Space-time finite element analysis of the advection-diffusion equation using Galerkin/least-square stabilization

Abstract

We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method on conforming meshes. The Galerkin/least-square method is employed to ensure stability of the discrete variational problem. In the full space-time formulation, time is considered another dimension, and the time derivative is interpreted as an additional advection term of the field variable. We derive a priori error estimates and illustrate spatio-temporal convergence with several numerical examples. We also derive a posteriori error estimates, which coupled with adaptive space-time mesh refinement provide efficient and accurate solutions. The accuracy of the space-time solutions is illustrated by comparing against analytical solutions as well as against numerical solutions using a conventional time-marching algorithm.