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  • 1990-1994  (13)
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  • 1
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    An Adaptive Multilevel Approach to Parabolic Equations I. General Theory & 1D-Implementation. (1990)
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    Publication Date: 2014-02-26
    Description: A new adaptive multilevel approach for parabolic PDE's is presented. Full adaptivity of the algorithm is realized by combining multilevel time discretization, better known as extrapolation methods, and multilevel finite element space discretization. In the theoretical part of the paper the existence of asymptotic expansions in terms of time-steps for single-step methods in Hilbert space is established. Finite element approximation then leads to perturbed expansions, whose perturbations, however, can be pushed below a necessary level by means of an adaptive grid control. The theoretical presentation is independent of space dimension. In this part I of the paper details of the algorithm and numerical examples are given for the 1D case only. The numerical results clearly show the significant perspectives opened by the new algorithmic approach.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 2
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    An Adaptive Multilevel Approach to Parabolic Equations III. (1991)
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    Publication Date: 2014-02-26
    Description: Part III of the paper is devoted to the construction of an adaptive FEM solver in two spatial dimensions, which is able to handle the singularly perturbed elliptic problems arising from discretization in time. The problems of error estimation and multilevel iterative solution of the linear systems - both uniformly well behaved with respect to the time step - can be solved simultaneously within the framework of preconditioning. A multilevel nodal basis preconditioner able to handle highly nonuniform meshes is derived. As a numerical example an application of the method to the bioheat-transfer equation is included. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65M50, 65M60, 65N30.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 3
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    Interpolation Spaces and Optimal Multilevel Preconditioners. (1993)
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    Publication Date: 2014-02-26
    Description: This paper throws light on the connection between the optimal condition number estimate for the BPX method and constructive approximation theory. We provide a machinery, which allows to understand the optimality as a consequence of an approximation property and an inverse inequality in $H^{1+\epsilon}$, $\epsilon 〉 0$. This machinery constructs so-called {\em approximation spaces}, which characterize a certain rate of approximation by finite elements and relates them with interpolation spaces, which characterize a certain smoothness.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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    OPUS
  • 4
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    An Adaptive Multilevel Approach to Parabolic Equations in Two Space Dimensions. (1991)
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    Publication Date: 2014-02-26
    Description: A new adaptive multilevel approach for linear partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction. The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A Multilevel nodal basis preconditioner is derived, which allows the use of highly nonuniform triangulations. Implementation issues are discussed in detail. Numerous numerical examples in one and two space dimensions clearly show the significant perspectives opened by the new algorithmic approach. Finally an application of the method is given in the area of hyperthermia, a recent clinical method for cancer therapy.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 5
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    A Sharpened Condition Number Estimate for the BPX Preconditioner of Elliptic Finite Element Problems on Highly Nonuniform Triangulations. (1991)
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    Publication Date: 2014-02-26
    Description: In this paper it is shown that for highly nonuniformly refined triangulations the condition number of the BPX preconditioner for elliptic finite element problems grows at most linearly in the depth of refinement. This is achieved by viewing the computational available version of the BPX preconditioner as an abstract additive Schwarz method with exact solvers. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65N20, 65N30.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 6
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    An Adaptive Multilevel Approach to Parabolic Equations II. (1990)
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    Publication Date: 2014-02-26
    Description: In continuation of part I this paper develops a variable-order time discretization in Hilbert space based on a multiplicative error correction. Matching of time and space errors as explained in part I allows to construct an adaptive multilevel discretization of the parabolic problem. In contrast to the extrapolation method in time, which has been used in part I, the new time discretization allows to separate space and time errors and further to solve fewer elliptic subproblems with less effort, which is essential in view of the application to space dimension greater than one. Numerical examples for space dimension one are included which clearly indicate the improvement.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 7
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    A Basic Norm Equivalence for the Theory of Multilevel Methods. (1992)
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    Publication Date: 2014-02-26
    Description: Subspace decompositions of finite element spaces based on $L2$-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors proved the equivalence of the associated discrete norms with the $H^1$-norm. The present report gives an elementary, self-contained derivation of this result which is based on the use of $ K$-functionals known from the theory of interpolation spaces. {\bf Keywords:} multilevel methods, nonuniform meshes, optimal convergence rates. {\bf AMS(MOS) Subject classifications:} 65N55, 65N30, 65N50.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
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  • 8
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    On the Convergence of Cascadic Iterations for Elliptic Problems. (1994)
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    Publication Date: 2014-02-26
    Description: We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them {\em cascadic iterations}. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments that the CCG method works within a few iterations if the linear systems on coarser triangulations are solved accurately enough. Shaidurov subsequently proved multigrid complexity for the CCG method in the case of $H^2$-regular two-dimensional problems with quasi-uniform triangulations. We show that his result still holds true for a large class of smoothing iterations as basic iteration procedure in the case of two- and three-dimensional $H^{1+\alpha}$-regular problems. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new termination criterion for the CCG method.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 9
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    A Posteriori Error Estimates for Elliptic Problems. (1993)
    Details
    Publication Date: 2019-05-10
    Description: {\def\enorm {\mathop{\mbox{\boldmath{$|\!|$}}}\nolimits} Let $u \in H$ be the exact solution of a given self--adjoint elliptic boundary value problem, which is approximated by some $\tilde{u} \in {\cal S}$, $\cal S$ being a suitable finite element space. Efficient and reliable a posteriori estimates of the error $\enorm u - \tilde{u}\enorm $, measuring the (local) quality of $\tilde{u}$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well--known class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.}
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 10
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    Numerische Mathematik. II (1994)
    Details
    Publication Date: 2020-05-04
    Language: English
    Type: book , doc-type:book
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