Local error indicator in finite element analysis of Laplacian fields based on the green integration formula

Abstract

SUMMARYThe paper presents a simple and practical method for the local error assessment in the finite element solution of fields in linear media governed by the Laplace equation. The local error is estimated for each node or element of the mesh model by applying the Green integration formula, with the error distribution being used for the adaptive mesh refinement. Several alternatives of the local error estimation algorithm are compared and numerical examples are given for a typical %' -section potential problem. These confirm the efficiency of the proposed method.

Estimating accuracy of electrostatic finite element models

“…Owing to the impracticality of exhaustive laboratory evaluation, ÿnite element algorithms are commonly employed [3,4]. With this arises the requirement for a reliable system of estimating the error in these approximations at minimal expense [5]. Although ÿnite element techniques are well established, as the true electric ÿeld distribution is usually unknown and the reliability of ÿnite element results can vary greatly between models, solution accuracy is di cult to quantify [5,6].…”

“…There are several established approaches to estimating the error in ÿnite element results [5,7]. Methods are often based on discontinuity across elements.…”

“…These generally provide a numerical measure of discretization error, but tend to gauge comparative accuracy of regions, rather than give the absolute solution error [7]. Other methods to estimate accuracy include comparing results with independent or reÿned approximations [5,7]. Potentially the most meaningful error estimates are based on dual schemes.…”

“…For example, in electrostatic analyses, Maxwell's equations are exploited to derive D and E and the error estimated by their comparison using D = E [6]. Although this may provide a boundary of the exact error [5,7], to attain high ÿeld accuracy is very computationally demanding and, in reality, for example, the iteration is aborted before the speciÿed ÿeld accuracy is met, based on the convergence of other quantities e.g. energy [6,8].…”

Meshing methods and adaptive algorithms in two and three dimensions for solving closed electromagnetic problems by means of the finite element method

“…Estos estimadores, en su mayor parte, pueden catalogarse también como estimadores de recuperación. • Basados en la fórmula de integración de Green [Ada92], que estiman el error cometido comparando el valor de la solución en un nodo con el obtenido a partir de la fórmula de integración de Green en un contorno alrededor del nodo. • Basados en la variación de alguna magnitud en dos soluciones MEF, y asumiendo que una de ellas es más precisa que la otra [Gol93].…”