HIERARCHICAL ERROR ESTIMATORS FOR LOWEST-ORDER MIXED FINITE ELEMENT METHODS

Kwang-Yeon Kim

doi:10.11568/kjm.2014.22.3.429

Abstract

In this work we study two a posteriori error estimators of hierarchical type for lowest-order mixed finite element methods.One estimator is computed by solving a global defect problem based on the splitting of the lowest-order Brezzi--Douglas--Marini space, and the other estimator is locally computable by applying the standard localization to the first estimator. We establish the reliability and efficiency of both estimators by comparing them with the standard residual estimator. In addition, it is shown that the error estimator based on the global defect problem is asymptotically exact under suitable conditions.

Highlights

In this paper we consider the second-order elliptic problem on a bounded polygonal domain− div(a∇u) = f in Ω ⊂ R2 (1)u = −g on ∂Ω for given f ∈ L2(Ω) and g ∈ H1/2(∂Ω)
In the past two decades much effort has been devoted to development of a posteriori error estimators for mixed finite element methods of (2); see, for example, [1, 2, 4, 5, 6, 7, 8, 10, 11, 13, 14] and references therein
In this work we study two a posteriori error estimators of hierarchical type for the lowest-order mixed finite element methods mentioned above

Summary

Introduction

In this paper we consider the second-order elliptic problem on a bounded polygonal domain. Key words and phrases: a posteriori error estimation, mixed finite element method, hierarchical error estimator, asymptotic exactness. Adaptive refinement based on a posteriori error estimators is a well-established tool for efficient implementation of finite element methods. In the past two decades much effort has been devoted to development of a posteriori error estimators for mixed finite element methods of (2); see, for example, [1, 2, 4, 5, 6, 7, 8, 10, 11, 13, 14] and references therein. In this work we study two a posteriori error estimators of hierarchical type for the lowest-order mixed finite element methods mentioned above. The first estimator based on the global defect problem is asymptotically exact under suitable conditions.

Preliminaries

Hierarchical Error Estimators

Reliability and Efficiency

Asymptotic Exactness