Finite element problems can often naturally be divided into subproblems which correspond to subregions into which the region has been partitioned or from which it was originally assembled. A class of iterative methods is discussed in which these subproblems are solved by direct methods, while the interaction across the curves or surfaces which divide the region is handled by a conjugate gradient method.A mathematical framework for this work is provided by regularity theory for elliptic finite element problems and by block Gaussian elimination. A full development of the theory, which shows that certain of these methods are optimal, is given for Lagrangian finite element approximations of second order linear elliptic problems in the plane. Results from numerical experiments are also reported.

In this paper, we review the classical nonoverlapping domain decomposition (NODD) preconditioners, together with the newly developed multiscale control volume (MSCV) method. By comparing the formulations, we observe that the MSCV method is a special case of a NODD preconditioner. We go on to suggest how the more general framework of NODD can be applied in the multiscale context to obtain improved multiscale estimates.

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A new method for the numerical solution of the first biharmonic Dirichlet problem in a rectangular domain is presented. For an N x N mesh the complexity of this algorithm is on the order of N arithmetic operations. Only one array of order N and a workspace of size less than 10N are required.These results are therefore optimal and the algorithm is an order of magnitude more efficient than previously known methods with the possible exception of multi-grid. The method has an iterative part where a problem with different boundary conditions is used to precondition the original problem. It is shown that any initial error will be reduced by a factor e after at most k In (2/e) iterations using the conjugate gradient method.The conjugate gradient method is also shown to have a superlinear rate of convergence when applied to this formulation of the problem. The purpose of this paper is to provide a description and analysis of the new method.

An algorithm is described which computes the conformal mapping from the unit disk onto an arbitrary polygon having circular arcs as sides. This generalizes the Schwarz-Christoffel program of Trefethen (SIAM J. Sci. Star. Comp., (1980), pp. 82-102). Our algorithm must also determine certain parameters by solving a nonlinear least squares problem. Instead of using Gauss-Jacobi quadrature to evaluate the SchwarzoChristoffel integral, however, an ordinary differential equation solver is applied to a nonsingular formulation of the Schwarzian differential equation. The construction of a conformal mapping reduces simple elliptic partial differential equations on an irregular region to similar problems on a disk, for which existing programs can compute solutions very efficiently. Typical examples arise in the modeling of conductivity past an array of conducting cylinders and electrical fields inside a waveguide.