Anisotropic Mesh Refinement Considering a Recovery-Based Error Estimator and Metric Tensors

This paper proposes a new anisotropic h-adaptive technique applied to the finite element method considering triangular finite elements. The proposed technique, named anisotropic error density recovery, utilizes the concept of the energy error density function and the analytical solution of an optimization problem by Lagrange’s multipliers to obtain a metric tensor that represents the dimensions of the new elements. This process aims to limit and equidistribute discretization errors throughout the mesh utilizing finite elements with large aspect ratios. The anisotropic h-adaptive technique is computationally implemented and tested in two-dimensional elliptic problems. The results demonstrate the ability of this approach to generate meshes of elements with high aspect ratios and oriented according to the principal directions of discretization errors. Thus, when compared with isotropic refinement techniques, the anisotropic error density recovery approach shows a significant reduction in the number of vertices for the same permissible error value.