On Adaptivity for Diffusion Problems Using Triangular Elements

Abstract

In this work an adaptive scheme to solve diffusion problems using linear and quadratic triangles is presented. The densification algorithm, based on the subdivision of the selected elements, and the error estimator used are first described. We pay special attention to the behavior of the estimator. It has two contributions: the residual term and the flux-jump term. Babuska and co-workers have shown that for bilinear quadrilaterals, the first term is
negligible, but for biquadratic, it is the dominant term. We show evidence suggesting that these results can not be extended to triangular clements when the problem has a singular solution. We found in that if the case that if the flux-jump term is neglected, the expected rate of
convergence can not be obtained. Finally, some remarks about the whole adaptive process are discussed.