New approaches for error estimation and adaptivity for 2D potential boundary element methods

摘要

This work presents two new error estimation approaches for the BEM applied to 2D potential problems. The first approach involves a local error estimator based on a gradient recovery procedure in which the error function is generated from differences between smoothed and non-smoothed rates of change of boundary variables in the local tangential direction. The second approach involves the external problem formulation and gives both local and global measures of error, depending on a choice of the external evaluation point. These approaches are post-processing procedures. Both estimators show consistency with mesh refinement and give similar qualitative results. The error estimator using the gradient recovery approach is more general, as this formulation does not rely on an 'optimal' choice of an external parameter. This work presents also the use of a local error estimator in an adaptive mesh refinement procedure. This r-refinement approach is based on the minimization of the standard deviation of the local error estimate. A non-linear programming procedure using a feasible-point method is employed using Lagrange multipliers and a set of active constraints. The optimization procedure produces finer meshes close to a singularity and results that are consistent with the problem physics.