Report Number: CS-TR-76-585
Institution: Stanford University, Department of Computer Science
Title: Numerical solution of nonlinear elliptic partial differential
equations by a generalized conjugate gradient method
Author: Concus, Paul
Author: Golub, Gene H.
Author: O'Leary, Dianne Prost
Date: December 1976
Abstract: We have studied previously a generallized conjugate gradient
method for solving sparse positive-definite systems of linear
equations arising from the discretization of ellilptic
partial-differential boundary-value problems. Here,
extensions to the nonlinear case are considered. We split the
original discretized operator into the sum of two operators,
one of which corresponds to a more easily solvable system of
equations, and accelerate the associated iteration based on
this splitting by (nonlinear) conjugate gradients. The
behavior of the method is illustrated for the minimal surface
equation with splittings corresponding to nonlinear SSOR, to
approximate factorization of the Jacobian matrix, and to
elliptic operators suitable for use with fast direct methods.
The results of numerical experiments are given as well for a
mildly nonlinear example, for which, in the corresponding
linear case, the finite termination property of the conjugate
gradient algorithm is crucial.
http://i.stanford.edu/pub/cstr/reports/cs/tr/76/585/CS-TR-76-585.pdf