# Applications of Computer Algebra Methods for High Order Finite Elements

## Dr.^{in} Veronika Pillwein

**March 20, 2007, 2:30 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.