Discrete invariant imbedding and elliptic boundary-value problems over irregular regions

Abstract

The numerical solution of elliptic boundary-value problems has been the subject of a great deal of effort. At the present, the most popular methods for obtaining numerical results to these problems have been the finite difference techniques [ 11, [2]. S ince the system of linear algebraic equations arising from the finite difference approximation of a linear elliptic equation is sparse, many efficient iterative methods [3], [4] h ave been found to solve these equations. However, there are some difficulties associated with the iterative techniques which demonstrate the need for a practical direct (noniterative) technique. First, the rate of convergence of the more efficient iterative techniques such as successive overrelaxation [3] and alternating direction implicit methods [4] depends critically upon one or more parameters. The determination of optimal or near optimal parameters is a nontrivial mathematical problem since these parameters depend not only on the particular partial differential equation but also on the particular boundary values. In the nonlinear case, which is usually handled by quasilinearization [Sj, this problem is compounded since the partial differential equation changes in each qua&near iteration. Second, if the region is more general than a rectangle, the iterative techniques may either fail or have their rates of convergence decrease. The most serious problem, however, is concerned with what is meant by efficiency. In many situations, we are content with two or three place accuracy. However, we must solve the same problem with many different boundary conditions. A serious fault of the iterative techniques is that a given solution furnishes no information about other solutions. In this more general sense, the iterative techniques can be very inefficient. For these reasons, a direct method is desirable. We will show that a discrete version of invariant imbedding [6], [7] will provide such a method. We will develop the technique, first for a rectangular region. Then we will demonstrate how easily the method extends to irregular regions. Although our example will be Laplace’s equation, we will make no use of the theory of harmonic functions.