A multiple-scale higher order polynomial collocation method for 2D and 3D elliptic partial differential equations with variable coefficients

In this paper, we present a multiple-scale higher order polynomial collocation method for the numerical solution of 2D and 3D elliptic partial differential equations (PDEs) with variable coefficients. The collocation method with higher order polynomial approximation is very simple for solving PDEs, but it has not become the mainstream method. The main reason is that its resultant algebraic equations have highly ill-conditioned behavior. In our scheme, the multiple-scale coefficients are introduced in the polynomial approximation to overcome the ill-conditioned problem. Based on the concept of the equilibrate matrix, the multiple scales are automatically determined by the collocation points. We find these scales can largely reduce the condition number of the coefficient matrix. Numerical results confirm the accuracy, effectiveness and stability of the present method for smoothed and near-singular 2D and 3D elliptic problems on various irregular domains.