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

Quentin Carayol; Francis Collino

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Error estimates in the fast multipole method for scattering problems. Part 2 : truncation of the Gegenbauer series

Quentin Carayol, Francis Collino (2005)

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 Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, $\frac{e^{i | \vec{u} - \vec{v} |}}{4 π i | \vec{u} - \vec{v} |}$ , which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices $ℓ \leq L$ . 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 $L + \frac{1}{2} ≃ v + C W^{\frac{2}{3}} (K (α) ϵ^{- δ} v^{γ}) v^{\frac{1}{3}}$ where $W$ is the Lambert function, $K (α)$ depends only on...

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...

A comparison of some a posteriori error estimates for fourth order problems

Segeth, Karel

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A lot of papers and books analyze analytical a posteriori error estimates from the point of view of robustness, guaranteed upper bounds, global efficiency, etc. At the same time, adaptive finite element methods have acquired the principal position among algorithms for solving differential problems in many physical and technical applications. In this survey contribution, we present and compare, from the viewpoint of adaptive computation, several recently published error estimation procedures...

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...

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...

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 $0.5$ .