Adaptive Mixed Finite Element Method for Elliptic Problems with Concentrated Source Terms

An adaptive mixed finite element method using the Lagrange multiplier technique is used to solve elliptic problems with delta Dirac source terms. The problem arises in the use of Chow-Anderssen linear functional methodology to recover coefficients locally in parameter estimation of an elliptic equation from a point-wise measurement. In this article, we used a posterior error estimator based on averaging technique as refinement indicators to produce a cycle of mesh adaptation, which is experimentally shown to capture singularity phenomena. Our numerical results showed that the adaptive refinement process successfully refines elements around the center of the source terms. The results also showed that the global error estimation is better than uniform refinement process in terms of computation time.