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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 49 (1986), S. 379-412 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10, 65F35, 65N20, 65N30 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log κ)2) where κ is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like $$O\left( {\left( {\log \frac{1}{h}} \right)^2 } \right)$$ instead of $$O\left( {\left( {\frac{1}{h}} \right)^2 } \right)$$ for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 50 (1986), S. 123-123 
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 54 (1989), S. 719-734 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10, 65N20, 65N30 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can also be used for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 41 (1983), S. 223-253 
    ISSN: 0945-3245
    Keywords: AMS: 65L05 ; 65L10 ; 34D15 CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we consider singular perturbation problems for ordinary differential operators of ordern and their discrete counterparts on arbitrary nonuniform grids. We prove that the singularly perturbed initial value problem is stable uniformly in the perturbation parameter ε in both the continuous and the discrete case. We use this result to characterize the stability of the corresponding continuous and discrete boundary value problems. If the continuous problem is stable and if the consistency error is smaller than a certain constant, the discrete problem is also stable.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 64 (1993), S. 455-476 
    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
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 76 (1997), S. 111-142 
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 76N99, 76M25, 65M12, 65M99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Fluid mechanics describes the motion of mass in space under the influence of internal and external forces. The particle model presented in this article is based on this fact. The fluid is subdivided into a finite number of small mass packets, the particles. These mass packets have a finite extension and share all properties with the fluid, except for the restriction that they cannot get deformed and can perform only rigid body motions. The forces acting upon the particles are identical to those acting on a part of a fluid. The exact conservation of mass and, for the case of adiabatic flows, also of entropy is automatically guaranteed by the approach. When the particle size tends to zero, the mean local displacement of the particles converges in the weak sense. In the inviscid case, the resulting flows can be regarded as solutions of the Euler equations.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 42 (1983), S. 119-123 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Many difference methods for the numerical solution of elliptic boundary value problems lead to systems of linear equations whose matrices areM-matrices and which therefore have nonnegative inverses. In this paper it is shown, that these difference methods are at most consistent of second order.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 34 (1980), S. 171-187 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called “Mehrstellenformeln’ or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 52 (1988), S. 427-458 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; 65F35 ; 65N20 ; 65N30 ; CR:G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 58 (1990), S. 163-184 
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65F10 ; 65F35 ; 65N20 ; 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The hierarchical basis preconditioner and the recent preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; exclusively local properties like the shape regularity of the finite elements are utilized.
    Type of Medium: Electronic Resource
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