Metric Generation for a Given Error Estimation

The mesh adaptation is a classical method for accelerating and improving the PDE finite element computation. Two tools are generally used: the metric [10] to define the mesh size and the error indicator to know if the solution is accurate enough. A lot of algorithms used to generate adapted meshes suitable for a PDE numerical solution for instance [15, 1] for discrete metrics, or [15, 2] for the continuous one, use those tools for the local specification of the mesh size. Lot of PDE softwares like Freefem++ [14] use metrics to build meshes, the edges sizes of which are equal with respect to the metric field. The construction of metrics from the hessian matrix [16, 18, 13] is only justify for the piecewise linear Lagrange finite element. So there is the problem of metric generation when we have another interpolation error estimator [6, 11] that could be used for instance when the Lagrange interpolation needed is a k degree polynomial, k > 1. To answer that question, we propose in this paper an algorithm whose complexity is quasi-linear, in two spacial dimensions; assuming that the error is locally described by a closed curve representing the error level set. Some efficient numerical examples are given. This algorithm allows us to obtain the analytical metric [1], when the error indicator is based on the hessian matrix. We have also done one comparison in the software Freefem++ of mesh adaptation with metrics computed using this algorithm with respect to the interpolation error estimation described in [11], and the method with metrics based on the hessian. The results seem to be better for the maximal error.