Analysis of geometric discretization methods

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The theme of the proposed project is the theoretical understanding of compatible discretization techniques for geometric partial differential equations. More specifically, my interests are in discretization methods that inherit fundamental geometric and topological properties of the underlying differential equation. My long term goals are to help invent discretization methods for the Einstein field equations with nice geometric properties, and to make contributions to the design and analysis of lattice quantum chromodynamics. Within the framework of this project, we would like to investigate convergence properties of discrete exterior calculus (DEC). DEC is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics and scientific computing. However, a rigorous convergence analysis of DEC has always been lacking. Recently, the applicant and his student proved that DEC solutions to the Poisson problem in arbitrary dimensions converge. We plan to study convergence for general k-forms. This is a short term goal that can be accomplished in 23 years by a PhD student. The analogue of DEC for the Yang-Mills equations is the so called lattice gauge theories, whose quantum version include lattice quantum chromodynamics. Thus, a good understanding of DEC will certainly give insights into lattice gauge theories. Making use of this connection, I plan to study convergence properties of (classical) lattice gauge theory for the Yang-Mills equations. The analogue of DEC for the Einstein equations is the so called Regge calculus. Very recently, the Regge discretization of the Einstein field equations has been found to be numerically unstable. We would like to extend the Regge calculus to general tessellations including quadrilaterals, prisms, etc, instead of only triangles. Our hope is that this would alleviate the aforementioned instability. The projects described in the last 2 paragraphs are open-ended long term projects. The second subset is on finite element exterior calculus (FEEC). This is a theoretical framework to handle mixed finite element type discretizations of abstract Hilbert complexes, together with a growing body of theory on concrete realizations. The first project I want to work on in this direction is to develop an Lp-theory of FEEC. This is a well-defined question with known methodologies and I expect to have concrete results in 12 years. An important pending issue here is adaptivity. The biggest obstacle in this direction has been that the so called quasi-orthogonality property (which is a generalization of Galerkin orthogonality) is hard to establish in the mixed finite element setting. However, very recently, this problem has been solved in the context of the Stokes problem. We hope to adapt their techniques into the FEEC setting. This project is technically challenging, but not so much open ended, as general ideas on its resolution are just beginning to emerge.