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Equilibrated residual error estimators applied to high order finite elements are analyzed. The estimators provide always a true upper bound for the energy error. We prove that also the efficiency estimate is robust with respect to the polynomial degrees. The result is complete for tensor product elements. In the case of simplicial elements, the theorem is… (More)

- Sven Beuchler, Veronika Pillwein
- Computing
- 2007

In this paper, we investigate the discretization of an elliptic boundary value problem in 3D by means of the hp-version of the finite element method using a mesh of tetrahedrons. We present several bases based on integrated Jacobi polynomials in which the element stiffness matrix has $${\mathcal{O}}(p^3)$$ nonzero entries, where p denotes the polynomial… (More)

The goal of this paper is to discuss the application of computer algebra methods in the design of a high order finite element solver. The finite element method is nowadays the most popular method for the computer simulation of partial differential equations. The performance of iterative solution methods depends on the condition number of the system matrix,… (More)

- I. Georgieva, C. Hofreither, +14 authors Walter Zulehner
- 2011

Given information about a harmonic function in two variables, consisting of a nite number of values of its Radon projections, i.e., integrals along some chords of the unit circle, we study the problem of interpolating these data by a harmonic polynomial. With the help of symbolic summation techniques we show that this interpolation problem has a unique… (More)

- Christos Koukouvinos, Veronika Pillwein, Dimitris E. Simos, Zafeirakis Zafeirakopoulos
- Inf. Process. Lett.
- 2011

The average complexity analysis for a formalism pertaining pairs of compatible sequences is presented. The analysis is done in two levels, so that an accurate estimate is achieved. The way of separating the candidate pairs into suitable classes of ternary sequences is interesting, allowing the use of fundamental tools of symbolic computation, such as… (More)

- A. Becirovic, Peter Paule, Veronika Pillwein, Axel Riese, Carsten Schneider, Joachim Schöberl
- Computing
- 2006

High-order finite elements are usually defined by means of certain orthogonal polynomials. The performance of iterative solution methods depends on the condition number of the system matrix, which itself depends on the chosen basis functions. The goal is now to design basis functions minimizing the condition number, and which can be computed efficiently. In… (More)

We present a computer-assisted proof of positivity of sums over kernel polynomials for ultraspherical Jacobi polynomials.

holds for all v ∈ H Γ1(Ω). Problem (1) will be discretized by means of the hp-version of the finite element method using triangular/tetrahedral elements △s, s = 1, . . . , nel, see e.g. Schwab [1998], Solin et al. [2003]. Let △̂d, d = 2, 3 be the reference triangle (tetrahedron) and Fs : △̂ → △s be the (possibly nonlinear) isoparametric mapping to the… (More)

H(curl) conforming finite element discretizations are a powerful tool for the numerical solution of the system of Maxwell’s equations in electrodynamics. In this paper we construct a basis for conforming high-order finite element discretizations of the function space H(curl) in 3 dimensions. We introduce a set of hierarchic basis functions on tetrahedra… (More)

- Sven Beuchler, Veronika Pillwein, Sabine Zaglmayr
- Numerische Mathematik
- 2012

This paper deals with conforming high-order finite element discretizations of the vectorvalued function space H(div) in 2 and 3 dimensions. A new set of hierarchic basis functions on simplices with the following two main properties is introduced. When working with an affine, simplicial triangulation, first, the divergence of the basis functions is… (More)