Post-processing for spatial accuracy-enhancement of pure Lagrange-Galerkin schemes applied to convection-diffusion equations

We analyze a technique to improve the spatial accuracy, by the single application at the end of the simulation of a local post-processing, for pure Lagrange-Galerkin (PLG) methods applied to evolutionary convection-diffusion (possibly pure convection/diffusion) equations with time-dependent domains. The post-processing technique is based on a simple convolution that extracts the 'hidden accuracy' of Galerkin schemes, and it is used and rigorously analyzed in a fully discrete context.We prove that, when applied to the numerical solution of PLG schemes, it improves the spatial accuracy in the l∞ (L2(ω0))-norm from order k+1 to at least order 2k, where k is the degree of the polynomials defining the finite element space and ω0 any interior region of the computational domain meshed with translation-invariant meshes. For pure convection, a spatial accuracy enhancement in the l∞ (L2(ω0))-norm from order k+1 to order 2k+2 is obtained by post-processing the numerical solution of PLG schemes. Numerical tests are presented that confirm these theoretical results.