Applications of Computer Algebra Methods for High Order Finite Elements

Dr.in Veronika Pillwein

March 20, 2007, 3:30 p.m. T 1010

In this talk we present the interaction of numerical and symbolic computation in the context of high order finite element method. The first part will consist of a review on some of the past activities of the SFB project F1301. In the second part we will describe in more detail some recent work.

First, the construction of interior shape functions for tetrahedra (joint work with S. Beuchler), which yield,in the case of a polygonally bounded domain, a sparse system matrix

$\hat K = \int_{\hat \tau} (\nabla \Phi(x,y,z))^T C (\nabla \Phi(x,y,z))\, d (x,y,z),$

where $\Phi$ denotes the vector of element bubbles $\phi_{i,j,k}$ and $C \in \mathbb{R}^{3\times 3}$ the constant coefficient matrix.

The sparsity of $\hat K$ was proven by explicitely computing the matrix entries using the computer algebra software Mathematica. The identities used in this computation can be generated using the RISC-symbolic summation package MultiSum.

When working on a convergence proof for a certain higher order finite element scheme, J. Schöberl was led to conjecture that the inequality

$\sum_{j=0}^{n} (4j+1)(2n-2j+1) P_{2j}(0) P_{2j}(x) \ge 0$

holds for $x \in [−1, 1]$ and $n \ge 0$, where $P_k (x)$ denotes the $k$th Legendre polynomial. This sum can be viewed as a sum over so called kernel polynomials. With the aid of the summation package SumCracker we have found a different representation of the above sum which makes it better treatable. We will present these recent findings and positivity results on related sums of kernel polynomials.