Dedicated to ProfessorJoachim Nitsche on the occasion of the sixtieth anniversary of his birthday. Abstract.We consider the problem of solving the algebraic system of equations which arise from the discretization of symmetric elliptic boundary value problems via finite element methods. A new class of preconditioners for these discrete systems is developed based on substructuring (also known as domain decomposition). The resulting preconditioned algorithms are well suited to emerging parallel computing architectures.The proposed methods are applicable to problems on general domains involving differential operators with rather general coefficients. A basic theory for the analysis of the condition number of the preconditioned system (which determines the iterative convergence rate of the algorithm) is given. Techniques for applying the theory and algorithms to problems with irregular geometry are discussed and the results of extensive numerical experiments are reported.1. Introduction. The aim of this series of papers is to propose and analyze methods for efficiently solving the equations resulting from finite element discretizations of second-order elliptic boundary value problems on general domains in R2 and R3. In particular, we shall be concerned with constructing easily invertible and "effective" preconditioners for the resulting system of discrete equations which can be used in a preconditioned iterative algorithm to achieve a rapid solution method. The methods to be presented are well suited to parallel computing architectures.In this paper we shall restrict ourselves to boundary value problems in R2. Let fî be a bounded domain in R2 with a piecewise smooth boundary dû. As a model problem for a second-order uniformly elliptic equation we shall consider the

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Abstract.Existence, uniqueness and error estimates for Ritz-Galerkin methods are o discussed in the case where the associated bilinear form satisfies a Carding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given.In this note, we would like to discuss existence, uniqueness and estimates over the whole domain for some Ritz-Galerkin methods where the bilinear form satisfies o a Garding type inequality, i.e., it is indefinite in a special way. We shall first illustrate the problem by an example.For simplicity, let £2 be a simply connected convex region in the plane with a polygonal boundary d£2 and consider the Dirichlet problem

Interior a priori error estimates in Sobolev norms are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain ÍÍ can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain ii. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given. 0. Introduction. There are presently many methods which are available for computing approximate solutions of elliptic boundary value problems which may be classified as Ritz-Galerkin type methods. Many of these methods differ from each other in some respects (for example, in how they treat the boundary conditions) but have much in common in that they have what may be called "interior Ritz-Galerkin equations" which are the same. Here we shall be concerned with finding interior estimates for the rate of convergence for such a class of methods which are consequences of these interior equations. Let us briefly describe, in a special case, the type of question we wish to consider. Let £2 be a bounded domain in R^ with boundary 9£2 and consider, for simplicity, the problem of finding an approximate solution of a boundary value problem (0.1) Am =/ in £2, (0.2) Au =g on 9£2, where A is some boundary operator. Suppose now that we are given a one-parameter family of finite-dimensional subspaces Sh (0 < A < 1) of an appropriate Hilbert space in which u lies and that, for each A, we have computed an approximate solution un GSh to u using some Ritz-Galerkin type method. Here we have in mind, for example, methods such as the "engineer's" finite element method [8], [22], the Aubin-BabusTca penalty method [2], [4], the methods of Nitsche [12], [13] or the

Abstract.Existence, uniqueness and error estimates for Ritz-Galerkin methods are o discussed in the case where the associated bilinear form satisfies a Carding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given.In this note, we would like to discuss existence, uniqueness and estimates over the whole domain for some Ritz-Galerkin methods where the bilinear form satisfies o a Garding type inequality, i.e., it is indefinite in a special way. We shall first illustrate the problem by an example.For simplicity, let £2 be a simply connected convex region in the plane with a polygonal boundary d£2 and consider the Dirichlet problem