Gauß' adaptive relaxation for the multilevel solution of partial differential equations on sparse grids

Abstract

In combination with the multilevel principle, relaxation methods are among the most efficient numerical solution techniques for elliptic partial differential equations. Typical methods used today are derivations of the Gauß-Seidel or Gauß-Jacobi method. Recently it has been recognized that in the context of multilevel algorithms, the original method suggested by Gauß has specific advantages. For this method the iteration is concentrated on unknowns where fast convergence can be obtained by intelligently monitoring the residuals. We will present this algorithm in the context of a sparse grid multigrid algorithm. Using sparse grids the dimension of the discrete approximation space can be reduced additionally. 1 Introduction In this paper we study the numerical solution of elliptic partial differential equations (PDE). We assume the equation is given in the weak form a(u; v) = Z \Omega fv dz for all v 2 H 1 0 (\Omega\Gamma ; (1) where u 2 H 1 0 and f 2 L 2 and a : H 1 0 (\Omeg..