An irregular grid for the numerical solution of linear elliptic partial differential equations

This paper first presents a symmetrically geometric proportion grid with respect to the boundary for linear elliptic partial differential equations with the homogeneous Dirichlet conditions. Then a symmetric scaling technique is proposed for the coefficient matrix derived from the second-order centered difference discretization on the irregular grid. It is proved that the condition number of the symmetrically scaled system is bounded by a constant independent of the matrix order for one-dimensional problem. The numerical results also indicate that the same conclusion holds for a two-dimensional problem.