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Optimal error estimates for finite elements on meshes containing bands of caps

Kučera, Václav, Szotkowski, Jiří (2025)

Programs and Algorithms of Numerical Mathematics

In this short note we provide an optimal analysis of finite element convergence on meshes containing a so-called band of caps. These structures consist of a zig-zag arrangement of ‘degenerating’ triangles which violate the maximum angle condition. A necessary condition on the geometry of such a structure for various H 1 -convergence rates was previously given by Kučera. Here we prove that the condition is also sufficient, providing an optimal analysis of this special case of meshes. In the special...

Optimal error Estimates for the Stokes and Navier–Stokes equations with slip–boundary condition

Eberhard Bänsch, Klaus Deckelnick (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a finite element discretization by the Taylor–Hood element for the stationary Stokes and Navier–Stokes equations with slip boundary condition. The slip boundary condition is enforced pointwise for nodal values of the velocity in boundary nodes. We prove optimal error estimates in the H1 and L2 norms for the velocity and pressure respectively.

Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem

A. Kadir Aziz, Donald A. French, Soren Jensen, R. Bruce Kellogg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the analysis and numerical solution of a forward-backward boundary value problem. We provide some motivation, prove existence and uniqueness in a function class especially geared to the problem at hand, provide various energy estimates, prove a priori error estimates for the Galerkin method, and show the results of some numerical computations.

Penalty method and extrapolation for axisymmetric elliptic problems with Dirichlet boundary conditions

Ivan Hlaváček (1990)

Aplikace matematiky

A second order elliptic problem with axisymmetric data is solved in a finite element space, constructed on a triangulation with curved triangles, in such a way, that the (nonhomogeneous) boundary condition is fulfilled in the sense of a penalty. On the basis of two approximate solutions, extrapolates for both the solution and the boundary flux are defined. Some a priori error estimates are derived, provided the exact solution is regular enough. The paper extends some of the results of J.T. King...

Plane wave discontinuous Galerkin methods: Analysis of the h-version

Claude J. Gittelson, Ralf Hiptmair, Ilaria Perugia (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that ...

Postprocessing of a finite volume element method for semilinear parabolic problems

Min Yang, Chunjia Bi, Jiangguo Liu (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we study a postprocessing procedure for improving accuracy of the finite volume element approximations of semilinear parabolic problems. The procedure amounts to solve a source problem on a coarser grid and then solve a linear elliptic problem on a finer grid after the time evolution is finished. We derive error estimates in the L2 and H1 norms for the standard finite volume element scheme and an improved error estimate in the H1 norm. Numerical results demonstrate the accuracy...

Quasi-Interpolation and A Posteriori Error Analysis in Finite Element Methods

Carsten Carstensen (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

One of the main tools in the proof of residual-based a posteriori error estimates is a quasi-interpolation operator due to Clément. We modify this operator in the setting of a partition of unity with the effect that the approximation error has a local average zero. This results in a new residual-based a posteriori error estimate with a volume contribution which is smaller than in the standard estimate. For an elliptic model problem, we discuss applications to conforming, nonconforming and mixed...

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