A Post-processing Technique for Streamline Upwind/Petrov-Galerkin for Advection Dominated Partial Differential Equations

TL;DR

This paper introduces a simple post-processing method to construct locally conservative fluxes from finite element solutions of advection diffusion equations, improving flux conservation in numerical simulations.

Contribution

It presents a novel post-processing technique applicable to both standard and SUPG finite element methods for advection-dominated problems, ensuring flux conservation.

Findings

The technique produces locally conservative fluxes on a dual mesh.

Convergence analysis confirms the method's effectiveness.

Numerical examples demonstrate improved flux accuracy.

Abstract

We consider the construction of locally conservative fluxes by means of a simple post-processing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields non-conservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element (CGFEM) for solving non dominating advection diffusion equations and the streamline upwind/Petrov-Galerkin (SUPG) for solving advection dominated problems. We then describe the post-processing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The post-processing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the post-processing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift-diffusion equations.