Lumped Element Method - A Discrete Calculus Approach for Solving Elliptic and Parabolic PDEs

In this report, we introduce a novel discrete calculus method for obtaining the numerical solution of parabolic and elliptic type partial differential equations. The discrete operators applied during the process of obtaining the numeric solution have the same advantageous properties as those of their continuous counterparts: orthogonality, conservation laws, and minimum-maximum principle. The results of our proposed solution method are interpreted in terms of the underlying physics and the material properties on the same graph. This can significantly simplify the solution of the discretised system that originates from the advantageous properties of discrete operators defined on weighted graphs. We demonstrate the applicability of the presented approach by using it to calculate the numeric solutions of an elliptic and a parabolic model problem and compare these results to the solutions of the same problems calculated using a well-known FEM solver.