Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes, a new preprocessing approach, called multineighboring grids (MNG), is proposed for solving PDE eigen-problems on an existing grid . The linear or multi-linear element, based on box-splines, are taken as the first stage K 1 h U h = λ 1 h M 1 h U h . In this paper, the j -th stage neighboring-grid scheme is defined as K j h = λ j h M j h U h , where K j h := M j −1 h ⊗ K 1 h and M j h U h is to be found as a better mass distribution over the j -th stage neighboring-grid , and K j h can be seen as an expansion of K 1 h on the j -th neighboring-grid with respect to the ( j − 1)-th mass distribution M j −1 h . It is shown that for an ODE model eigen-problem, the j -th stage scheme with 2 j -th order B-spline basis can reach 2 j -th order accuracy and even (2 j +2)-th order accuracy by perturbing the mass matrix. The argument can be extended to high dimensions with separable variable cases. For Laplace eigen-problems with some 2-D and 3-D structured uniform grids, some 2 j -th order schemes are presented for j ⩽ 3.