A unified approach to singular problems arising in the membrane theory

Irena Rachůnková; Gernot Pulverer; Ewa B. Weinmüller

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Displaying similar documents to “A unified approach to singular problems arising in the membrane theory”

Analysis and numerical solutions of positive and dead core solutions of singular Sturm-Liouville problems.

Pulverer, Gernot, Staněk, Svatoslav, Weinmüller, Ewa B. (2010)

Advances in Difference Equations [electronic only]

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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.

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...

$h p$ -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...

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....

Computable error bounds for collocation methods.

Ahmed, A.H. (1995)

International Journal of Mathematics and Mathematical Sciences

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Singular nonlinear problem for ordinary differential equation of the second order

Irena Rachůnková, Jan Tomeček (2007)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The paper deals with the singular nonlinear problem $u^{''} (t) + f (t, u (t), u^{'} (t)) = 0, u (0) = 0, u^{'} (T) = ψ (u (T)),$ where $f \in 𝐶𝑎𝑟 ((0, T) \times D)$ , $D = (0, \infty) \times$ . We prove the existence of a solution to this problem which is positive on $(0, T]$ under the assumption that the function $f (t, x, y)$ is nonnegative and can have time singularities at $t = 0$ , $t = T$ and space singularity at $x = 0$ . The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.

Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method

Abdallah Bradji, Jürgen Fuhrmann (2014)

Mathematica Bohemica

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We consider a family of conforming finite element schemes with piecewise polynomial space of degree $k$ in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is $h^{k} + τ^{2}$ in the discrete norms of $ℒ^{\infty} (0, T; ℋ^{1} (Ω))$ and $𝒲^{1, \infty} (0, T; ℒ^{2} (Ω))$ , where $h$ and $τ$ are the mesh size of the spatial and temporal discretization,...