Evaluation of the GMRES Method Applied on the Stabilized Dynamic Diffusion Method

Abstract

In this work, the computational performance of the Dynamic Diffusion method is addressed when the GMRES method is used to solve the resulting linear system. The DD method, introduced by Arruda, Almeida and Dutra do Carmo (Dynamic Diffusion Formulations for Advection Dominated Transport Problems, to appear), is a two-scale model for transport problem, obtained by adding to the Galerkin formulation a nonlinear dissipative operator acting isotropically in all scales. The amount of the artificial diffusion is determined by the solution of the resolved scale at the element level yielding a self adaptive free parameter method. The discrete problem is solved by using the well known element-byelement and edge-based storage local data schemes to optimize the matrix-vector product in the GMRES algorithm. Comparisons between these two storage schemes are addressed for a variety of numerical experiments covering advection dominated regimes. Our experiments have shown that the edge-based storage scheme leads to less CPU time and, since the resulting matrix is not well conditioned for some problems, the GMRES algorithm might fail for some dimensions of restart vectors.