A Weighted $H(div)$ Least-Squares Method for Second-Order Elliptic Problems

Abstract

This paper presents analysis of a weighted-norm least squares finite element method for elliptic problems with boundary singularities. We use $H(div)$ conforming Raviart–Thomas elements and continuous piecewise polynomial elements. With only a rough estimate of the power of the singularity, we employ a simple, locally weighted $L^2$ norm to eliminate the pollution effect and recover better rates of convergence. Theoretical results are carried out in weighted Sobolev spaces and include ellipticity bounds of the homogeneous least-squares functional, new weighted Raviart–Thomas interpolation results, and error estimates in both weighted and nonweighted norms. Numerical tests are given to confirm the theoretical estimates and to illustrate the practicality of the method