Jan Brandts

On Nonobtuse Simplicial Partitions

This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

Superconvergence of mixed finite element semi-discretizations of two time-dependent problems

We will show that some of the superconvergence properties for the mixed finite element method for elliptic problems are preserved in the mixed semi-discretizations for a diffusion equation and for a Maxwell equation in two space dimensions. With the help of mixed elliptic projection we will present estimates global and pointwise in time. The results for the Maxwell equations form an extension of existing results. For both problems, our results imply that post-processing and a posteriori error estimation for the error in the space discretization can be performed in the same way as for the underlying elliptic problem.

On the equivalence of ball conditions for simplicial finite elements in R-d

We prove that the inscribed and circumscribed ball conditions, commonly used in finite element analysis, are equivalent in any dimension.

ON ANGLE CONDITIONS IN THE FINITE ELEMENT METHOD

Angle conditions play an important role in the analysis of the finite element method. They enable us to derive the optimal interpolation order and prove convergence of this method, to derive various a posteriori error estimates, to perform regular mesh refinements, etc. In 1968, Miloš Zlámal introduced the minimum angle condition for triangular elements. From that time onward many other useful geometric angle conditions on the shape of elements appeared. In this paper, we shall give a survey of various generalizations of the minimum and also maximum angle condition in the finite element method and present some of their applications.

A geometric toolbox for tetrahedral finite element partitions

In this work we present a survey of some geometric results on tetrahedral partitions and their refinements in a unified manner. They can be used for mesh generation and adaptivity in practical calculations by the finite element method (FEM), and also in theoretical finite element (FE) analysis. Special emphasis is laid on the correspondence between relevant results and terminology used in FE computations, and those established in the area of discrete and computational geometry (DCG).

Manifestations of dark energy in the dynamics of the Solar system

The expansion speed of the Universe is increasing (Glanz 1998). This acceleration is attributed to dark energy which acts almost uniformly everywhere (including the Solar system) and thus essentially influences the Hubble constant. Its current value on a distance of 1 AU is H0 = 10 m/(yr AU). This is quite a large number and thus, the impact of dark energy should be detectable in the Solar system. We will illustrate it by several examples. Dark energy may partially be caused by gravitational aberration of the Sun, planets and other bodies.

The strengthened cauchy-Bunyakowski-Schwarz inequality for n -simplicial linear finite elements

It is known that in one, two, and three spatial dimensions, the optimal constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality for the Laplacian for red-refined linear finite element spaces, takes values zero, and , respectively. In this paper we will conjecture an explicit relation between these numbers and the spatial dimension, which will also be valid for dimensions four and up. For each individual value of n, it is easy to verify the conjecture. Apart from giving additional insight into the matter, the result may find applications in four dimensional finite element codes in the context of computational relativity and financial mathematics.

An Alternative to the Least-Squares Mixed Finite Element Method for Elliptic Problems

In this paper we derive a strengthened Cauchy-Schwarz inequality that enables us to formulate a short and transparant proof of the coercivity of a Least Squares Mixed Finite Element bilinear form. Also, it shows that the coupling between H 0 1 (Ω) and H(div; Ω) is weak enough to be neglected. This results in an alternative way to compute approximations of both the scalar variable and its gradient for second order elliptic problems.

Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

Abstract The convex hull of n + 1 affinely independent vertices of the unit n-cube In is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.

Paradoxes in Numerical Calculations

When solving problems of mathematical physics using numerical methods we always encounter three basic types of errors: modeling error, discretization error, and round-off errors. In this survey, we present several pathological examples which may appear during numerical calculations. We will mostly concentrate on the influence of round-off errors.