasian O. A., Pascal F., Adaptive discontinuous Galerkin approximations of second-order elliptic problems. In: Neittaanmäki P., Rossi T., Korotov S., Oñate E., Périaux J., and Knörzer D. (eds.) European Congress on Computational Metods in Applied Sciences and Engineering, ECCOMAS 2004, University of Jyväskyläs, (2004). Karakasian O. A., Pascal F., Convergence of adaptive discontinuous Galerkin approximations of secondorder elliptic problems, SIAM J. Numer. Anal. 45 (2007), Rivière B., Weeler M. F., A posteriori error estimates for a discontinuous Galerkin metod applied to elliptic problems, Comput. Mat. Appl., 46 (2003), Šebestová, I., A posteriori error estimates of te discontinuous Galerkin metod for convection-diffusion equations, Master Tesis, Carles University in Prague, Verfürt R., A Review of A Posteriori Error Estimation and Adaptive Mes-Refinement Tecniques, Wiley-Teubner, New York,