Multilevel Methods for Exponential Spline Discretizations of Convection-Diffusion Problems

Gerhard Starke

Institut für Praktische Mathematik, Universität Karlsruhe, 76128 Karlsruhe, Germany.


We present a study of multilevel algorithms for boundary value problems of convection-dominated convection-diffusion equations which are discretized by a Petrov-Galerkin scheme using exponential splines. Our approximation is based on a nested sequence of finite element spaces based on a quasi-uniform sequence of triangulations. It is well-known that the Galerkin method using standard linear finite elements on triangles does not produce physically meaningful approximations, in general, for convection-dominated problems. The use of exponential splines (representing local element-wise solutions of the convection-diffusion equation) as trial functions, however, leads to good approximations on rather coarse grids. In the context of the multilevel solution of the resulting linear system of equations this leads to stable coarse-grid operators naturally associated with the hierarchy of trial and test spaces. We investigate the convergence of multilevel (V-cycle) algorithms associated with different variants of Petrov-Galerkin schemes using exponential splines. This includes a set of computational experiments for some test problems on general triangulations.