Study of the enriched mixed finite element method comparing errors and computational cost with classical FEM and mixed scheme on quadrilateral meshes

Abstract

Mixed finite element formulations are used to approximate stress and displacement variables simultaneously for Poisson problems. The purpose of this article is to analyze a new discrete mixed approximation based on the application of the enriched version of classic Poisson-compatible spaces. With that purpose we decided to measure the computational cost of applying three formulations for two Poisson problems with known exact solution. The objective is not to compare which formulation is better, but rather to highlight characteristics of computational cost and the errors obtained for both the primal and dual variables. Weak formulations corresponding to the application of the classical H1-conforming, the mixed method and the enriched mixed method. In the developed algorithms, we computed the error of the primal variable and we measured the computational cost of the assembly and the solving processes. When analyzing these costs together with the errors obtained, we visualized that the cost of the enriched version of order k, is less expensive computationally than the non-enriched version of order k+1, however getting them the same approximation errors.