Advisor: Dr. Clayton Shonkwiler
Committee: Dr. Alexander Hulpke, Dr. Nathaniel Blanchard

Title: Discrete Exterior Calculus on Polygonal Meshes and an Extension To Subdivision Surfaces

Abstract: Exterior calculus is a modern, coordinate-free approach to classical vector calculus. Discrete Exterior Calculus (DEC) is a discrete analogue which serves as a numerical framework for solving geometric differential equations and acts as a discrete theory of geometry unto itself. DEC goes back to Anil Hirani’s 2003 PhD Thesis on the subject, but related results appear in earlier works on computational electromagnetism. It is desirable to extend this theory for higher quality numerical computation.

Finite Element Exterior Calculus (FEEC) is one such extension, fusing DEC with finite element theory. It enjoys a substantial amount of expository material. Another generalization, Subdivision Exterior Calculus (SEC), was pioneered by Desbrun et al. and lacks expository material. Subdivision surfaces evolved from spline functions out of a need to represent smooth surfaces of arbitrary topology. SEC utilizes subdivision surfaces to build accurate operators for solving differential equations. The original SEC papers are concise reference materials for subject experts, but newcomers may find themselves searching for needles in haystacks attempting to answer nitty-gritty theoretical questions.      

We aim to:

(1) Introduce DEC on polygonal meshes (common expositions focus on the simplicial setting).

(2) Summarize key results in the theory of splines and subdivision surfaces, enabling readers to navigate relevant literature confidently. Proofs are mostly not replicated.

(3) Exposit on SEC catering to theoretically-minded newcomers.