In order to have reliable numerical simulations it is very important to preserve basic qualitative properties of solutions of mathematical models by computed approximations. For scalar second-order elliptic equations, one of such properties is the maximum principle. In our work, we give a short review of the most important results devoted to discrete counterparts of the maximum principle (called discrete maximum principles, DMPs), mainly in the framework of the finite element method, and also present our own recent results on DMPs for a class of second-order nonlinear elliptic problems with mixed boundary conditions.
On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions Brandts, J.; Korotov, S.; Krizek, M.
Link to publicationCitation for published version (APA): Brandts, J., Korotov, S., & Krizek, M. (2008). On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions.
AbstractIn this note we examine several regularity criteria for families of simplicial finite element partitions in R d , d ∈ {2, 3}. These are usually required in numerical analysis and computer implementations. We prove the equivalence of four different definitions of regularity often proposed in the literature. The first one uses the volume of simplices. The others involve the inscribed and circumscribed ball conditions, and the minimal angle condition.
Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For 3 ≤ n ≤ 7 this socalled simplexity of the unit cube I n is now known to be 5, 16, 67, 308, 1493, respectively. In this paper, we study triangulations of I n with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into n! simplices. In this paper we show that, surprisingly, for each n ≥ 3 there is essentially only one other nonobtuse triangulation of I n , and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than n!(e − 2).
MSC: 65M50 65N30 65N50Keywords: Zlámal's minimum angle condition Finite element method Nested triangulations Conforming longest-edge bisection algorithm High aspect ratio elements a b s t r a c tWe examine a generalized conforming bisection (GCB-)algorithm which allows both global and local nested refinements of the triangulations without generating hanging nodes. It is based on the notion of a mesh density function which prescribes where and how much to refine the mesh. Some regularity properties of generated sequences of refined triangulations are proved. Several numerical tests demonstrate the efficiency of the proposed bisection algorithm. It is also shown how to modify the GCB-algorithm in order to generate anisotropic meshes with high aspect ratios.
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