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Two-sided bounds of the discretization error for finite elements

Michal Křížek, Hans-Goerg Roos, Wei Chen (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis. ...

Two-sided symmetric condition in the analysis of magnetic fields with permanent magnets

František Melkes (2005)

Applications of Mathematics

Mathematical treatment of a planar magnetic field excited by permanent magnets is presented. A special two-sided condition for differential magnetic reluctivity is introduced to prove the unique existence of both the weak and the approximate solutions and also a certain error estimate. Notes to numerical algorithm and practical applications are given.

Un problema di omogeneizzazione bidimensionale

Stefano Mortola, Sergio Steffè (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this note we study the periodic homogenization problem for a particular bidimensional selfadjoint elliptic operator of the second order. Theoretical and numerical considerations allow us to conjecture explicit formulae for the coefficients of the homogenized operator.

Unconditional stability of difference formulas

Tomáš Roubíček (1983)

Aplikace matematiky

The paper concerns the solution of partial differential equations of evolution type by the finite difference method. The author discusses the general assumptions on the original equation as well as its discretization, which guarantee that the difference scheme is unconditionally stable, i.e. stable without any stability condition for the time-step. A new notion of the A n -acceptability of the integration formula is introduced and examples of such formulas are given. The results can be applied to ordinary...

Une méthode nodale appliquée à un problème de diffusion à coefficients généralisés

Abdelkader Laazizi, Nagib Guessous (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we consider second order neutrons diffusion problem with coefficients in L∞(Ω). Nodal method of the lowest order is applied to approximate the problem's solution. The approximation uses special basis functions [1] in which the coefficients appear. The rate of convergence obtained is O(h2) in L2(Ω), with a free rectangular triangulation.

Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation

Huangxin Chen, Ronald H. W. Hoppe, Xuejun Xu (2013)

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

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal...

Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation∗

Huangxin Chen, Ronald H.W. Hoppe, Xuejun Xu (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal...

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