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  • 1
    Electronic Resource
    Electronic Resource
    A basic norm equivalence for the theory of multilevel methods (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 455-476 
    Details
    ISSN: 0945-3245
    Keywords: 65N55 ; 65N50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Subspace decompositions of finite element spaces based onL 2-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors have proved the equivalence of the associated discrete norms with theH 1-norm. The present paper gives an elementary, self-contained derivation of this result which is based on the use ofK-functionals known from the theory of interpolation spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01388699
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  • 2
    Electronic Resource
    Electronic Resource
    A mathematical investigation of the Car-Parrinello method (1998)
    Springer
    Numerische Mathematik 78 (1998), S. 359-376 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 34C15, 34C40, 70F20, 81Q15, 81V55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the quantitative influence of the involved parameter $\mu$ , the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order ${\cal O}(\mu^{1/2})$ – provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain non-degeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050316
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  • 3
    Electronic Resource
    Electronic Resource
    Adaptive accuracy control for Car-Parrinello simulations (1999)
    Springer
    Numerische Mathematik 83 (1999), S. 179-186 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65M99, 34C15, 34C40, 70F20, 81Q15, 81V55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The Car-Parrinello (CP) approach to ab initio molecular dynamics serves as an approximation to time-dependent Born-Oppenheimer (BO) calculations. It replaces the explicit minimization of the energy functional by a fictitious Newtonian dynamics and therefore introduces an artificial mass parameter $\mu$ which controls the electronic motion. A recent theoretical investigation shows that the CP-error, i.e., the deviation of the CP–solution from the BO-solution decreases like $\mu^{1/2}$ asymptotically. Since the computational effort increases like $\mu^{-1/2}$ , the choice of $\mu$ has to find a compromise between efficiency and accuracy. The asymptotical result is used in this paper to construct an easily implemented algorithm which automatically controls $\mu$ : the parameter $\mu$ is repeatedly adapted during the simulation by choosing $\mu$ as large as possible while pushing an error measure below a user-given tolerance. The performance and reliability of the algorithm is illustrated by a typical example.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050445
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  • 4
    Electronic Resource
    Electronic Resource
    The cascadic multigrid method for elliptic problems (1996)
    Springer
    Numerische Mathematik 75 (1996), S. 135-152 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65F10, 65N30, 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, whichused the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050234
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  • 5
    Electronic Resource
    Electronic Resource
    Quantum-classical molecular dynamics as an approximation to full quantum dynamics (1996)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 105 (1996), S. 1074-1083 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a partial classical limit of the full Schrödinger equation. This limit is achieved in two steps: separation of the full wave function and short wave asymptotics for its "classical'' part. Both steps can be rigorously justified under the same smallness assumptions. This throws some light on the time-dependent self-consistent-field method and on mixed quantum-semiclassical models, which also depend on the separation step. On the other hand, the theory leads to a characterization of the critical situations in which the QCMD model is in danger of largely deviating from the solution of full Schrödinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of a classical particle with a harmonic quantum oscillator. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.471952
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  • 6
    Electronic Resource
    Electronic Resource
    An adaptive Rothe method for the wave equation (1998)
    Springer
    Computing and visualization in science 1 (1998), S. 137-144 
    Details
    ISSN: 1433-0369
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus perturbing the virtual function space method. The admissible size of that perturbation can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s007910050013
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  • 7
    Electronic Resource
    Electronic Resource
    Adaptive multivlevel methods in three space dimensions (1993)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 36 (1993), S. 3187-3203 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider the approximate solution of self-adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. As the performance of hierarchical bases preconditioners deteriorates in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.
    Additional Material: 16 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620361808
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  • 8
    Book
    Book
    Numerische Mathematik II: Integration gewöhnlicher Differentialgleichungen; II (1994)
    Berlin [u. a.] :de Gruyter,
    Details
    Title: Numerische Mathematik II: Integration gewöhnlicher Differentialgleichungen; II
    Author: Deuflhard, Peter
    Contributer: Bornemann, Folkmar
    Publisher: Berlin [u. a.] :de Gruyter,
    Year of publication: 1994
    Pages: XV, 383 S.
    Series Statement: Numerische Mathematik II
    ISBN: 3-11-013937-5
    Type of Medium: Book
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  • 9
    Book
    Book
    Gewöhnliche Differentialgleichungen; [Bd.] II, 2 (2002)
    Berlin u.a. :de Gruyter,
    Details
    Title: Gewöhnliche Differentialgleichungen; [Bd.] II, 2
    Author: Deuflhard, Peter
    Contributer: Bornemann, Folkmar
    Edition: 2. vollst. überarb. erw. Aufl.
    Publisher: Berlin u.a. :de Gruyter,
    Year of publication: 2002
    Pages: 509 S.
    Series Statement: Numerische Mathematik 2
    Type of Medium: Book
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  • 10
    Book
    Book
    Numerische Mathematik /
    Berlin :de Gruyter
    Details
    Title: Numerische Mathematik /
    Author: Deuflhard, Peter
    Contributer: Hohmann, Andreas , Bornemann, Folkmar , Weiser, Martin
    Publisher: Berlin :de Gruyter
    Series Statement: De-Gruyter-Lehrbuch
    Type of Medium: Book
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