Topic
Maximal element
About: Maximal element is a research topic. Over the lifetime, 759 publications have been published within this topic receiving 9456 citations.
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Papers
TL;DR: It is shown that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
Abstract: We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
606 citations
Book•
2 May 1997
TL;DR: In this article, the stability of functional equations and complementarity problems on cones has been studied under variational principles, maximal element principles, and zero-epi mappings variational principle.
Abstract: Stability of functional equations isometric mappings cones and complementarity problems metrics on cones zero-epi mappings variational principles maximal element principles.
228 citations
TL;DR: In this article, it was shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2] and a classification of these partially-ordered sets is given.
Abstract: It is shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2]. A classification of these partially ordered sets is given.
171 citations
TL;DR: In this paper, the authors generalize the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem of Ky Fan by introducing a class of generalized closedness and continuity conditions, which are called the Transfer Closedness and Transfer Continuity.
157 citations
TL;DR: In this article, the authors prove fixed point theorems 2 and 4, as well as Corollaries 2, 4, and 5 as a corollary to the fixed point Theorem 1.
Abstract: In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries. Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P, a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element e greater than or not comparable to f(e), W(a), although constructed differently, is identical with the set A of Bourbaki (3) determined by a, f , and P1: {x|x ∈ P, x ≤ f(x)}. Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new. Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4). The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3). The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice.
126 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 2 |
| 2023 | 7 |
| 2022 | 7 |
| 2021 | 18 |
| 2020 | 16 |
| 2019 | 20 |