Stabilized Finite Elements on Anisotropic Meshes: A Priori Error Estimates for the Advection-Diffusion and the Stokes Problems

Abstract

Stabilized finite elements on strongly anisotropic meshes are considered. The design of the stability coefficients is addressed for both the advection-diffusion and the Stokes problems when using continuous piecewise linear finite elements on triangles. Using the polar decomposition of the Jacobian of the affine mapping from thereference triangle to the current one, K, and from a priori error estimates, a new definition of the stability coefficients is proposed. Our analysis shows that these coefficients do not depend on the element diameter hK but on a characteristic length associated with K via the polar decomposition. A numerical assessment of the theoretical analysis is carried out.