#### Surface PDEs

I have developed a high-order accurate fast direct solver for variable-coefficient partial differential equations on surfaces, based on spectral collocation and the hierarchical Poincare–Steklov scheme. The method may be used to accelerate implicit time-stepping schemes as repeated solves require only \(\mathcal{O}(N \log N)\) work on a mesh with \(N\) elements, after an \(\mathcal{O}(N^{3/2})\) precomputation. I have applied it to a range of scalar- and vector-valued problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace–Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent reactionâ€“diffusion systems.