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  • 1
    Electronic Resource
    Electronic Resource
    Multilevel iterative methods for mixed finite element discretizations of elliptic problems (1992)
    Springer
    Numerische Mathematik 63 (1992), S. 503-520 
    Details
    ISSN: 0945-3245
    Keywords: 65N30 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385872
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Stabilizing the Hierarchical Basis by Approximate Wavelets, I: Theory (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 4 (1997), S. 103-126 
    Details
    ISSN: 1070-5325
    Keywords: hierarchical basis ; multilevel methods ; preconditioning ; fine element elliptic equations ; approximate wavelets ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper proposes a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L2-projections onto finite element spaces of relatively coarse levels. The corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners, and one action of such a preconditioner is of optimal order computationally. The results are regularity-free for the continuous problem (second order elliptic) and can be applied to problems with rough coefficients and local refinement. © 1997 by John Wiley & Sons, Ltd.
    Type of Medium: Electronic Resource
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    Paper (German National Licenses)
  • 3
    Electronic Resource
    Electronic Resource
    Preconditioning Mixed Finite Element Saddle-point Elliptic Problems (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 1-20 
    Details
    ISSN: 1070-5325
    Keywords: preconditioning saddle-point problems ; eigenvalue estimation ; mixed finite element method ; minimum residual method ; second-order elliptic problems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider saddle-point problems that typically arise from the mixed finite element discretization of second-order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form(δ is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditioner for the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
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    Paper (German National Licenses)
  • 4
    Electronic Resource
    Electronic Resource
    Domain embedding preconditioners for mixed systems (1998)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 5 (1998), S. 321-345 
    Details
    ISSN: 1070-5325
    Keywords: second order elliptic problems ; Dirichlet boundary conditions ; mixed finite elements ; preconditioning ; domain embedding ; auxiliary space method ; non-conforming finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we study block diagonal preconditioners for mixed systems derived from the Dirichlet problems for second order elliptic equations. The main purpose is to discuss how an embedding of the original computational domain into a simpler extended can be utilized in this case. We show that a family of uniform preconditioners for the corresponding problem on the extended, or fictitious, domain leads directly to uniform preconditioners for the original problem. This is in contrast to the situation for the standard finite element method, where the domain embedding approach for the Dirichlet problem is less obvious. Copyright © 1999 John Wiley & Sons, Ltd.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 5
    Book
    Book
    Iterative methods in linear algebra II. Proceedings of the 2nd IMACS international symposium on iterative methods in linear algebra, Blagoevgrad, Bulgaria, June 17-20, 1995; 3 (1996)
    Piscataway, NJ :IMACS,
    Details
    Title: Iterative methods in linear algebra II. Proceedings of the 2nd IMACS international symposium on iterative methods in linear algebra, Blagoevgrad, Bulgaria, June 17-20, 1995; 3
    Contributer: Margenov, Svetozar D. , Vassilevski, Panayot S. , International Association for Mathematics & Computers in Simulation
    Publisher: Piscataway, NJ :IMACS,
    Year of publication: 1996
    Pages: 493 S.
    Series Statement: IMACS series in computational and applied mathematics 3
    Type of Medium: Book
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