An h-adaptive SUPG-FEM solution of the pure advection equation

Abstract

A standard h-adaptive finite element procedure based on a-posteriori error estimation is described. The first order wave equation (pure advection) is solved using the SUPG (streamline upwind Petrov—Galerkin) form of the finite element method. A benchmark problem is solved which has a uniform flow advecting a function with a boundary discontinuity. The SUPG method on its own is insufficient to resolve the sharp discontinuity present in the benchmark problem when used with a uniform mesh. Although the solution is a significant improvement on other methods, it still suffers with sharp overshoots and undershoots on either side of the discontinuity. The well-known plague of numerical models of advection, false diffusion, is also evident downstream of the discontinuity boundary. The h-adaptive procedure is then used in combination with the SUPG formulation and after a number of adaptive cycles (depending upon a preset tolerance value of error) produces a high quality solution.