Incremental Unknowns for Solving Partial Differential Equations

Abstract

Incremental unknowns have been proposed in Temam (1990) as a method to approximate fractal attractors by using finite difference approximations of evolution equations. In the case of linear elliptic problems, the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization; the method is similar but different from the classical multigrid method.

In this article we describe the application of incremental unknowns for solving Laplace equations in dimensions one and two. We provide theoretical results concerning two-level approximations and we report on numerical tests done with multi-level approximations.