Multidimensional Astrophysical Structural and Dynamical Analysis. I. Development of a Nonlinear Finite Element Approach

A new field of numerical astrophysics is introduced which addresses thesolution of large, multidimensional structural or slowly-evolving problems(rotating stars, interacting binaries, thick advective accretion disks, fourdimensional spacetimes, etc.). The technique employed is the Finite ElementMethod (FEM), commonly used to solve engineering structural problems. Theapproach developed herein has the following key features: 1. The computational mesh can extend into the time dimension, as well asspace, 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 thatroutinely solved by most engineering finite element codes. 3. The transformations that occur naturally in the four-dimensional FEMpossess 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 simplecoordinate 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- Eulerianadvective derivatives automatically. 4. The complex difference equations on the arbitrary curvilinear grid aregenerated automatically from encoded differential equations. This first paper concentrates on developing a robust and widely-applicableset of techniques using the nonlinear FEM and presents some examples.Comment: 28 pages, 9 figures; added integral boundary conditions, allowing very rapidly-rotating stars; accepted for publication in Ap.