Optimal Convergence Analysis for Convection Dominated Diffusion Problems

M A Mohamed Ali

doi:10.4236/jamp.2013.13004

Abstract

In classical mixed finite element method, the choice of the finite element approximating spaces is restricted by the imposition of the LBB consistency condition. The method of H1-Galerkin mixed finite element method avoids completely the imposition of such a condition on the approximating spaces. In this article, we discuss and analyze error estimates for Convection-dominated diffusion problems using H1-Galerkin mixed finite element method, along with the method of characteristics. Optimal order of convergence has been achieved for the error estimates of a two-step Euler backward difference scheme.

Highlights

The convection-dominated diffusion problems have been treated heavily using finite element methods [1,2,3,4]
In classical mixed finite element method, the choice of the finite element approximating spaces is restricted by the imposition of the LBB consistency condition
An H1-Galerkin mixed finite element method has been discussed for a class of second order Schrödinger equation by LIU et al [12]

Summary

Introduction

The convection-dominated diffusion problems have been treated heavily using finite element methods [1,2,3,4]. Mixed finite element method has been proposed by Douglas et al [5] These methods need to satisfy the Ladyzhenskaya-Babuska-Brezzi (LBB), consistency condition [6,7,8,9], on the approximating spaces which restrict the choice of the finite element spaces. Other standard techniques without unwinding produce unacceptable oscillations in the approximations These difficulties can be reduced substantially by using the Modified Method of Characteristics (MMOC). This procedure was introduced and analyzed for a single parabolic equation by Douglas [13] using backward single-step in the direction of characteristic. To utilize the above advantages for the convection dominated diffusion problems, we shall propose an H1-Galerkin mixed finite element method combined with the method of characteristics, and examine the rate of convergence for a Two-Step Euler backward difference scheme

Consider the Convection-Dominated Diffusion Problem c x u t

Second-Order Backward Euler Method

A Priori Error Estimates for Double-Step Backward Euler Method

C hk 1 v k 1

Conclusion