Uniform estimate for characteristics-mixed finite element method for two-dimensional advection-dominated transport problems

Lei Gao,Fuji Xie

doi:10.1186/s13662-017-1324-x

Abstract

In this paper, we prove uniform optimal-order error estimates for characteristics-mixed finite element methods for two-dimensional convection-dominated diffusion equations. The generic constants in the error estimates do not explicitly depend on the scaling diffusion parameter ε, but depend linearly on certain Sobolev norms of the true solution. Combining the estimates with the stability estimates of the true solution, we prove that these constants depend only on the initial and the right-hand side data. Numerical experiments are presented to confirm our theoretical findings.

Highlights

The characteristics-mixed finite element method is an efficient numerical scheme frequently used to solve the two-dimensional advection-dominated transport problem and tends to generate accurate numerical solutions for both the concentration and the flux if large time steps are used
Much research has been conducted on the error estimates of the characteristics-mixed finite element method in the context of linear convection-diffusion equations [ – ] or coupled systems for flow and transport in porous media [ – ]
The characteristics-mixed finite element methods proposed in [, ] are mass-conservative, which is crucial in the practical applications

Summary

Introduction

The characteristics-mixed finite element method is an efficient numerical scheme frequently used to solve the two-dimensional advection-dominated transport problem and tends to generate accurate numerical solutions for both the concentration and the flux if large time steps are used. In order to overcome these difficulties, in this article, enlightened by the ideas in [ – ] for the Eulerian-Lagrangian localized adjoint method and the modified method of characteristics, we use an interpolation operator and Raviart-Thomas projection to replace the mixed elliptic projections and prove the uniform error estimates for characteristics-mixed finite element schemes for time dependent convection-diffusion equations with a periodic boundary condition. Of [ ], an H -function ζhn was found to approximate enh and the error estimate of ζhn – enh in the L ( )-norm was derived We introduce this result by the following lemma. We rewrite ρn – ρn– as the sum (ρn – ρn– ) + (ρn– – ρn– ) and apply the triangle inequalities to get ε enh , ρn

Qε enh

To prove the bound of

Note that