A discontinuous enrichment method for the finite element solution of high Péclet advection–diffusion problems

Abstract

A discontinuous enrichment method (DEM) for the efficient finite element solution of the two-dimensional advection–diffusion equation is presented. Following the general DEM, the standard Galerkin polynomial field is locally enriched with free-space solutions of the homogeneous and constant-coefficient version of the governing partial differential equation. For the advection–diffusion equation, the free-space solutions are exponential functions that exhibit a steep gradient in the advection direction. The continuity of the solution across the element boundaries is weakly enforced by a carefully discretized Lagrange multiplier field. Preliminary results for previously published benchmark problems reveal that the DEM elements proposed in this paper are significantly more competitive than their Galerkin and stabilized Galerkin counterparts, especially in advection-dominated (high Péclet number) flows. Whereas spurious oscillations are known to pollute the standard Galerkin solution unless a very fine mesh is used, the DEM solution is shown to deliver an impressive accuracy at low mesh resolution.