Computing upper bounds on Friedrichs’ constant

Vejchodský, Tomáš

Displaying similar documents to “Computing upper bounds on Friedrichs’ constant”

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

Vejchodský, Tomáš

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This contribution presents a general numerical method for computing lower and upper bound of the optimal constant in Friedrichs’ inequality. The standard Rayleigh-Ritz method is used for the lower bound and the method of $𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 - 𝑎 𝑝𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟𝑖 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑖𝑒𝑠$ is employed for the upper bound. Several numerical experiments show applicability and accuracy of this approach.

Explicit estimation of error constants appearing in non-conforming linear triangular finite element method

Xuefeng Liu, Fumio Kikuchi (2018)

Applications of Mathematics

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The non-conforming linear ( $P_{1}$ ) triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuška-Aziz maximum angle condition is required just as in the case of the conforming $P_{1}$ triangle. Some...

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...

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...

Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Peter Oswald (2017)

Applications of Mathematics

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Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^{1}$ norm best approximation error estimates for $H^{2}$ functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation....

Explicit finite element error estimates for nonhomogeneous Neumann problems

Qin Li, Xuefeng Liu (2018)

Applications of Mathematics

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The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error estimate has the convergence rate as $0.5$ .

Error estimates in the fast multipole method for scattering problems. Part 1 : truncation of the Jacobi-Anger series

Quentin Carayol, Francis Collino (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave $e^{i \hat{s} \cdot \vec{v}}$ in terms of spherical harmonics ${Y_{ℓ, m} (\hat{s})}_{| m | \leq ℓ \leq \infty}$ . We consider the truncated series where the summation is performed over the $(ℓ, m)$ ’s satisfying $| m | \leq ℓ \leq L$ . We prove that if $v = | \vec{v} |$ is large enough, the truncated series gives rise to an error lower than $ϵ$ as soon as $L$ satisfies $L + \frac{1}{2} ≃ v + C W^{\frac{2}{3}} (K ϵ^{- δ} v^{γ}) v^{\frac{1}{3}}$ where $W$ is the Lambert function and $C, K, δ, γ$ are pure positive constants. Numerical experiments show that this asymptotic is optimal....

Complementarity - the way towards guaranteed error estimates

Vejchodský, Tomáš

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This paper presents a review of the complementary technique with the emphasis on computable and guaranteed upper bounds of the approximation error. For simplicity, the approach is described on a numerical solution of the Poisson problem. We derive the complementary error bounds, prove their fundamental properties, present the method of hypercircle, mention possible generalizations and show a couple of numerical examples.