This paper is about -triangles, which are the simplest nontrivial examples of -polytopes: convex hulls of a subset of vertices of the unit -cube . We consider the subclasses of right -triangles, and acute -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of .
We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between 2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive...
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...
We will investigate the possibility to use superconvergence results for the mixed finite element discretizations of some time-dependent partial differential equations in the construction of a posteriori error estimators. Since essentially the same approach can be followed in two space dimensions, we will, for simplicity, consider a model problem in one space dimension.
Cottle’s proof that the minimal number of -simplices needed to triangulate the unit -cube equals uses a modest amount of computer generated results. In this paper we remove the need for computer aid, using some lemmas that may be useful also in a broader context. One of the -simplices involved, the so-called antipodal simplex, has acute dihedral angles. We continue with the study of such acute binary simplices and point out their possible relation to the Hadamard determinant problem.
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...
V článku budeme studovat třídu duálních simplexů v -rozměrném eukleidovském prostoru. Dokážeme, že tato třída je stejná jako třída tzv. dobře centrovaných simplexů. Dále ukážeme, že jisté přirozené konvergenční vlastnosti duálních trojúhelníků nelze přímo zobecnit do trojrozměrného prostoru. K tomuto účelu představíme rovnostěnné čtyřstěny, což je speciální podtřída dobře centrovaných čtyřstěnů.
A symmetric positive semi-definite matrix is called completely positive if there exists a matrix with nonnegative entries such that . If is such a matrix with a minimal number of columns, then is called the cp-rank of . In this paper we develop a finite and exact algorithm to factorize any matrix of cp-rank . Failure of this algorithm implies that does not have cp-rank . Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that...
A -simplex is the convex hull of affinely independent vertices of the unit -cube . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute -simplices in can be represented by -matrices of size whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part of the transposed inverse of is doubly stochastic and has the same support...
We outline a solution method for mixed finite element discretizations based on dissecting the problem into three separate steps. The first handles the inhomogeneous constraint, the second solves the flux variable from the homogeneous problem, whereas the third step, adjoint to the first, finally gives the Lagrangian multiplier. We concentrate on aspects involved in the first and third step mainly, and advertise a multi-level method that allows for a stable computation of the intermediate and final...
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