High precision differentiation of FEM approximate solutions

Abstract

This paper presents the high precision differentiation method based on Green's second identity. The technique is compared to several recent methods based on local smoothing and superconvergent patch recovery (SPR). The methodology is extended to 3D problems described by scalar Poisson equation, using the sphere as a base domain for extraction of derivatives. Analytic verification and error sensitivity analysis is performed. The alternative approach employing fundamental solutions to the Dirichlet problem in place of Green's functions is also outlined. The technique is suited to postprocessing of finite element solutions, or may be applied to other numerical approximate solutions.