Anisotropic meshes and stabilized parameters for the stabilized finite element methods

TL;DR

This paper introduces a numerical strategy for generating anisotropic meshes and selecting stabilized parameters simultaneously for convection-dominated convection-diffusion equations using stabilized finite elements, improving accuracy and stability.

Contribution

It presents a novel method to derive mesh metrics and stabilization parameters based on the Hessian and flow properties, coupling them for enhanced finite element solutions.

Findings

Validated the stability of the proposed strategy

Demonstrated improved accuracy in numerical tests

Showed efficiency in mesh and parameter selection

Abstract

We propose a numerical strategy to generate the anisotropic meshes and select the appropriate stabilized parameters simultaneously for two dimensional convection-dominated convection-diffusion equations by stabilized continuous linear finite elements. Since the discretized error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective fields, and the geometric properties such as directed edges and the area of the triangle. Based on this observation, the shape, size and equidistribution requirements are used to derive the corresponding metric tensor and the stabilized parameters. It is easily found from our derivation that the optimal stabilized parameter is coupled with the optimal metric tensor on each element. Some numerical results are also provided to validate the stability and efficiency of the proposed numerical strategy.