Projected successive overrelaxation method for finite-element solutions to the Dirichlet problem for a system of nonlinear elliptic equations

Abstract

In this paper, we consider the projected successive overrelaxation (SOR) method for obtaining finite-element solutions applied to the Dirichlet problem for a system of nonlinear elliptic equations. These equations arise in gas dynamics and chemical reactions. The Jacobian matrix for the nonlinear equations is not symmetric, so that there is no longer an associated minimization problem. A convergence proof of the projected SOR method is established by using a contraction argument, because minimization techniques are not applicable. We also discuss the optimum relaxation parameter, based upon the linear SOR theory. Finally, we show some numerical examples to indicate the effectiveness of the projected SOR method.