The PI will work on the numerical analysis aspects of discrete exterior calculus. This is a class of numerical methods for solving partial differential equations (PDEs) and these methods attempt to preserve the geometric and algebraic structures of the physics being modeled. One ingredient of this project will be the use of algebraic topology and differential geometry to create and analyze discretizations of objects and operators of exterior calculus that appear in important PDEs. Several discretizations will be derived and the convergence and stability of the resulting numerical methods will be studied. In addition to this, the PI will develop algorithms and software to make it easier to conduct experimental studies involving such discretizations. All of this will be done in the context of several physical problems.
Numerical solution of partial differential equations is a core part of numerical analysis and its applications in engineering and science. Some of the equations that the PI will work on have applications in ground water contamination, oil exploration, weather modeling, nuclear fusion, star and planet formation and formation of solar flares and storms. The research in this project will use pure mathematics topics like differential geometry and algebraic topology as well as computational mathematics. Thus it will be a unique vehicle with which to introduce these topics to students in mathematics as well as computer science. Due to the wide appeal of the applications mentioned above, the PI will use animations, images and concepts produced in this research to create interest in mathematics and science amongst primary school students.
In this project we developed some connections between two fields of mathematics and computation that were generally considered in the past to be unrelated. These fields are topology and numerical solution of partial differential equations. Topology, roughly speaking, is the study of shapes that does not involve drastic changes like tearing or gluing. Partial differential equations are equations that model many physical phenomena important in science and engineering. A class of methods that have been successful for solving such equations are various forms of the finite element method and these have been intensively studied for many decades. However when the domains are topologically complicated or curved much less is known about how to solve the equations. Traditional approaches can get bogged down in the coordinates needed to represent operations of calculus from which the equations are built. Luckily there exists a language in mathematics called exterior calculus which allows one to write the equations in a manner that is intrinsic, in the sense of being written without the use of coordinates. This translates surprisingly well to computers. The domain is broken up into pieces called simplices (for example triangles or tetrahedra) and the calculus is made combinatorial or discrete. In this project we developed, analyzed, and related two such discretizations -- discrete exterior calculus, and finite element exterior calculus. This translation from the smooth physical world to the discrete world of computations requires the use of basic concepts from a subfield of topology called algebraic topology. The basic objects of exterior calculus called differential forms turn into the combinatorial objects of algebraic topology called cochains. Similarly the smooth physical domains where the equations live turn into algebraic topology objects called chains. While this is a simple connection between the fields of topology and numerical partial differential equations, deeper connections exist in what is called Hodge theory. Even to make the equations have a unique solution requires the computation of certain objects called harmonic forms which have a direct relationship with the topological characteristics of the space such as holes, handles, and tunnels. Since the domain has to be broken up (discretized) into pieces, this breaking up process of the domain naturally played an important role in this project. We studied certain special techniques for breaking up the domain into simplices, a process which is called triangulation or meshing. These special triangulations or meshes are optimal in some sense and are called well-centered and dihedral acute triangulations. These triangulations impart efficiencies to the process of solving the equations using discrete exterior calculus. In addition, such meshes often provide a level of physical fidelity that is not available in more general meshes that do not satisfy these special properties. Before this project, very little was known about such meshes. In fact a problem that had been open for many years was whether even a simple cube-shaped domain has a dihedral acute triangulation. We were able to resolve this long-standing open problem by producing a triangulation with this property. It requires 1370 tetrahedra to achieve this triangulation. With sufficiently small cubes, any domain could be triangulated to a desired accuracy by repeating our triangulation after reflections. Once we started exploring the connections between topology and partial differential equations we asked questions about relaxing various assumptions. For example, what if the domain is more general, such as a graph, or what is called an abstract simplicial complex. The answer to this led to a method for ranking alternatives for which some comparison data between some pairs is known. This was an unexpected outcome of working in the exterior calculus framework for partial differential equations. Using this method we ranked men's college basketball teams based on the scores from an entire season. Our ranking could predict with about 70% accuracy, which team would win any given game, using only the scores of all games played to date in that season. Another unexpected outcome came from studying the solution of some of the partial differential equations as an optimization problem. We asked what would happen if the type of optimization was different. Starting from this we were able to reformulate an important problem of finding the tightest holes, handles, and tunnels etc. which was being considered intractable before we did our work. We were able to find an efficient method for solving this class of problems. We later applied this to resolve an open problem in combinatorial soap films. The results of this project are likely to have an impact on many science and engineering fields since many problems in these fields are modeled by partial differential equations. In addition, a path that connects two seemingly different fields of mathematics and computation was discovered and that has led to a series of unexpected benefits and algorithms and is likely to lead to many more.
