Figure 1. Heat map within a 2D domain featuring an insulating triangle. Temperature is high (value 1; red color) near the top of the domain (Dirichlet boundary) and cold (value 0; blue color) near the bottom. The insulating triangle in the middle of the domain creates an intriguing temperature profile in the elliptic (equilibrium) problem results shown here.
A lot of my dissertation work (and papers that branched out from it) involved numerical solutions to diffusion and wave problems in which model parameters (density, wave speed, thermal diffusivity, etc.) change suddenly between two or more different materials. Many of the test cases I presented dealt with mildly-curved interfaces, and although I presented a few cases with cornered interfaces, I didn't have a high-order (highly accurate) approach prepared and vetted for the latter scenario.
Over the past month or so, I've been working on incorporating some singular basis functions into radial basis function-generated finite difference (RBF-FD) collocation schemes in a manner that significantly increases the accuracy of elliptic and parabolic (diffusion) problems with cornered interfaces. I've also been combing through existing literature to see where this approach may fit. It seems as though a number of investigators have used similar methods and analysis for elliptic (equilibrium) problems before, but the number of articles covering parabolic (time-dependent) problems is far fewer.
I've had a decent amount of success so far in both the elliptic and parabolic cases, but a lot of challenges remain. One exciting aspect of the approach I'm using is that it would translate fairly readily (at least in theory and on paper; I'm not sure at all about its stability) to simple hyperbolic problems of recent interest (acoustic and electromagnetic wave transport). Keep a lookout for more updates on this soon!