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  • 1
    Electronic Resource
    Electronic Resource
    Local refinement techniques for elliptic problems on cell-centered grids (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 431-452 
    Details
    ISSN: 0945-3245
    Keywords: 65F10 ; 65N20 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Algebraic multilevel analogues of the BEPS preconditioner designed for solving discrete elliptic problems on grids with local refinement are formulated, and bounds on their relative condition numbers, with respect to the composite-grid matrix, are derived. TheV-cycle and, more generally,v-foldV-cycle multilevel BEPS preconditioners are presented and studied. It is proved that for 2-D problems theV-cycle multilevel BEPS is almost optimal, whereas thev-foldV-cycle algebraic multilevel BEPS is optimal under a mild restriction on the composite cell-centered grid. For thev-fold multilevel BEPS, the variational relation between the finite difference matrix and the corresponding matrix on the next-coarser level is not necessarily required. Since they are purely algebraically derived, thev-fold (v〉1) multilevel BEPS preconditioners perform without any restrictionson the shape of subregions, unless the refinement is too fast. For theV-cycle BEPS preconditioner (2-D problem), a variational relation between the matrices on two consecutive grids is required, but there is no restriction on the method of refinement on the shape, or on the size of the subdomains.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385790
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  • 2
    Electronic Resource
    Electronic Resource
    Superconvergence analysis of approximate boundary-flux calculations (1992)
    Springer
    Numerische Mathematik 63 (1992), S. 483-501 
    Details
    ISSN: 0945-3245
    Keywords: 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Certain projection post-processing techniques have been proposed for computing the boundary flux for two-dimensional problems (e.g., see Carey, et al. [5]). In a series of numerical experiments on elliptic problems they observed that these post-processing formulas for approximate fluxes were almost (O(h 2)-accurate for linear triangular elements. In this paper we prove that the computed boundary flux isO(h 2 ln 1/h)-accurate in the maximum norm for the partial method of [5]. If the solutionuφH 3(Ω) then the boundary flux error isO(h 3/2) in theL 2-norm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385871
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  • 3
    Electronic Resource
    Electronic Resource
    Convergence analysis of least-squares mixed finite elements (1993)
    Springer
    Computing 51 (1993), S. 111-123 
    Details
    ISSN: 1436-5057
    Keywords: 65N30 ; 65N15 ; 65L10 ; Least-squares ; mixed finite element method ; superconvergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Methode der kleinsten Fehlerquadrate wird bei gemischten finiten Elementen für die Differentialgleichungen erster Ordnung angewandt, die den linearen elliptischen 1-D Randwertaufgaben zweiter Ordnung entsprechen. Es werden theoretische Untersuchungen vorgestellt und Fehlerabschätzungen vorgenommen. Diese sind mit den früther von Carey and Shen [5] veröffentlichten numerischen Analysen konsistent. Die LBB Bedingung ist nicht notwendig und es werden Abschätzungen für verschiedene Grade der Austatz-Polynome vorgenommen. Superkonvergente Abschätzungen werden ebenso vorgestellt.
    Notes: Abstract The least-squares finite element method for first order systems corresponding to second order linear two-point boundary value problems is considered. A theoretical analysis and error estimates are developed and the estimates are seen to be consistent with our previous numerical studies in Carey and Shen [5]. The method is not subject to the LBB condition and we consider, in particular, the estimates when the polynomial degree differs for the corresponding variables. Superconvergence estimates are also developed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243846
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  • 4
    Electronic Resource
    Electronic Resource
    Finite difference schemes on grids with local refinement in time and space for parabolic problems I. Derivation, stability, and error analysis (1990)
    Springer
    Computing 45 (1990), S. 193-215 
    Details
    ISSN: 1436-5057
    Keywords: AMS (MOS) ; 65M05 ; 65M10 ; 65M15 ; Finite difference scheme, cell-centered grids ; local refinement ; refinement in time ; error estimates ; parabolic problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es werden Differenzenverfahren für parabolische Anfangswertprobleme für zellenorientierte räumliche Gitter (rechteckig für zwei Raumdimensionen) mit regulärer lokaler Verfeinerung bzgl. Zeit und Raum vorgestellt. Ihre Stabilitäts- und Konvergenzeigenschaften werden studiert. Die Differenzenverfahren basieren auf der Idee der finiten Volumina durch Approximation der Bilanzgleichungen und erhalten die Masse (bzw. die Energie). Die Approximation an den Gitterpunkten nahe der Überschneidung von feineren und gröberen Gittern verwendet eine frühere Idee der Autoren für selbstadjungierte elliptische Operatoren. Die vorgeschlagenen Verfahren sind implizit vom Typ “Euler rückwärts” und unkonditioniert stabil. Eine Fehleranalyse ist angeschlossen.
    Notes: Abstract Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat). The approximation at the grid points near the fine and coarse grid interface is based on the approach proposed by the authors in a previous paper for selfadjoint elliptic equations. The proposed schemes are implicit of backward Euler type and are shown to be unconditionally stable. Error analysis is also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02250633
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  • 5
    Electronic Resource
    Electronic Resource
    Finite Volume Element Approximations of Nonlocal in Time One-Dimensional Flows in Porous Media (2000)
    Springer
    Computing 64 (2000), S. 157-182 
    Details
    ISSN: 1436-5057
    Keywords: AMS Subject Classifications: 65M12, 65M60, 65N40. ; Key Words: Finite volume method, parabolic equation, integro-differential equation.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract Various finite volume element schemes for parabolic integro-differential equations in 1-D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowest-order (linear and L-splines) finite volume elements, although higher-order volume elements can be considered as well under this framework. It is proved that finite volume element approximations are convergent with optimal order in H 1-norms, suboptimal order in the L 2-norm and super-convergent order in a discrete H 1-norm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s006070050007
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  • 6
    Electronic Resource
    Electronic Resource
    Local refinement techniques for elliptic problems on cell-centered grids; II. Optimal order two-grid iterative methods (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 337-368 
    Details
    ISSN: 1070-5325
    Keywords: Local refinement ; Preconditioning ; Two-grid method ; Cell-centered grids ; Finite differences ; Elliptic problem ; Optimal rate of convergence ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Two preconditioning techniques for solving difference equations arising in finite difference approximation of elliptic problems on cell-centered grids are studied. It is proven that the BEPS and the FAC preconditioners are spectrally equivalent to the corresponding finite difference schemes, including a nonsymmetric one, which is of higher-order accuracy. Numerical experiments that demonstrate the fast convergence of the preconditioned iterative methods (CG and GCG-LS in the nonsymmetric case) are presented.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010403
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  • 7
    Electronic Resource
    Electronic Resource
    Natural and postprocessed superconvergence in semilinear problems (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 245-259 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Superconvergence error estimates are established for a class of semilinear problems defined by a linear elliptic operator with a nonlinear forcing term. The analysis is for rectangular biquadratic elements, and we prove superconvergence of the derivative components along associated lines through the Gauss points. Derivative postprocessing formula and formulas for integrals are also considered and similar superconvergence estimates proven.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070304
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