| It is well known that the solution of an elliptic equation has singularities on non-smooth domains and when the boundary condition changes. This talk is to provide a systematic introduction to singular solutions of this type and to problems that may rise both theoretically and numerically. In stead of usual Sobolev spaces, we will concentrate on the well-posedness and regularity property of the solution in weighted Sobolev spaces, which captures the behavior of the solution well. A general construction of a sequence of finite element spaces will be presented accordingly to recover the quasi-optimal rate of convergence of the numerical solution. In addition, based on the method of subspace corrections, a convergence estimate will show the optimality of the multigrid method on these special subspaces. An interesting software package for handling singular solutions will be introduced at the end. |
