Mixed Finite Element Method for Elliptic Partial Differential Equations

Abstract

This report has the main aim of comparing the Mixed Finite Element Method to the standard Finite Element Method. The other aim is to let the reader understand what these methods entail. The latter is done by first journeying through the theory behind the FEM. It is first explored in one dimension to keep the setting simple. Next, the Mixed FEM is explored. A new way of approximating the gradient from the found solution is constructed, using the basis of Finite Differences and the ideas from the Finite Volume Method. Following the theory is the implementation of the mentioned methods and the analysis of theresults. It yields that, when looking at the L2-norm, the classic FEM has a convergence order of 2, comparable to that of similar numerical methods. The Mixed FEM seemed to converge with an order of 4 in the same norm. Our constructed method only had a measly first order convergence, implying much greater accuracy of the Mixed FEM.