Gaussian basis representation FIR filtering for the numerical differentiation of Laplacian FE solutions

An innovative approach for an accurate numerical evaluation of derivatives of Laplacian finite-element (FE) solutions is presented. The technique is based on a suitable numerical filtering of the FE solution values. The filtering scheme is based on Gaussian basis representation (GBR) finite impulse response (FIR) filters and employs three kinds of filters, whose main specifications are related to the kernels of the Poisson integrals (PIs) for the Laplace problem. Even from rough (first order) FE discretizations, very accurate estimates for derivatives are obtained with a moderate computational effort. Comparisons against analytical solutions are provided in order to show the achievable degree of accuracy. An example of application to a practical problem is also given, and the obtained results are compared with standard numerical differentiation techniques.