Numerische Mathematik

Items from 1 to 20 out of 27 results

article

Wavelet stabilization of the Lagrange multiplier method

Silvia Bertoluzza

Numerische Mathematik > 2000 > 86 > 1 > 1-28

Summary. We propose here a stabilization strategy for the Lagrange multiplier formulation of Dirichlet problems. The stabilization is based on the use of equivalent scalar products for the spaces $H^{1/2}(\partial\Omega)$ and $H^{-1/2}(\partial\Omega)$ , which are realized by means of wavelet functions. The resulting stabilized bilinear form is coercive with respect to the natural norm associated to...

article

Discrete boundary element methods on general meshes in 3D

I.G. Graham, W. Hackbusch, S.A. Sauter

Numerische Mathematik > 2000 > 86 > 1 > 103-137

Abstract. This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in $\mathbb{R}^3$ . Our theory covers equations with very general operators, provided the associated weak form is bounded and elliptic on $H^\mu$ , for some $\mu \in [-1,1]$ . In contrast to other studies on this topic, we do not assume...

article

The conformal bratwurst mapsand associated Faber polynomials

Tino Koch, Jrg Liesen

Numerische Mathematik > 2000 > 86 > 1 > 173-191

Summary. Conformal maps from the exterior of the closed unit disk onto the exterior of bratwurst shape sets in the complex plane are constructed. Using these maps, coefficients for the computation of the corresponding Faber polynomials are derived. A bratwurst shape set is the result of deforming an ellipse with foci on the real axis, by conformally mapping the real axis onto the unit circle. Such...

article

Nonlinear functionals of wavelet expansions adaptive reconstruction and fast evaluation

Wolfgang Dahmen, Reinhold Schneider, Yuesheng Xu

Numerische Mathematik > 2000 > 86 > 1 > 49-101

Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective...

article

Hybrid Galerkin boundary elements: theory and implementation

I.G. Graham, W. Hackbusch, S.A. Sauter

Numerische Mathematik > 2000 > 86 > 1 > 139-172

Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting...

article

On the construction of stable curvilinear $p$ version elements for mixed formulations of elasticity and Stokes flow

Lawrence Chilton, Manil Suri

Numerische Mathematik > 2000 > 86 > 1 > 29-48

Summary. The use of mixed finite element methods is well-established in the numerical approximation of the problem of nearly incompressible elasticity, and its limit, Stokes flow. The question of stability over curved elements for such methods is of particular significance in the p version, where, since the element size remains fixed, exact representation of the curved boundary by (large) elements...

article

Minimal decompositions and iterative methods

Marko Huhtanen, Olavi Nevanlinna

Numerische Mathematik > 2000 > 86 > 2 > 257-281

Summary. We study how singular values and approximation numbers by algebraic operators of a bounded linear operator A on a separable Hilbert space H can be related to the speed of convergence of iterative methods. In infinite dimensional H these two quantities lead us to consider those A that can be split as $A=N+K$ with N normal andK compact. Then the spectrum of N, via a classical polynomial approximation...

article

An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition

Jae Ryong Kweon

Numerische Mathematik > 2000 > 86 > 2 > 305-318

Summary. We formulate the compressible Stokes system given in (1.1) into a (new) weak formulation (2.1). A finite element method for this is presented. Existence and uniqueness of the finite element method is shown. An optimal error estimate for the numerical approximation is obtained. Numerical examples are given, showing its efficiency and rates of convergence of the approximate solutions that results...

article

Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes

G. Kunert, R. Verfrth

Numerische Mathematik > 2000 > 86 > 2 > 283-303

Summary. Both for the $H^1$ - and $L^2$ -norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation.

article

Generalized p-FEM in homogenization

A.M. Matache, I. Babuka, C. Schwab

Numerische Mathematik > 2000 > 86 > 2 > 319-375

Summary. A new finite element method for elliptic problems with locally periodic microstructure of length $\varepsilon 0$ is developed and analyzed. It is shown that the method converges, as $\varepsilon \rightarrow 0$ , to the solution of the homogenized problem with optimal order in $\varepsilon$ and exponentially in the number of degrees of freedom independent of $\varepsilon 0$ . The computational work...

article

Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition

A. Cohen, R. Masson

Numerische Mathematik > 2000 > 86 > 2 > 193-238

Summary. Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of non-homogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the setting of space refinement...

article

A characteristic-Galerkin approximation to a system of shallow water equations

Clint N. Dawson, Mnica L. Martnez-Canales

Numerische Mathematik > 2000 > 86 > 2 > 239-256

Summary. Characteristic methods are known to handle advective flow better than traditional Galerkin methods and allow large time steps to be taken when compared to standard time-stepping methods. In this paper, we investigate a characteristic-Galerkin approximation to the 2-dimensional system of shallow water equations. We derive ${\cal L}^{\infty}\left ((0,T);{\cal L}^2(\Omega)\right )$ bounds for...

article

Error analysis for a potential problem on locally refined grids

T. Washio, C.W. Oosterlee

Numerische Mathematik > 2000 > 86 > 3 > 539-563

Summary. For the simulation of biomolecular systems in an aqueous solvent a continuum model is often used for the solvent. The accurate evaluation of the so-called solvation energy coming from the electrostatic interaction between the solute and the surrounding water molecules is the main issue in this paper. In these simulations, we deal with a potential problem with jumping coefficients and with...

article

Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

Georgij M. Kobelkov, Maxim A. Olshanskii

Numerische Mathematik > 2000 > 86 > 3 > 443-470

Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized Stokes problem (the Stokes problem with the additional term $\alpha\boldmath{u}$ in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution of the Poisson equation with Neumann boundary conditions. For a wide...

article

Accurate computation of the zeros of the generalized Bessel polynomials

L. Pasquini

Numerische Mathematik > 2000 > 86 > 3 > 507-538

Summary. A general method for approximating polynomial solutions of second-order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized Bessel polynomials, and an algorithm is derived for simultaneously finding their zeros. Then a comparison with several alternative algorithms is carried out....

article

Finite element approximationof a nonlinear elliptic equation arising from bimaterial problemsin elastic-plastic mechanics

Wenbin Liu

Numerische Mathematik > 2000 > 86 > 3 > 491-506

Summary. In [13], a nonlinear elliptic equation arising from elastic-plastic mechanics is studied. A well-posed weak formulation is established for the equation and some regularity results are further obtained for the solution of the boundary problem. In this work, the finite element approximation of this boundary problem is examined in the framework of [13]. Some error bounds for this approximation...

article

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Steinar Evje, Kenneth Hvistendahl Karlsen

Numerische Mathematik > 2000 > 86 > 3 > 377-417

Summary. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form $\begin{array} c \displaystyle{\partial_t u}+{\partial_x} f(u)= {\partial_x} A(b(u){\partial_x}u),\qquad \quad u(x,0)=u_0(x),\\[3mm] \displaystyle A(s)=\displaystyle\int_0^s a(\xi)\,d\xi,\; a(s)\geq 0,\;b(s)\geq 0, \end{array} \leqno(P)$ ...

article

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

Javier de Frutos, Bosco Garca-Archilla, Julia Novo

Numerische Mathematik > 2000 > 86 > 3 > 419-442

Summary. We present an approximate-inertial-manifold-based postprocess to enhance Chebyshev or Legendre spectral Galerkin methods. We prove that the postprocess improves the order of convergence of the Galerkin solution, yielding the same accuracy as the nonlinear Galerkin method. Numerical experiments show that the new method is computationally more efficient than Galerkin and nonlinear Galerkin...

article

An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes

Gerd Kunert

Numerische Mathematik > 2000 > 86 > 3 > 471-490

Summary. A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only. The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that...

article

Finite element analysis of the vibration problem of a plate coupled with a fluid

R.G. Durn, L. Hervella-Nieto, E. Liberman, R. Rodrguez, more

Numerische Mathematik > 2000 > 86 > 4 > 591-616

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas...

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2000

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MATHEMATICS SUBJECT CLASSIFICATION (1991): 65N30 (6)
65F10 (3)
65N30 (3)
65N55 (3)
65F05 (2)
65F30 (2)
65N15 (2)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65N35 (2)
MATHEMATICS SUBJECT CLASSIFICATION (1991):65N30 (2)
MATHEMATICS SUBJECT CLASSIFICATION (1991):65N38 (2)
33A65 (1)
35J05 (1)
35K65 (1)
35L65 (1)
41A25 (1)
41A80 (1)
45B05 (1)
46E35 (1)
65D05 (1)
65D15 (1)
65H20 (1)
65L10 (1)
65N22 (1)
65N25 (1)
65N50 (1)
65Y20 (1)
76M10 (1)
76N10 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 30C20,30E10 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 41A63 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65D15,65D32 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65D32 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65F10 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65F15 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65M12 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65M55 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65M60 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65N06 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65N12 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65N15 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65N22 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65N55 (1)
MATHEMATICS SUBJECT CLASSIFICATION (1991): 65R20 (1)
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INFONA - science communication portal

Numerische Mathematik

Wavelet stabilization of the Lagrange multiplier method

Discrete boundary element methods on general meshes in 3D

The conformal bratwurst mapsand associated Faber polynomials

Nonlinear functionals of wavelet expansions adaptive reconstruction and fast evaluation

Hybrid Galerkin boundary elements: theory and implementation

On the construction of stable curvilinear $p$ version elements for mixed formulations of elasticity and Stokes flow

Minimal decompositions and iterative methods

An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition

Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes

Generalized p-FEM in homogenization

Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition

A characteristic-Galerkin approximation to a system of shallow water equations

Error analysis for a potential problem on locally refined grids

Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

Accurate computation of the zeros of the generalized Bessel polynomials

Finite element approximationof a nonlinear elliptic equation arising from bimaterial problemsin elastic-plastic mechanics

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes

Finite element analysis of the vibration problem of a plate coupled with a fluid

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