TITLE:

Linear Algebra and Geometry of Simplicial Finite Elements 
in Four Space Dimensions

ABSTRACT:

The entanglement of space and time as a consequence of the 
relativity principle shows that absolute space nor absolute 
time exists. Therefore, any realistic theoretical model of 
spacetime should be at least truly four-dimensional, and 
a corresponding computational model would involve the approximation 
of four-dimensional manifolds by simplicial meshes. In particular, 
differential equations governing relevant physical behavior and their
numerical approximation would not be formulated as 3+1 time-dependent. 
However, one of the major objections to such models has always been 
the limitation in speed and memory of computational resources. Since 
nowadays, these run in the teraflops and terabytes, the time has come 
to seriously investigate the possibility of four-dimensional 
computational methods for relevant problems.

We will address questions in the meshing of four-dimensional polytopes
with non-trivial geometries and to consider feasible non-degenerate 
refinement strategies, and briefly mention the canonical construction 
of standard and mixed finite elements on the resulting partitions, 
using tools from computational and differential geometry. In particular 
the role of differential geometry in numerical analysis as a unifying 
mathematical language to address the design of stable families of mixed 
finite element methods is only recently, though not yet widely, 
appreciated.