A weighted-norm least-squares method is considered for the numerical approximation of solutions that have singularities at the boundary. While many methods suffer from a global loss of accuracy due to boundary singularities, the least-squares method can be particularly sensitive to a loss of regularity. The method we describe here requires only a rough lower bound on the power of the singularity and can be applied to a wide range of elliptic equations. Optimal order discretization accuracy is achieved in weighted H 1, and functional norms and L 2 accuracy are retained for boundary value problems with a dominant div/curl operator. Our analysis, including interpolation bounds and several Poincaré-type inequalities, are carried out in appropriately weighted Sobolev spaces. Numerical results confirm the error bounds predicted in the analysis.
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics