Computational Mathematics and Scientific Computing Seminar

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Geometry can play a crucial role in the dynamics of physical systems, yet numerically computing with complex geometry can hinder the speed and accuracy of traditional algorithms. In this talk, we present new methods for the fast, high-order accurate solution of elliptic partial differential equations (PDEs) on embedded surfaces in 3D and multiscale domains in 2D. We introduce a new fast direct solver for variable-coefficient PDEs on surfaces based on the hierarchical Poincaré–Steklov method. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in O(N log N) operations for an unstructured mesh with N elements. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. We apply the method to a range of 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 nonlinear reaction–diffusion systems.

On multiscale domains in 2D, we present a new potential-theoretic framework for the adaptive solution of inhomogeneous elliptic boundary value problems. We avoid function extension and cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary where a truncated volume potential may be applied, and a boundary-conforming strip region where spectral methods may be applied. Solutions in each region are patched together using layer potentials to yield a globally smooth particular solution defined everywhere in the domain. The resulting solver has an optimal computational complexity of O(N) for a discretization with N degrees of freedom and allows for the adaptive resolution of volumetric data, boundary data, and geometric features across a range of length scales.