ALMOST OPTIMAL INTERIOR PENALTY DISCONTINUOUSAPPROXIMATIONS OF SYMMETRIC ELLIPTIC PROBLEMS ONNON-MATCHING GRIDSR.D. LAZAROV, J.E. PASCIAK, J. SCH¨OBERL, AND P.S. VASSILEVSKIAbstract. We consider an interior penalty discontinuous approximation for sym-metric elliptic problems of second order on non–matching grids in this paer. Themain result is an almost optimal error estimate for the interior penalty approxi-mation of the original problem based on the partition of the domain into a finitenumber of subdomains. Further, an error analysis for the finite element approxi-mation of the penalty formulation is given. Finally, numerical experiments on aseries of model second order problems are presented.1. IntroductionIn this paper, we propose and analyze a simple strategy to construct compositediscretizations of self-adjoint second order elliptic equations on non–matching grids.The need for discretizations on non–matching grids is motivated partially from thedesire for parallel discretization methods (including adaptive) for PDEs, which is amuch easier task if non–matching grids are allowed across the subdomain boundaries.Another situation may arise when different discretizations techniques are utilized indifferent parts of the subdomains and there is no aprioriguarantee that the mesheswill be aligned.Our method can be described as interior penalty approximation based on partiallydiscontinuous elements. The mortar method is a general technique for handling dis-cretizations on non–matching grids. However, our motivation for using the penaltyapproach is that it eliminates the need for additional (Lagrange multiplier or mor-tar) spaces. There is a vast number of publications devoted to the mortar finiteelement method as a general strategy for deriving discretization methods on non–matching grids. We refer the interested reader to the series of Proceedings of theInternational Conferences on Domain Decomposition Methods cf. e.g. [5], [11], [17](for more information see, http://www.ddm.org).In the present paper, we assume a model situation when the domain is splitinto a fixed number of non–overlapping subdomains and each subdomain is meshedDate: August 31, 2000–beginning; Today is August 24, 2001.1991 Mathematics Subject Classific ation. 65F10, 65N20, 65N30.Key words and phrases. non–matching grids, interior penalty discretization, error estimates.The work of the first and the second authors has been partially supported by the NationalScience Foundation under Grant DMS-9973328. The work of the last author was performed underthe auspices of the U. S. Department of Energy by University of California Lawrence LivermoreNational Laboratory under contract W-7405-Eng-48.12 R.D. LAZAROV , J.E. PASCIAK, J. SCH¨OBERL, AND P.S. VASSILEVSKIindependently. This is a non-conforming method and the functions are discontinuousacross the subdomain interfaces. The jump in the values of the functions along theseinterfaces is penalized in the variational formulation, a standard approach in theinterior penalty method (cf. [2], [4], [14], [22]). For a recent comprehensive surveyon this subject see [3]. An important feature of this approach is that we skip theterm in the weak formulation that involves the co-normal derivative of the solutionto the interface boundaries since the latter may lead to non–symmetric discretization(cf. [22]) of the original symmetric positive definite problem. An interior penaltyfinite element approximation with optimal condition number was proposed, studied,and tested on various examples in [20]. The error estimates derived in [20] weresuboptimal with a loss of factor h1/2−δ, 0 <δ<1/2 for solutions in the Sobolevspace H2−δ(Ω). In this paper we present a refined analysis and get almost optimalerror estimates for linear finite element and solutions in H2−δ(Ω). In addition,we extend the analysis to decompositions with cross points. One can improve theaccuracy somewhat for problems with smooth solutions by increasing the weight inthe penalty term with the expense of increased condition number.In the case of matching grids, finite element Galerkin method with penalty for aclass of problems with discontinuous coefficients (interface problem) has been studiedin [4]. Similarly, in [10], the interface problem has been addressed by recasting theproblem as a system of first order (by introducing the gradient of the solution as anew vector variable) and applying the least–squares method for the system. Integralsof the squared jumps in the scalar and the normal component of the vector functionson the interface are added as penalty terms in the least–squares functional. In bothcases an optimal with respect to the error method leads to a non-optimal conditionnumber of the discrete problem.Other approaches for handling discretizations on non–matching grids can involvedifferent discretizations in the different subdomains. For example, mixed finite el-ement method in one subdomain and standard Galerkin on the other (proposed in[25] and studied further in [18]), mixed finite element method and discontinuousGalerkin method cf. e.g., [13], or mixed finite element discretizations in both sub-domains, cf. e.g., [1], [19]. Similarly, coupling finite volume and Galerkin methodshas been proposed and studied in [15].The structure of the present paper is as follows. In Section 2, we formulate theproblem and its discretization. In Section 3, we introduce the primal and dualpenalty formulations of the problem split into subproblems on nonoverlapping sub-domains. In order to get an optimal estimate for the error in Section 4, we introducethe mixed formulation of the penalty problem and derive a fundamental apriorier-ror estimate for its solution. In Section 5, we analyse the difference between thesolution of the original problem and the solution of the penalty formulation. Theerror is shown to be of almost optimal order for u ∈ H2−δ(Ω) for δ ≥ 0. For meth-ods without cross-points the error is oprimal for 1/2 >δ>0. Finally, the finiteelement discretization and its error analysis is presented in Section 6. Numericaltest illustrating the accuracy of the method are given for two model problems.INTERIOR PENALTY APPROXIMATIONS 32. Notations and problem formulationIn this paper we use the standard notations for Sobolev spaces of functions definedin a bounded domain Ω ⊂Rd, d =2, 3. For example, Hs(Ω) for s integer denotesthe Hilbert space of