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  • Articles  (25)
  • Articles: DFG German National Licenses  (25)
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  • Mathematics  (25)
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  • Articles: DFG German National Licenses  (25)
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  • Mathematics  (25)
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  • 1
    Electronic Resource
    Electronic Resource
    Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 345-371 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418330
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  • 2
    Electronic Resource
    Electronic Resource
    A mixed finite element approximation of the Navier-Stokes equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 381-404 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399007
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  • 3
    Electronic Resource
    Electronic Resource
    A convergence condition for the quadrilateral Wilson element (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 349-361 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper deals with the convergence properties of the nonconforming quadrilateral wilson element which violates the patch test. The convergence of the element is proved under a certain condition on mesh subdivisions without any modifications of the variational formulation. This result extends the range of applicability of Wilson's element. The necessity of the proposed condition is also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405567
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  • 4
    Electronic Resource
    Electronic Resource
    Approximation de quelques problèmes de type Stokes par une méthode d'éléments finis mixtes (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 75-91 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5-17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary We present in this paper a mixed-finite element approximation of Stokes equations with boundary conditions of Fourier's or Neumann's type. For this approximation we prove that the error estimates for the velocity-vector and for the pressure are optimal. These results of convergence generalize to this kind of boundary conditions the Glowinski-Pironneau's approximation of Stokes problem with Dirichlet's boundary conditions [9, 10].
    Notes: Résumé Nous présentons dans cet article une méthode d'éléments finis mixtes qui permet la résolution des équations de Stokes avec des conditions aux limites de type Fourier ou Neumann. Pour cette méthode nous démontrons que les estimations de l'erreur d'approximation sont optimales; en vitesse et en pression. Ces résultats de convergences généralisent à ce type non standard de conditions aux limites les travaux de Glowinski-Pironneau [9, 10] pour le probleme de Stokes avec des conditions aux limites de Dirichlet.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01379662
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  • 5
    Electronic Resource
    Electronic Resource
    Estimations des erreurs de meilleure approximation polynomiale et d'interpolation de Lagrange dans les espaces de Sobolev d'ordre non entier (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 301-321 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: G18
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary Explicit bounds for the best polynomial approximation error, explicit and non-explicit bounds for the Lagrange interpolation error are derived for functions belonging to fractional order Sobolev spaces defined over a bounded open set in ℝ n . Thus the classical results of the integer order Sobolev spaces are extended.
    Notes: Résumé On établit des majorations explicites de I'erreur de meilleure approximation polynomiale ainsi que des majorations explicites et nonexplicites de I'erreur d'interpolation de Lagrange, lorsque la fonction considérée appartient à un espace de Sobolev d'ordre non entier défini sur un ouvert borné de ℝ n . Les résultats obtenus généralisent les résultats connus dans le cas des espaces de Sobolev d'ordre entier.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389473
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  • 6
    Electronic Resource
    Electronic Resource
    Error estimates for the combinedh andp versions of the finite element method (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 257-277 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398256
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  • 7
    Electronic Resource
    Electronic Resource
    The finite element method for parabolic equations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 339-371 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396451
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  • 8
    Electronic Resource
    Electronic Resource
    Error analysis for Laplace transform —Finite element solution of hyperbolic equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 55-62 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The finite element method with Laplace transform of time variable is proposed for the solution of hyperbolic equations. Error estimates in Hardy spaces of functions with values in Sobolev spaces are derived. Due to the isometric isomorphism of Hardy spaces with weighted Hilbert spaces these estimates are valid also for original formulations of hyperbolic equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396305
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  • 9
    Electronic Resource
    Electronic Resource
    An analysis of thep-version of the finite element method for nearly incompressible materials (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 39-53 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; 73K25 CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we analyze the behavior of the so-calledp-version of the finite element method when applied to the equations of plane strain linear elasticity. We establish optimal rate error estimates that are uniformly valid, independent of the value of the Poisson ratio,v, in the interval ]0, 1/2[. This shows that thep-versiondoes not exhibit the degeneracy phenomenon which has led to the use of various, only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396304
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  • 10
    Electronic Resource
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    On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 173-194 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395310
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  • 11
    Electronic Resource
    Electronic Resource
    Eine Aussage zurL ∞-Stabilität und zur genauen Konvergenzordnung derH 0 1 -Projektionen (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 393-405 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: G.1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we settle a question concerning the necessity of the logh-factor appearing in error estimates of linear finite element solutions. Let Ω⊂ℝ2 be a bounded convex polygon and {T h } a regular family of subdivisions of Ω into triangles. Be furtherS 0 h ⊂H 0 1 (Ω) the corresponding linear finite element spaces andP h :H 0 1 (Ω)→S 0 h theH 0 1 -projections. With respect to the uniform norm asymptotically estimates of the form $$\parallel u - P_h u\parallel _\infty \leqq c|logh|dist(u,S_0^h ), u \in C(\bar \Omega ) \cap H_0^1 (\Omega )$$ hold, as pointed out by Schatz [1980]. For functionsu with bounded second derivatives this implies $$\parallel u - P_h u\parallel _\infty \leqq ch^2 |\log h||u|_{2,\infty } ,$$ a result which was proved by Nitsche [1975] and Scott [1976] with different methods. It was an open question if the factor |logh| is necessary or only due to the methods of proof. In this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if |logh| is replaced by a termo(|logh|).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405570
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  • 12
    Electronic Resource
    Electronic Resource
    Approximation of tricomi problem with Neumann boundary condition (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 371-391 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The Tricomi problem with Neumann boundary condition is reduced to a degenerate problem in the elliptic region with a non-local boundary condition and to a Cauchy problem in the hyperbolic region. A variational formulation is given to the elliptic problem and a finite element approximation is studied. Also some regularity results in weighted Sobolev spaces are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405569
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  • 13
    Electronic Resource
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    Convergence of a penalty-finite element approximation for an obstacle problem (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 105-120 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This study establishes an error estimate for a penalty-finite element approximation of the variational inequality obtained by a class of obstacle problems. By special identification of the penalty term, we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality. The rate of convergence of the penalization is ɛ where ɛ is the penalty parameter. To obtain the error of finite element approximation, we apply the results obtained by Brezzi, Hager and Raviart for the mixed finite element method to the variational inequality.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396189
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  • 14
    Electronic Resource
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    Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 365-382 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396438
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  • 15
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    The numerical solutions of interface problems by infinite element method (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 39-50 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems. Furthermore, approximations of stress intensity factors are given. The infinite element method may be considered as a certain scheme of mesh refinement, but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently, and that an approximate solution which has a singularity at the singular point can be also obtained.
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    URL: http://dx.doi.org/10.1007/BF01399310
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  • 16
    Electronic Resource
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    Analysis of finite element methods for second order boundary value problems using mesh dependent norms (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 41-62 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a new approach to the analysis of finite element methods based onC 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL 2 norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL 2 norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01463997
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  • 17
    Electronic Resource
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    Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 419-437 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem Δu=bu 2, based on piecewise linear polynomials and the lumping operator. These iterations construct the monotonically decreasing and increasing sequences, and convergence proofs are given. Finally, we present some numerical examples verifying the effectiveness of the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390183
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  • 18
    Electronic Resource
    Electronic Resource
    Free boundary problems with nonlinear source terms (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 463-483 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390185
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  • 19
    Electronic Resource
    Electronic Resource
    Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 33-52 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A modified variational formulation, recently introduced by Taylor, Beresford and Wilson for solving second order problems, using the nonconforming Wilson element is here analysed. It is shown that the Patch Test is satisfied and that stresses and displacements are respectively first and second order accurate for arbitrary quadrilateral meshes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395987
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  • 20
    Electronic Resource
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    On mixed finite element methods for first order elliptic systems (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 29-48 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the Galerkin finite element solution of these systems. Using this theory in conjunction with a certain hypothesis concerning approximation spaces, optimal error estimates for Galerkin type approximations are demonstrated. An example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396185
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  • 21
    Electronic Resource
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    Combined finite element and spectral approximation of the Navier-Stokes equations (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 201-217 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Resumé On présente une méthode d'approximation numérique des équations de Navier-Stokes possédant une direction de périodicité. Dans cette direction une méthode pseudospectrale basée sur des développements en série de Fourier est utilisée, dans les deux autres on applique une méthode d'éléments finis standard. On montre que la convergence est optimale et que les deux paramètres de discrétisation peuvent être choisis de façon indépendante.
    Notes: Summary We present a method for the numerical approximation of Navier-Stokes equations with one direction of periodicity. In this direction a Fourier pseudospectral method is used, in the two others a standard F.E.M. is applied. We prove optimal rate of convergence where the two parameters of discretization intervene independently.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01410105
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  • 22
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    On the stability of bilinear-constant velocity-pressure finite elements (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 219-222 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An analysis of the Babuška stability of bilinear/constant finite element pairs for viscous flow calculations is given. An unstable mode not of the checkerboard type is given for which the stability constant turns out to beO(h). Thus, the indicated spaces are not stable in general for numerical calculation.
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    URL: http://dx.doi.org/10.1007/BF01410106
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  • 23
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    On numerical asymptotic behavior of finite element solutions for parabolic equations (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 285-300 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, we investigate the numerical asymptotic behavior of the finite element solutions for linear parabolic equations under some appropriate conditions. We also give some results of numerical experiments in the two dimensional problems to indicate the effectiveness of our results.
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    URL: http://dx.doi.org/10.1007/BF01410112
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  • 24
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    A finite element lumped mass scheme for solving eigenvalue problems of circular arches (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 267-290 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.
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    URL: http://dx.doi.org/10.1007/BF01396655
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  • 25
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    Finite dimensional approximation of nonlinear problems (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 1-30 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Kármán equations.
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    URL: http://dx.doi.org/10.1007/BF01395805
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