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

TL;DR

This paper develops a stable, full space-time finite element method for solving the advection-diffusion equation, incorporating Galerkin/least-square stabilization, and demonstrates its accuracy and efficiency through error estimates and numerical examples.

Contribution

It introduces a full space-time finite element approach with Galerkin/least-square stabilization for the advection-diffusion equation, including new a priori and a posteriori error estimates.

Findings

The method achieves stable and accurate solutions.

Numerical examples confirm convergence and effectiveness.

Adaptive mesh refinement improves efficiency.

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 against analytical solutions as well as against numerical solutions using a conventional time-marching algorithm.