  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.
