Metric tensor recovery for adaptive meshing

Adaptive computation is now recognized as essential for solving complex PDE problems. Conceptually, such a computation requires at each step the definition of a continuous metric field (mesh size and direction) to govern the generation of adapted meshes. In practice, in the adaptive computation, an appropriate a posteriori error estimation is used and an upper-bounding of the error is expressed in terms of discrete metrics associated with the element vertices. In order to obtain a continuous metric field, the discrete field is recovered in the whole domain mesh using an appropriate interpolation method on each element. In this paper, a new method for interpolating discrete metric fields, based on a so-called natural decomposition of metrics, is introduced. The proposed method uses known matrix decompositions and is computationally robust and efficient. Classical interpolation methods are recalled and, from numerical examples on simplicial mesh elements, some qualitative comparisons against the new methodology are made to show its relevance.