The incremental unknown method I

Abstract

Incremental unknowns were introduced in [Temam 1990] as a means to approximate fractal attractors by using finite differences. However incremental unknowns also provide a new way for solving linear elliptic problems using several levels of discretization; the method is similar but different from the classical multigrid methods (see [Brandt 1984], [Hackbusch 1985], [McCormick 1987]). It is efficient and easy to implement. We also expect the method to be suitable for problems for which the utilization of the standard multigrid methods is difficult (see [Chen(a) et al.], [McCormick 1987]). In this article we describe the utilization of incremental unknowns for solving Laplace operator in dimension two. We provide some theoretical results concerning two-level approximations and we present the numerical tests done with the multi-level approximations. The numerical tests show that for this problem, the efficiency of the Incremental Unknown Method is comparable to the V-cycle multigrid method.