Solving PDEs on Deconstructed Domains

Abstract

When finding analytical solutions to Partial Differential Equations (PDEs) becomes impossible, it is useful to approximate them via a discrete mesh of the domain. Sometimes a robust triangular (2D) or tetrahedral (3D) mesh of the whole domain is a hard thing to accomplish, and in those cases we advocate for breaking up the domain in various different subdomains with nontrivial intersection and to find solutions for the equation in each of them individually. Although this approach solves one issue,it creates another, i.e. what constraints to impose on the separate solutions in a way that they converge to true solution on their union. We present a method that solves this problem for the most common second and fourth order equations in graphics.