We de ne and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane. Our estimates show the error to be comparable to a standard Galerkin nite element method using piecewise linear polynomials.

Abstract. We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.1. Introduction. In this work, we will describe several methods for computing a posteriori error estimates for finite element calculations. That is, given some piecewise polynomial approximation U to uH, the solution of an elliptic partial differential equation, we seek some practical method for computing an estimate of \\\uH -U\\\ for an appropriate norm ||| • |||. A priori estimates can give asymptotic rates of convergence as the mesh parameter h tends to zero, but often cannot provide much practical information about the actual errors encountered on a given mesh with a fixed h. A posteriori estimates, on the other hand, attempt to provide the user of a finite element package with such information, enhancing the robustness of the package, and the reliability of the approximations it produces.There has been a great deal of recent work by Babuska and his co-workers on local mesh refinement strategies and the a posteriori error indicators necessary for their success [4]-[10], The indicator in [8], for example, is based on solving local Dirichlet problems in the patch of elements surrounding each vertex in the finite element mesh. In this scheme, the Dirichlet boundary conditions insure well-posedness of the local problems. Error indicators can also be based on the computation of the norm of the local residual of the elliptic equation and the jump in normal derivative of the computed solution at interelement boundaries (e.g., [9], [13]). Such schemes as a rule require less computation than the ones involving the solution of local problems. They also appear to give good results when used in local mesh refinement algorithms. However, with highly nonuniform triangular meshes, as arise with the finite element code PLTMG [11], it is sometimes difficult to weight the residual and boundary terms properly.

Abstract. A ¿-level iterative procedure for solving tbe algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed. The work estimate for this procedure is proportional to the number of unknowns, an optimal order result. General geometry is permitted for the underlying domain, but the shape plays a role in the rate of convergence through elliptic regularity. Finally, a short discussion of the use of this method for parabolic problems is presented.

Summary. We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric GauB-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigrid V-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In particular, we first show a superconvergence of the gradient of the finite element solution u h and to the gradient of the interpolant u I . We then analyze a postprocessing gradient recovery scheme, showing that Q h ∇u h is a superconvergent approximation to ∇u. Here Q h is the global L 2 projection. In Part II, we analyze a superconvergent gradient recovery scheme for general unstructured, shape regular triangulations. This is the foundation for an a posteriori error estimate and local error indicators. 1. Introduction. The study of superconvergence and a posteriori error estimates has been an area of active research; see the monographs by Verfürth [17], Chen and Huang [8], Wahlbin [18], Lin and Yan [16], and Babuška and Strouboulis [3] and a recent article by Lakhany, Marek, and Whiteman [13] for overviews of the field. In this two-part work we study some new superconvergence results. In Part I, we develop some superconvergence results for finite element approximations of a general class of elliptic partial differential equations (PDEs), based mainly on the geometry of the underlying triangular mesh. In Part II, we develop a gradient recovery technique that can force superconvergence on general shape regular meshes. Patch recovery techniques have been studied by Zienkiewicz and Zhu and this subject has itself evolved into an active subfield of research [25,14,23,24,9,22]. Although our algorithm in some respects resembles this and other similar schemes [12,19,4,6,2,10], it draws much of its motivation from multilevel iterative methods.Let Ω ⊂ R 2 be a bounded domain with Lipschitz boundary ∂Ω. For simplicity of exposition, we assume that Ω is a polygon. We assume that Ω is partitioned by a shape regular triangulation T h of mesh size h ∈ (0, 1). Let V h ⊂ H 1 (Ω) be the corresponding continuous piecewise linear finite element space associated with this triangulation T h , and u h ∈ V h be a finite element approximation to a second order elliptic boundary value problem.Our development has three main steps. In the first step, we prove a superconvergence result for |u h − u I | 1,Ω , where u I is the piecewise linear interpolant for u. In

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We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.