A localized Fourier collocation method for 2D and 3D elliptic partial differential equations:Theory and MATLAB code

A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems.The method first discretizes the entire domain into a set of overlapping small subdomains,and then in each of the subdomains,the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation method.The key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale simulations.The present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape geometries.Preliminary numerical experiments involving Poisson,Helmholtz,and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.