Robustness of an Elementwise Parallel Finite Element Method for Convection-Diffusion Problems

We consider an elementwise data-parallel finite element procedure, recently proposed by Layton and Rabier [Appl. Math. Lett., 5 (1992), pp. 67–70], [J. Numer. Linear Algebra Appl., 2 (1995), pp. 363–394], applied to singularly perturbed convection-diffusion equations with possibly highly anisotropic diffusion. It is shown that the number of iterations required for the solution of the linear algebraic system is proportional to the inverse of the smallest grid element diameter, uniformly in the diffusion parameter and the degree of anisotropy. This is optimal, since the method can in some cases use only element matrices and load vectors and the algorithm requires only local communication on the physical mesh between adjacent elemental subdomains. Our analysis includes both the usual Galerkin formulation and the streamline upwind finite element formulations. The convergence result holds with conforming elements of any order or elements of arbitrary order from a new family of nonconforming elements.