An Adaptive Mixed Finite Element Method using the Lagrange Multiplier Technique

Adaptive methods in finite element analysis are essential tools in the efficient computation and error control of problems that may exhibit singularities. In this paper, we consider solving a boundary value problem which exhibits a singularity at the origin due to both the structure of the domain and the regularity of the exact solution. We introduce a hybrid mixed finite element method using Lagrange Multipliers to initially solve the partial differential equation for the both the flux and displacement. An a posteriori error estimate is then applied both locally and globally to approximate the error in the computed flux with that of the exact flux. Local estimation is the key tool in identifying where the mesh should be refined so that the error in the computed flux is controlled while maintaining efficiency in computation. Finally, we introduce a simple refinement process in order to improve the accuracy in the computed solutions. Numerical experiments are conducted to support the advantages of mesh refinement over a fixed uniform mesh