Hessian-free metric-based mesh adaptation via geometry of interpolation error

Generation of meshes adapted to a given function u requires a specially designed metric. For metric derived from the Hessian of u, optimal error estimates for the interpolation error on simplicial meshes have been proved in [2, 5, 8, 10, 11]. The Hessian-based metric has been successfully applied to adaptive solution of PDEs [4, 7, 9]. However, theoretical estimates have required to make an additional assumption that the discrete Hessian approximates the continuous one in the maximum norm. Despite the fact that this assumption is frequently violated in many Hessian recovery methods, the generated adaptive meshes still result in optimal error reduction. In this article we continue the rigorous analysis [1, 3] of an alternative way for generating a space tensor metric using the error estimates prescribed to mesh edges. The new methodology produces meshes resulting in the optimal reduction of the P1-interpolation error or its gradient. We define a tensor metric M such that the volume and the perimeter of a simplex measured in this metric control the norm of error or its gradient. The equidistribution principle, which can be traced back to D’Azevedo [6], suggests to balance Mvolumes and M-perimeters. This leads to meshes that are quasi-uniform in the metric M. The paper outline is as follows. In Section 2, we derive appropriate metrics from analysis of the interpolation errors. In Section 3, we present the algorithm for generating adaptive meshes and its application to a model problem.