Numerical Analysis of the Enriched Mixed Method, Evaluating Computational Costs and Errors through Comparisons with Formulations in Continuous Functions, Discontinuous Functions, and Mixed Finite Elements

In diffusion problems, mixed finite element formulation are implemented to simultaneously approximate primal and dual variables. A new discrete mixed approximation has been devised by Devloo, which utilizes an enhanced version of conventional Poisson-compatible spaces. The objective of this chapter was to assess the computational cost of implementing this new formulation in conjunction with three other classical formulations to solve three Poisson problems that have known exact solutions in two and three dimensions. The goal is not to determine the preferable formulation, but rather to stress the computational expense and the errors that are derived for both the primal and dual quantities. Weak formulations include the mixed method, the enriched mixed method, and the finite element method, which uses both continuous and discontinuous functions. The computational expenses of the assembly and solution procedures are evaluated, and the errors in and norms are computed. It is evident that the computational expenditure of the enriched version is lower than that of the non-enriched version when these costs are compared to the resulting errors. However, their approximation errors are quite similar. The improved version is perceived as an optimized iteration of the mixed finite element method. Three of these formulations exhibit a close equivalence.