Numerical Methods for Partial Differential Equations Seminar

Wednesday, December 11, 2019 at 4:30pm to 5:30pm

Room 2-136 -- MIT - Dept. of Mathematics, 77 Mass. Ave.., Cambridge, MA

SPEAKER:  Daniel Fortunato  (Harvard University)

TITLE:  The ultraspherical spectral element method


We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincare-Steklov scheme for solving general partial differential equations on polygonal unstructured meshes.  Whereas traditional finite element methods have a computational complexity that scales as O(p6) with polynomial degree p, our method scales as O(p4), allowing for hp-adaptivity to be based on physical considerations instead of computational ones. Properties of the ultraspherical spectral  method lead to almost banded well-conditioned linear systems, allowing for the element method to be competitive in the ultra-high-polynomial regime. The hierarchical Poincare-Steklov scheme enables precomputed solution operators to be reused, allowing for fast elliptic solves in implicit and semi-implicit time-steppers. We develop an open-source software system to demonstrate the  flexibility and robustness of the method. Joint work with Alex  Townsend and Nick Hale.


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