Convergence of an adaptive lowest-order Raviart–Thomas element method for general second-order linear elliptic problems

Abstract

In this article, the convergence of an adaptive mixed finite element method for general second-order linear elliptic problems defined in bounded polygonal domains is analysed. The main difficulty in the analysis is posed by the nonsymmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools used are a posteriori error estimators, a quasi-orthogonality property and a quasi-discrete reliability result. These results are established using a representation formula for the mixed lowest-order Raviart–Thomas solution in terms of the nonconforming Crouzeix–Raviart solution of the problem under the assumption that the initial mesh size is small enough. For the local refinement, in each step an adaptive marking is chosen which is based on the comparison of the edge residual and volume residual terms of the a posteriori estimator. Numerical experiments confirm the theoretical convergence.