Rapid Prototyping for the Construction of Higher Order Finite Element Methods on Sparse
Grids
Date: July 18th (Thursday)
Time: 11:10-11:35
Abstract
For the development of new algorithmic concepts in the area of numerical analysis
or, more generally, scientific computing, modern tools for an efficient rapid prototyping
like computer algebra programs like Maple or Mathematica and shell-script-type
interpreter languages like Perl gain more and more in importance. This is mainly due to the
fact that the period of time from an idea to a first programmed version of the corresponding
algorithm can be cut down significantly, since all the implementation and declaration
overhead coming along with standard (numerical) programming languages can be avoided.
As an example for that, we present a new unidirectional approach for $d$-dimensional
finite element methods of higher order on sparse grids that enables us to deal with problems
of an arbitrary number $d$ of dimensions and to use polynomial bases of an arbitrary degree
$p$ with the same storage requirements as and only a little bit more of computational work
than in the usual piecewise linear case. The code consists of two parts: a kind of setup phase
which is done in Maple and the iterative solver programmed in Perl. We report on both
numerical and implementational experiences and on interface problems resulting from the
use of tools originating from different ``worlds''.
