Space-Time Finite Elements

Solving differential equations is essential to a wide range of scientific applications including modeling, controlling and designing physical, environmental and biological systems. It is usually infeasible to analytically solve these equations, and have to be solved numerically. The governing equations are solved numerically on a discretized domain, called a mesh, to approximate the true solution. This allows researchers to approximate the solutions to many equations lacking an analytical solution. One such method is called the Finite Element Method. Current methodologies for this finite element method employ iterative time stepping to solve time-dependent problems. Due the size of the system being fixed, time-stepping approaches are limited in terms of computational scalability. In this work, we augmented the open-source finite element method framework, TalyFEM, to solve time-dependent problems simultaneously in space and time. We first added the capability to produce and read 4-dimensional meshes made of 4D simplex elements called pentatopes. Next, we implemented support for basis functions on pentatopes. Finally, we formulated the well-known heat equation to make it suitable for solving over a 4-dimensional space-time mesh. This approach enables finite element methods to adaptively refine the solution in space and time, as well as bypass scalability limitations of time-stepping approaches