A numerical investigation of Schwarz domain decomposition techniques for elliptic problems on unstructured grids

We consider a parallel implementation of the additive two-level Schwarz domain decomposition technique. The procedure is applied to elliptic problems on general unstructured grids of triangles and tetrahedra. A symmetric, positive-definite system of linear equations results from the discretization of the differential equations by a standard finite-element technique and it is solved with a parallel conjugate gradient (CG) algorithm preconditioned by Schwarz domain decomposition. The two-level scheme is obtained by augmenting the preconditioning system by a coarse grid operator constructed by employing an agglomeration-type algebraic procedure. The algorithm adopts an overlap of just a single layer of elements, in order to simplify the data-structure management involved in the domain decomposition and in the matrix-times-vector operation for the parallel conjugate gradient. Numerical experiments have been carried out to show the effectiveness of the procedure and they, in turn, show how even such a simple coarse grid operator is able to improve the scalability of the algorithm.