Convergence properties of the Lagrange-Galerkin method with and without exact integration

Abstract

The properties of the Lagrange-Galerkin finite element method are investigated for advection and advection-diffusion problems. In one dimension, we investigate convergence rates of the method for a variety of initial data of differing smoothness on uniform and non-uniform meshes, with linear and quadratic elements. In two dimensions, we carry out numerical experiments on two test problems. The first is the advection problem. The properties of area-weighting and the exactly integrated scheme are compared and a refinement of area-weighting called compound area-weighting introduced to improve accuracy. Comparison is also made between the performance of bilinear and biquadratic elements. The other test case is an advection diffusion problem - scalar transport in a reversed flow field.