Nodally exact ritz discretizations of 1D diffusion-Absorption and helmholtz equations by variatoinal FIC and modified equation methods

This article presents the first application of the Finite Calculus (FIC) in a ritz-FEM variational framework. FIC provides a steplength parametrization of mesh dimensions, which is used to modify the shape functions. This approach is applied to the FEM discretization of the steady-state, one-dimensional, diffusion-absorption and Helmholtz equations. Parametrized linear shape functions are directly inserted into a FIC functional. The resulting Ritz-FIC equatiolns are symmetric and carry a element-level free parameter coming from the function modification process. Both constant-and variable-coefficient cases are studied. It is shown that the parameter can be used to produce nodally exact solutions for the constant coefficient case. The optimal value is found by matching the finite-order modified differential equation (FOMoDE) of the Ritz-FIC equations with the original field equation. The three ingredients of the method (FIC, Ritz and MoDE) are extendible to multiple dimensions.