A finite difference method for solving the third boundary value problem for elliptic and parabolic equations in an arbitrary region. Iterative solution of finite difference equations. I

Abstract

DIFFERENCE schemes are constructed on a uniform mesh for solving the third boundary value problem for a second order elliptic equation in a two-dimensional region; the schemes are suitable for solving such equations by simple iterations and by the method of alternating directions. An estimate in the norm of W 2 1 was obtained in [1] for the rate of convergence of finite-difference schemes of the variational type for solving the first and third boundary value problems for elliptic equations in a two-dimensional region; the estimate was of order 0(h 1.5). Irregular triangulation at the region boundary was used for the schemes. The present paper also used the variational approach and obtains a similar rate of convergence estimate for the solutions of difference schemes employing irregular triangulation, though the latter is more natural, see [2], for the third boundary value problem. Notice that the difference schemes thus obtained (see [2–4]) have the following drawback: there is no limit to how ill posed the matrices of the sets of difference equations are. In Section 1 difference schemes are constructed, suitable for solving the equations by a modification of a well-known iterative method such as simple iterations or the method of alternating directions. Section 2 presents auxiliary propositions. In Section 3, the convergence rate of the solutions of the difference equations to the solution of the differential problem is investigated. Section 4 proves the convergence of the method of simple iterations and finds the number of iterations required to achieve a given accuracy. The convergence of the iterative process is proved under conditions on the differential operator of the problem such as are almost, necessary for the method of simple iterations to be applicable.