A metric-based adaptive mesh refinement criterion under constrain for solving elliptic problems on quad/octree grids

Abstract

In this work we propose and investigate the performance of a metric-based refinement criteria for adaptive meshing used for improving the numerical solution of an elliptic problem. We show that in general, when solving elliptic equations such as the Poisson-Helmholtz equation, the minimization of the interpolation error often used as local refinement criteria does not always guarantee the minimization of the total numerical error. Numerical and theoretical arguments are given to unveil the critical role of the mesh compression – the size aspect ratio between the finest cell size and the mean cell size of an adapted mesh – to determine whether the estimated error is purely local meaning that the interpolation error is a good enough error model for the total error or if other, non-local, sources of error need to be accounted for. We show through particular examples that a slightly sub-optimal mesh in terms of interpolation error may significantly reduce the total error of a numerical solution, depending on the value of the compression ratio and not on the number of grid points. Based on this observation, we propose a new method to exclude the grids where non-local errors can control the accuracy of the solution. This is achieved by an automatic estimation of the optimal compression ratio, which is imposed as an additional constraint in the minimal element size during the mesh adaptation process. The method is tested on quadtree and octree grids, showing very satisfactory performances in reducing the total numerical error despite the additional constrain imposed.