Relaxation Methods for Convex Problems

Abstract : Extensions and simplifications are made for convergence proofs of relaxation methods for nonlinear systems arising from the minimization of strictly convex functions. This work extends these methods to group relaxation, which includes an extrapolated form of Newton's method, for various orderings. A relatively simple proof is given for cyclic orderings, sometimes referred to as nonlinear overrelaxation, and for residual orderings where an error estimate is given. A less restrictive choice of relaxation parameter is obtained than that previously. Applications are indicated primarily to the solution of nonlinear elliptic boundary problems. (Author)