Multi-Dimensional Astrophysical Structural and Dynamical Analysis I.   Development of a Nonlinear Finite Element Approach
  A new field of numerical astrophysics is introduced which addresses the solution of large, multidimensional structural or slowly-evolving problems (rotating stars, interacting binaries, thick advective accretion disks, four dimensional spacetimes, etc.). The technique employed is the Finite Element Method (FEM), commonly used to solve engineering structural problems. The approach developed herein has the following key features:   1. The computational mesh can extend into the time dimension, as well as space, perhaps only a few cells, or throughout spacetime.   2. Virtually all equations describing the astrophysics of continuous media, including the field equations, can be written in a compact form similar to that routinely solved by most engineering finite element codes.   3. The transformations that occur naturally in the four-dimensional FEM possess both coordinate and boost features, such that   (a) although the computational mesh may have a complex, non-analytic, curvilinear structure, the physical equations still can be written in a simple coordinate system independent of the mesh geometry.   (b) if the mesh has a complex flow velocity with respect to coordinate space, the transformations will form the proper arbitrary Lagrangian- Eulerian advective derivatives automatically.   4. The complex difference equations on the arbitrary curvilinear grid are generated automatically from encoded differential equations.   This first paper concentrates on developing a robust and widely-applicable set of techniques using the nonlinear FEM and presents some examples. 
David L. Meier