Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 75, pp. 1-26.
Title: Constructive Sobolev gradient preconditioning
for semilinear elliptic systems
Author: Janos Karatson (ELTE Univ., Budapest, Hungary)
Abstract:
We present a Sobolev gradient type preconditioning for iterative
methods used in solving second order semilinear elliptic systems;
the n-tuple of independent Laplacians acts as a preconditioning
operator in Sobolev spaces. The theoretical iteration is done
at the continuous level, providing a linearization approach that
reduces the original problem to a system of linear Poisson
equations. The method obtained preserves linear convergence when
a polynomial growth of the lower order reaction type terms is involved.
For the proof of linear convergence for systems with mixed boundary
conditions, we use suitable energy spaces. We use Sobolev embedding
estimates in the construction of the exact algorithm.
The numerical implementation has focus on a direct and
elementary realization, for which a detailed discussion and
some examples are given.
Submitted March 18, 2004. Published May 21, 2004.
Math Subject Classifications: 35J65, 49M10.
Key Words: Sobolev gradient; semilinear elliptic systems;
numerical solution; preconditioning.