Optimal adaptive grids of least-squares finite element methods in two spatial dimensions

Abstract

a b s t r a c tThis article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least-squares finite element approximations. The procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms. The idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions. Model problems considered are the two dimensional convection-diffusion problems where boundary and interior layers occur. Numerical results of model problems illustrating the efficiency of the proposed scheme are presented. In addition, to avoid skewed mesh in the optimal grids generated by the algorithm, an unstructured local mesh smoothing will be considered in the leastsquares approximations. Comparisons with the Gakerkin finite element method will also be provided.

Least-squares finite element solution of Euler equations with H-type mesh refinement and coarsening on triangular elements

“…Taghaddosi et al (1999) and Guaily and Megahed (2010) used LSFEM on r-refined quadrilateral meshes. Chang and Chen (2011) coupled LSFEM with a re-triangulation algorithm to perform optimal adaptive solutions of advection-diffusion equation on 2D triangular meshes. But none of these previous studies used h-type mesh refinement and coarsening together.…”