Guaranteed and fully computable two-sided bounds of Friedrichs’ constant

Vejchodský, Tomáš

Displaying similar documents to “Guaranteed and fully computable two-sided bounds of Friedrichs’ constant”

Computing upper bounds on Friedrichs’ constant

Vejchodský, Tomáš

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This contribution shows how to compute upper bounds of the optimal constant in Friedrichs’ and similar inequalities. The approach is based on the method of $a p r i o r i - a p o s t e r i o r i i n e q u a l i t i e s$ [9]. However, this method requires trial and test functions with continuous second derivatives. We show how to avoid this requirement and how to compute the bounds on Friedrichs’ constant using standard finite element methods. This approach is quite general and allows variable coefficients and mixed boundary conditions. We use the...

Finite element analysis for a regularized variational inequality of the second kind

Zhang, Tie, Zhang, Shuhua, Azari, Hossein

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In this paper, we investigate the a priori and the a posteriori error analysis for the finite element approximation to a regularization version of the variational inequality of the second kind. We prove the abstract optimal error estimates in the $H^{1}$ - and $L_{2}$ -norms, respectively, and also derive the optimal order error estimate in the $L_{\infty}$ -norm under the strongly regular triangulation condition. Moreover, some residual–based a posteriori error estimators are established, which can provide the...

Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)

Olivier Ramaré (2013)

Acta Arithmetica

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We prove that the error term $\sum_{n \leq x} Λ (n) / n - l o g x + γ$ differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.

Diagonalization in proof complexity

Jan Krajíček (2004)

Fundamenta Mathematicae

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We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ∙ There is a function f: 0,1* → 0,1 computable in that has circuit complexity $2^{Ω (n)}$ . ∙ ≠ co . ∙ There is no p-optimal propositional proof system. We note that a variant of the statement (either ≠ co or ∩ co contains a function $2^{Ω (n)}$ hard on average) seems to have a bearing on the existence of good proof complexity generators. In particular, we prove that...

Anisotropic $h p$ -adaptive method based on interpolation error estimates in the $H^{1}$ -seminorm

Vít Dolejší (2015)

Applications of Mathematics

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We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^{1}$ -seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of...

On the Riesz means of n/ϕ(n) - III

Ayyadurai Sankaranarayanan, Saurabh Kumar Singh (2015)

Acta Arithmetica

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Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_{k} (x)$ where $1 / k! \sum_{n \leq x} n / ϕ (n) {(1 - n / x)}^{k} = M_{k} (x) + E_{k} (x)$ . For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying $k ≫ {(l o g x)}^{1 + ϵ}$ .

Mixed $A_{p} - A_{\infty}$ estimates with one supremum

Andrei K. Lerner, Kabe Moen (2013)

Studia Mathematica

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We establish several mixed $A_{p} - A_{\infty}$ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the $A_{\infty}$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, to a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily...

Low-discrepancy point sets for non-uniform measures

Christoph Aistleitner, Josef Dick (2014)

Acta Arithmetica

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We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on ${[0, 1]}^{d}$ there exists a point set $x_{1}, . . ., x_{N} \in {[0, 1]}^{d}$ whose star-discrepancy with respect to μ is of order $D_{N} * (x_{1}, . . ., x_{N}; μ) ≪ ({(l o g N)}^{(3 d + 1) / 2}) / N$ . For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear...

Uniform $L^{1}$ error bounds for semi-discrete finite element solutions of evolutionary integral equations

Lin, Qun, Xu, Da, Zhang, Shuhua

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In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_{t} + \int_{0}^{t} β (t - s) A u (s) d s = 0, u (0) = v, t > 0,$ where A is an elliptic partial-differential operator and $β (t)$ is positive, nonincreasing and log-convex on $(0, \infty)$ with $0 \leq β (\infty) < β (0^{+}) \leq \infty$ . Error estimates are derived in the norm of $L_{t}^{1} (0, \infty; L_{x}^{2})$ , and some estimates for the first order time derivatives of the errors are also given.

$h p$ -anisotropic mesh adaptation technique based on interpolation error estimates

Dolejší, Vít

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We present a completely new $h p$ -anisotropic mesh adaptation technique for the numerical solution of partial differential equations with the aid of a discontinuous piecewise polynomial approximation. This approach generates general anisotropic triangular grids and the corresponding degrees of polynomial approximation based on the minimization of the interpolation error. We develop the theoretical background of this approach and present a numerical example demonstrating the efficiency of...

Fourier analysis, linear programming, and densities of distance avoiding sets in $ℝ^{n}$

Fernando Mário de Oliveira Filho, Frank Vallentin (2010)

Journal of the European Mathematical Society

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We derive new upper bounds for the densities of measurable sets in $ℝ^{n}$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2, \dots, 24$ . This gives new lower bounds for the measurable chromatic number in dimensions $3, \dots, 24$ . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems...

On some a posteriori error estimation results for the method of lines

Segeth, Karel, Šolín, Pavel

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The paper is an attempt to present an (incomplete) historical survey of some basic results of residual type estimation procedures from the beginning of their development through contemporary results to future prospects. Recently we witness a rapidly increasing use of the $h p$ -FEM which is due to the well-established theory. However, the conventional a posteriori error estimates (in the form of a single number per element) are not enough here, more complex estimates are needed, and this...

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu Zhang, Shu Hua Zhang (2015)

Applications of Mathematics

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We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$ -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$ , the gradient approximation possesses the superconvergence: ${max}_{P \in S} | (\nabla u - \bar{\nabla} u_{h}) (P) | = O (h^{2}) {| ln h |}^{3 / 2}$ , where $\bar{\nabla}$ denotes the average gradient on elements containing vertex $P$ . Furthermore, by using the interpolation post-processing...

Upper bounds for the domination numbers of toroidal queens graphs

Christina M. Mynhardt (2003)

Discussiones Mathematicae Graph Theory

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We determine upper bounds for $γ (Q n^{t})$ and $i (Q ₙ^{t})$ , the domination and independent domination numbers, respectively, of the graph $Q ₙ^{t}$ obtained from the moves of queens on the n×n chessboard drawn on the torus.