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.
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...
The famous Zlámal’s minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in . In this paper we present and discuss its generalization to simplicial partitions in any space dimension.
We provide a comparative study of the Subspace Projected Approximate Matrix method, abbreviated SPAM, which is a fairly recent iterative method of computing a few eigenvalues of a Hermitian matrix . It falls in the category of inner-outer iteration methods and aims to reduce the costs of matrix-vector products with within its inner iteration. This is done by choosing an approximation of , and then, based on both and , to define a sequence of matrices that increasingly better approximate...
Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the...
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 .
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