论文信息 - Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions.

Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions.

We present the analysis for the local projection stabilization applied to convection-diffusion problems with mixed boundary conditions. We concentrate on the enrichment approach of the local projection methods. Optimal a-priori error estimates will be proved. Numerical tests confirm the theoretical convergence results. Moreover, the local projection stabilization leads to numerical schemes which work well for problems with several types of layers. Away from layers, the solution is captured very well. Key words. stabilized finite elements, convection-diffusion AMS subject classifications. 65N12, 65N30 1. Introduction. Convection-diffusionequations occur for instance if physical processes in chemical engineering are modelled. Depending on the problem, different types of boundary conditions are applied on different parts of the domain boundary. A common feature of these problems is the small diffusion coefficient, i.e., the process is convection and/or reaction dominant. Since standard Galerkin discretisations will produce unphysical oscillations for this type of problems, stabilization techniques have been developed. The streamline-upwind Petrov–Galerkin method (SUPG) has been successfully applied to convection-diffusion problems. It was proposed by Hughes and Brooks [19]. One fundamental drawback of SUPG is that several terms which include second order derivatives have to be added to the standard Galerkin discretisation in order to ensure consistency. Alternatively, continuous interior penalty methods [1, 6], residual free bubble methods [10, 11, 12], or subgrid modelling [8, 18] can be used for stabilizing the discretised convection-diffusion problems. We will focus in this paper on the local projection stabilization. This method has been proposed for the Stokes problem in [3]. The extension to the transport problem was given in [4]. The analysis of the local projection method applied to equal-order interpolation discretisation of the Oseen problem can be found in [5, 23]. We will apply the local projection method to convection-diffusion problems. The stabilization term of the local projection method is based on a projection of the finite element space which approximates the solution into a discontinuous space . The standard Galerkin discretisation is stabilized by adding a term which gives control over the fluctuation of the gradient of the solution. Originally, the local projection technique was proposed as a two-level method where the projection space is defined on a coarser grid. The drawback of this approach is an increased discretisation stencil. The general approach given in [13, 23] allows to construct local projection methods, such that the discretisation stencil is not increased compared to the standard Galerkin or the SUPG approach since the approximation space and the projection space are defined on the same mesh. In this case, the approximation space is Received November 27, 2007. Accepted for publication May 2, 2008. Published online on February 2, 2009. Recommended by A. Rösch. This work was partially supported by the German Research Foundation (DFG) through grants To143 and FOR 447. Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780 Bochum, Germany (Gunar.Matthies@ruhr-uni-bochum.de). Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany (piotr.skrzypacz, tobiska@mathematik.uni-magdeburg.de). 90 ETNA Kent State University etna@mcs.kent.edu LOCAL PROJECTION METHOD FOR CONVECTION-DIFFUSION PROBLEMS 91 enriched compared to standard finite element spaces. We will concentrate in this paper on the enrichment approach of the local projection method. The main objective of this paper is to provide a convergence theory for the local projection method applied to convection-diffusion problems with mixed boundary conditions. For sufficiently regular solutions the same a-priori error estimates which are known for SUPG are proven. Furthermore, several test problems with different types of interior and boundary layers will be presented. They show that the local projection stabilization allows to obtain numerical solutions which capture the solution away from layers. The plan of this paper is as follows. Section 2 introduces the considered problem class, the weak formulation, and the local projection stabilization. An a-priori error estimate for the stabilized discrete problem will be given in Section 3. Numerical results for problems with different type of layers will be presented in Section 4. Conclusions will be given in Section 5. We use the following notation in this paper. The convection-diffusion problem is considered in a bounded domain "! , $#&%('*) , with polygonal or polyhedral boundary +, . For a set which is either a -dimensional measurable subset of or a -. / 021 -dimensional measurable subset of +, , the fractional order spaces 3546-. 1 , 7$8:9 ;<'>= 1 with norm ? @A? 4*B C and seminorm DE@FD 4*B C will be used. For 7G# HI8KJ/L , the space 3M46-. 1 is defined as the Sobolev space 3N4O-. P1Q R# SUTVB -. P1 with norm ?,@*? 4*B C #W?,@>? TVB B C and seminorm D @*D 4*B C #XD @>D TVB B C . For non-integer 7Y#&H[Z \ with H]8MJ L and \58 -.;^'_021 , the space 3M4E-. P1 is defined as the Sobolev-Slobodeckij space 3 4 -` P1 R# S T a bcB -. P1 , which consists of all functions from the Sobolev space SdTVB -` P1 , such that D e D TQafbOB B C g# hidj k l k m T n C n C D l e -.o 1p q l e -.r(1_D ?so Mrt? !>a b co crFuvKwyx {z =}| The Sobolev-Slobodeckij space S T a bcB -. 1 is equipped with the norm ?~e ? T a bcB B C R#>?~e ? TVB B C Z&D e D TQafbOB B C wyx | As usual, we set ?@(? 4*B C #?@<? T a bcB B C and D@^D 4*B C #Dc@<D T a bcB B C . The inner product over & and : +, will be denoted by -@'@g1 and @'@g , respectively. In case #X the index will be omitted. For ; and a -dimensional subset , let "^-.Y1 denote the space of polynomials of degree less than or equal to while ^-.Y1 is the space of all polynomials of degree less than or equal to in each variable separately. Throughout this paper, will denote a generic constant which is independent of the mesh and the diffusion parameter . We will use the notation if there are positive constants w and , such that w q¡ ¡ holds. 2. Model problem and local projection method. 2.1. Weak formulation. We consider the scalar convection-diffusion problem with mixed boundary conditions (2.1) ¢££ ¤ ££¥ c¦$§$Z ̈G@2©$§Z:a~§ # « in ¬' § # C on C ' + § + ® # c ̄ on f ̄°' where N±W; is a small constant. The boundary + of consists of two disjoint parts, the Dirichlet part C and the Neumann part ̄ . Let ̄ be a relatively open w part of + ETNA Kent State University etna@mcs.kent.edu 92 G. MATTHIES, P. SKRZYPACZ, AND L. TOBISKA and C #+, 2 ̄ . The unit outer normal vector with respect to + is denoted by ® . We are looking for the distribution of concentration § in . The reaction coefficient a{8N 3 ́-` 1 is assumed to be non-negative. Let «X8 -` 1 , C 83 wx -` C 1 , ̄ 83¶μ wx -. ̄ 1 be given functions. Furthermore, we require that the convection field ̈P8 S w B 3·-` 1 ! and the reaction coefficient a fulfil for some a L ± ; the following condition (2.2) aO̧o,1/ 0 % ©X@2 ̈ ̧o 1a~LG± ; 1,o·8 ¬| We assume also that the inflow boundary is part of the Dirichlet boundary, i.e., (2.3) oco·8·+ } -» ̈G@®1̧o 1 z ; 1⁄41⁄2 C | We define the function spaces X# 3 w -» 1 and L #d3⁄4_e 8K ¿ {e,D À # ;(Á| A weak formulation of (2.1) reads: Find §N8Â with §pD À # C , such that (2.4) Ã,̧§Ä'ye(1 #d-»« 'ye(1 Z Åc ̄$'ye( AÆ 1Ve 8K ^LA' where the bilinear form ÃP c3 w -` 1 ÇÂ3 w -» 1 È is defined by (2.5) Ã,̧§Ä'ye(1"#É -»©°§Ä'*©$e(1 Z -` ̈G@2©°§Ä'ye(1 Z -.a~§Ä'ye(1~| The conditions (2.2) and (2.3) guarantee the L -coercivity of the bilinear form Ã . The existence and uniqueness of a weak solution of problem (2.4) can be concluded from the Lax– Milgram lemma. For details, we refer to [15]. It is well-known that for pure Dirichlet boundary data, the weak solution of a twodimensional problem belongs to 3 -` 1 provided that the domain is convex; see [16, 17]. However, in general, we can not expect that the weak solution of the problem (2.1) is in 3 -` 1 . Indeed, in the two-dimensional case, the solution of the Poisson equation ( ·#I0 , Ê #Wa°#W; ) with homogenuous Dirichlet data behaves in the neighbourhoood of a vertex of the boundary with inner angle like § w R#ËEÌ x l,Í -.Îp1 and with mixed Dirichlet-Neumann data like § g# Ë Ì xÏ lcÐ Í -.Îp1 . Here, -.ËE'Îp1 denotes a local system of polar coordinates where Ë is the distance to the boundary vertex, Î89 ;^' Ñ , and Í -@g1 is a smooth function. From [16, Theorem 1.4.5.3] we conclude that § w 8 3M4 locally if and only if 7 z fÒEMZX0 provided that fÒE¿Ó 8 J . Analogously, for fÒ(-»%O/1PÓ 8 J we get § 83M4 locally if and only if 7 z fÒ(-»%O/1tZÉ0 . 2.2. Local projection method. For the finite element discretisation of (2.4), we are given a shape regular family 3⁄46Ô< ^Á of decomposition of into -simplices, quadrilaterals, or hexahedra. The diameter of Õ will be denoted by Ö,× and the mesh size parameter Ö is defined by Ö& g#}Ø1⁄2ÙOÚ^×¬ÛÜ2Ý"Ö × . For Ô^ , let Þ^ B ̄ denote the set of all edges/faces of cells ÕÈ8ÂÔ^ which belong to ̄ . Let ß be a finite element space of continuous elements of order Ë1⁄2&0 . We fix the polynomial order Ë and the dependence of constants on Ë will not be elaborated in this paper. Let L B #3⁄42eà8Â {ec D AÀ # ;(Á be the discrete test space. ETNA Kent State University etna@mcs.kent.edu LOCAL PROJECTION METHOD FOR CONVECTION-DIFFUSION PROBLEMS 93 Since the standard Galerkin discretisation of (2.4) lacks generally stability in the convection dominated regime Gáâ0 , unphysical oscillations will appear in the discrete solution. To circumvent this problem, we consider the sta