Boundary element procedure for computation of internal directional derivatives in homogeneous Laplace's problems solved by the finite element method

Abstract

Using the standard approach of the Finite Element Method, numerical values of the spatial derivatives of the primal variable present low accuracy. It is due to the order reduction in the interpolation functions used for the approximation of the derivatives. Aiming to overcome this problem, this work uses the integral equation of the Boundary Element Method to recalculate new values of internal variables, using the nodal boundary values obtained by the Finite Element solution. The mathematical foundation for this procedure is based on the idea that the boundary integral equation is equivalent to a weighted residual sentence, and its reuse implies a new minimization of numerical errors. This strategy was previously used within the Boundary Element context to successfully recalculate nodal variables in scalar problems governed by Laplace and Poisson Equations, also used successfully for solving linear elastic problems expressed by Navier's Equation. Here, to confirm the consistency of the proposed model, computational tests are performed, in which the standard FEM results are compared with those obtained by the proposed procedure. Internal potential derivatives and also internal potential values were recalculated. To assess the quality of the solution, the results are compared to a benchmark.