Veronika Pillwein

Sparse shape functions for tetrahedral p -FEM using integrated Jacobi polynomials

SummaryIn 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

Sparsity optimized high order finite element functions for H(div) on simplices

{\mathcal{O}}(p^3)

A local Fourier convergence analysis of a multigrid method using symbolic computation

nonzero entries, where p denotes the polynomial degree. The proof of the sparsity requires the assistance of computer algebra software. Several numerical experiments show the efficiency of the proposed bases for higher polynomial degrees p.

When can we detect that a P-finite sequence is positive?

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 this paper, we demonstrate the application of recently developed computer algebra algorithms for hypergeometric summation to derive cheap recurrence relations allowing a simple implementation for fast basis function evaluation.

Completions to Sparse Shape Functions for Triangular and Tetrahedral p-FEM

This paper deals with conforming high-order finite element discretizations of the vector-valued 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 L2-orthogonal, and secondly, the L2-inner product of the interior basis functions is sparse with respect to the polynomial degree p. The construction relies on a tensor-product based construction with properly weighted Jacobi polynomials as well as an explicit splitting of the higher-order basis functions into solenoidal and non-solenoidal ones. The basis is suited for fast assembling strategies. The general proof of the sparsity result is done by the assistance of computer algebra software tools. Several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter-dependent Div - Div problem. Even on curved elements a tremendous improvement in condition numbers is observed. The precomputed mass and stiffness matrix entries in general form are available online.

Dominance in the family of Sugeno--Weber t-norms

This article reports several results on sparsity optimized basis functions for hp-FEM on triangular and tetrahedral finite element meshes obtained within the Special Research Program “Numerical and Symbolic Scientific Computing” and within the Doctoral Program “Computational Mathematics” both supported by the Austrian Science Fund FWF under the grants SFB F013 and DK W1214, respectively. We give an overview on the sparsity pattern for mass and stiffness matrix in the spaces L 2, H 1, H({ div}) and H(curl). The construction relies on a tensor-product based construction with properly weighted Jacobi polynomials.

Harmonic interpolation based on Radon projections along the sides of regular polygons

Cylindrical algebraic decomposition (CAD) is a standard tool in symbolic computation. In this paper we use it to compute a bound for the convergence rate for a numerical method that usually is merely resolved by numerical interpolation. Applying CAD allows us to determine an exact bound, but the given formula is too large to be simply plugged in. Hence a combination of reformulating, guess and prove and splitting into subproblems is necessary. In this paper we work out the details of a symbolic local Fourier analysis for a particular multigrid solver applied to a particular optimization problem constrained to a partial differential equation (PDE-constrained optimization problem), even though the proposed approach is applicable to different kinds of problems and different kinds of solvers. The approach is based on local Fourier analysis (or local mode analysis), a widely-used straight-forward method to analyze the convergence of numerical methods for solving discretized systems of partial differential equations (PDEs). Such an analysis requires to determine the supremum of some rational function, for which we apply CAD.

Smoothing Analysis of an All-at-Once Multigrid Approach for Optimal Control Problems Using Symbolic Computation

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed on a great many examples, there is no guarantee for them to terminate, and they do in fact not terminate for every input. For some restricted classes of P-finite recurrence equations of order up to three we provide a priori criteria that assert the termination of the algorithms.