Convergence and stability of finite difference schemes for some elliptic equations

Abstract

The problem of convergence and stability of finite difference schemes used for solving boundary value problems for some elliptic partial differential equations has been studied in this thesis. Generally a boundary value problem is first replaced by a discretized problem whose solution is then found by numerical computation.

The difference between the solution of the discretized problem and the exact solution of the boundary value problem is called the discretization error. This error is a measure of the accuracy of the numerical solution, provided the roundoff error is negligible. Estimates of the discretization error are obtained for a class of elliptic partial differential equations of order 2m (M ≥ 1) with constant coefficients in a general n-dimensional domain. This result can be used to define finite difference approximations with an arbitrary order of accuracy.

The numerical solution of a discretized problem is usually obtained by solving the resulting system of algebraic equations by some iterative procedure. Such a procedure must be stable in order to yield a numerical solution. The stability of such an iteration scheme is studied in a general setting and several sufficient con­ditions of stability are obtained.

When a higher order differential equation is solved numeri­cally, roundoff error can accumulate during the computations. In order to reduce this error the differential equation is sometimes replaced by several lower order differential equations. The method of splitting is analyzed for the two-dimensional biharmonic equation and the convergence of the discrete solution to the exact solution is discussed. An iterative procedure is presented for obtaining the numerical solution. It is shown that this method is also applicable to non-rectangular domains.

The accuracy of numerical solutions of a nonselfadjoint elliptic differential equation is discussed when it is solved with a finite non-zero mesh size. This equation contains a parameter which may take large values. Some extensions to the two-dimensional Navier-Stokes equations are also presented.