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1 
Efficient time step parallelization of full multigrid techniquesWeickert, J., Steidten, T. 30 October 1998 (has links)
This paper deals with parallelization methods for timedependent
problems where the time steps are shared out among the
processors. A Full Multigrid technique serves as solution
algorithm, hence information of the preceding time step and of
the coarser grid is necessary to compute the solution at each new
grid level. Applying the usual extrapolation formula to process
this information, the parallelization will not be very efficient.
We developed another extrapolation technique which causes a much
higher parallelization effect. Test examples show that no
essential loss of exactness appears, such that the method
presented here shall be wellapplicable.

2 
Local inequalities for anisotropic finite elements and their application to convectiondiffusion problemsApel, Thomas, Lube, Gert 30 October 1998 (has links)
The paper gives an overview over local inequalities for anisotropic simplicial Lagrangian finite elements. The main original contributions are the estimates for higher derivatives of the interpolation error, the formulation of the assumptions on admissible anisotropic finite elements in terms of geometrical conditions in the threedimensional case, and an anisotropic variant of the inverse inequality. An application of anisotropic meshes in the context of a stabilized Galerkin method for a convectiondiffusion problem is given.

3 
NavierStokes equations as a differentialalgebraic systemWeickert, J. 30 October 1998 (has links)
Nonsteady NavierStokes equations represent a differentialalgebraic system of strangeness index one after any spatial discretization. Since such systems are hard to treat in their original form, most approaches use some kind of index reduction. Processing this index reduction it is important to take care of the manifolds contained in the differentialalgebraic equation (DAE). We investigate for several discretization schemes for the NavierStokes equations how the consideration of the manifolds is taken into account and propose a variant of solving these equations along the lines of the theoretically best index reduction. Applying this technique, the error of the time discretisation depends only on the method applied for solving the DAE.

4 
A note on anisotropic interpolation error estimates for isoparametric quadrilateral finite elementsApel, Th. 30 October 1998 (has links)
Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions.
The case of affine elements (parallelepipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multilinear map leads to estimates for certain classes of more general, isoparametric elements.
As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.

5 
Twopoint boundary value problems with piecewise constant coefficients: weak solution and exact discretizationWindisch, G. 30 October 1998 (has links)
For twopoint boundary value problems in weak formulation with piecewise constant coefficients and piecewise continuous righthand side functions we derive a representation of its weak solution by local Green's functions. Then we use it to generate exact threepoint discretizations by Galerkin's method on essentially arbitrary grids. The coarsest possible grid is the set of points at which the piecewise constant coefficients and the right hand side functions are discontinuous. This grid can be refined to resolve any solution properties like boundary and interior layers much more correctly. The proper basis functions for the Galerkin method are entirely defined by the local Green's functions. The exact discretizations are of completely exponentially fitted type and stable. The system matrices of the resulting tridiagonal systems of linear equations are in any case irreducible Mmatrices with a uniformly bounded norm of its inverse.

6 
Variable preconditioning procedures for elliptic problemsJung, M., Nepomnyaschikh, S. V. 30 October 1998 (has links)
For solving systems of grid equations approximating elliptic boundary value problems a method of constructing variable preconditioning procedures is presented. The main purpose is to discuss how an efficient preconditioning iterative procedure can be constructed in the case of elliptic problems with disproportional coefficients, e.g. equations with a large coefficient in the reaction term (or a small diffusion coefficient). The optimality of the suggested technique is based on fictitious space and multilevel decom position methods. Using an additive form of the preconditioners, we intro duce factors into the preconditioners to optimize the corresponding conver gence rate. The optimization with respect to these factors is used at each step of the iterative process. The application of this technique to twolevel $p$hierarchical precondi tioners and domain decomposition methods is considered too.

7 
A new method for computing the stable invariant subspace of a real Hamiltonian matrix or Breaking Van Loans curse?Benner, P., Mehrmann, V., Xu., H. 30 October 1998 (has links)
A new backward stable, structure preserving method of complexity
O(n^3) is presented for computing the stable invariant subspace of
a real Hamiltonian matrix and the stabilizing solution of the
continuoustime algebraic Riccati equation. The new method is based
on the relationship between the invariant subspaces of the
Hamiltonian matrix H and the extended matrix /0 H\ and makes use
\H 0/
of the symplectic URVlike decomposition that was recently
introduced by the authors.

8 
Rankrevealing topdown ULV factorizationsBenhammouda, B. 30 October 1998 (has links)
Rankrevealing ULV and URV factorizations are useful tools to determine the rank and to compute bases for nullspaces of a matrix. However, in the practical ULV (resp. URV ) factorization each left (resp. right) null vector is recomputed from its corresponding right (resp. left) null vector via triangular solves. Triangular solves are required at initial factorization, refinement and updating. As a result, algorithms based on these factorizations may be expensive, especially on parallel computers where triangular solves are expensive. In this paper we propose an alternative approach. Our new rankrevealing ULV factorization, which we call ¨topdown¨ ULV factorization ( TDULV factorization) is based on right null vectors of lower triangular matrices and therefore no triangular solves are required. Right null vectors are easy to estimate accurately using condition estimators such as incremental condition estimator (ICE). The TDULV factorization is shown to be equivalent to the URV factorization with the advantage of circumventing triangular solves.

9 
The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edgesApel, Th., Nicaise, S. 30 October 1998 (has links)
This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones.

10 
Grundlagen der DifferentialgeometrieMeyer, Arnd, Steinbrecher, Andreas 30 November 2000 (has links)
Vorlesungsscript zur Lehrveranstaltung ¨Differentialgeometrie¨ fuer Mathematiker, Lehramtsstudenten/Gymnasiallehrer Mathematik, Ingenieure und andere Interessenten.
Dieses Vorlesungsscript gibt Einblicke in die Grundlagen der Vektorrechnung, der Kurventheorie und der Flaechentheorie im zwei bzw. dreidimensionalen Raum. Die entwickelten Theorien werden durch Anwendungsbeispiele untermauert. Weiterhin ist eine Formelsammlung zur Kurven und Flaechentheorie enthalten.

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