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.