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  • 1
    Electronic Resource
    Electronic Resource
    Optimal error estimation for Petrov-Galerkin methods in two dimensions (1992)
    Springer
    Numerische Mathematik 61 (1992), S. 359-372 
    Details
    ISSN: 0945-3245
    Keywords: 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We examine the optimality of conforming Petrov-Galerkin approximations for the linear convection-diffusion equation in two dimensions. Our analysis is based on the Riesz representation theorem and it provides an optimal error estimate involving the smallest possible constantC. It also identifies an optimal test space, for any choice of consistent norm, as that whose image under the Riesz representation operator is the trial space. By using the Helmholtz decomposition of the Hilbert space [L 2(Ω)]2, we produce a construction for the constantC in which the Riesz representation operator is not required explicitly. We apply the technique to the analysis of the Galerkin approximation and of an upwind finite element method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385514
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The stability of boundary conditions for an angled-derivative difference scheme (1996)
    Springer
    Advances in computational mathematics 6 (1996), S. 263-279 
    Details
    ISSN: 1572-9044
    Keywords: numerical stability ; Godunov-Ryabenkii conditions ; initial boundary value problem ; angled-derivative difference scheme ; tidally-forced shallow water equations ; 64N99 ; 65L10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Tidal forcing of the shallow water equations is typical of a class of problems where an approximate equilibrium solution is sought by long time integration of a differential equation system. A combination of the angled-derivative scheme with a staggered leap-frog scheme is sometimes used to discretise this problem. It is shown here why great care then needs to be taken with the boundary conditions to ensure that spurious solution modes do not lead to numerical instabilities. Various techniques are employed to analyse two simple model problems and display instabilities met in practical computations; these are then used to deduce a set of stable boundary conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02127707
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  • 3
    Electronic Resource
    Electronic Resource
    Exterior flow with an isoparametric Hermite cubic element (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 1499-1509 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An isoparametric Hermite cubic element is developed for calculating compressible flows past complex aerofoils in two dimensions. A simple intermediate transformation is used to derive an isoparametric transform whose Jacobian is always non-zero and which attains very high accuracy on the boundary. Application of the method and assessment of the error is illustrated by flow past a circular cylinder.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620141006
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  • 4
    Electronic Resource
    Electronic Resource
    A numerical case study of a non-Newtonian flow problem (1988)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 26 (1988), S. 695-704 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper addresses some of the theoretical aspects involved in the numerical study of non-Newtonian flow problems. We consider the second-order Rivlin-Erickson constitutive model due to the simple differential form that emerges for the system of equations that govern the flow when expressed in stream function-vorticity variables. This model describes slightly elastic fluids that exhibit a constant viscosity behaviour. A steady two-dimensional flow is studied through a planar contraction geometry.An auxiliary variable is introduced into the problem formulation producing a non-linear system of differential equations comprising two elliptic equations and one hyperbolic equation. This system is discretized by finite difference methods and the resulting system of non-linear algebraic equations is solved iteratively by successive substitutions. The simple structure of this iteration permits a convergence analysis which is presented in Section 2. This analysis is performed prior to the spatial discretization and establishes the dependence of the iteration upon the material parameters. At the discrete linearized equation level a combination of inner iterations for elliptic equations and direct marching for the hyperbolic equation is used. The stability of the marching scheme is considered in Section 4.3 and a discussion on the results is given in Section 5.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620260312
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  • 5
    Electronic Resource
    Electronic Resource
    Optimal finite element solutions to diffusion-convection problems in one dimension (1980)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 15 (1980), S. 1457-1474 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Petrov-Galerkin methods have been proposed by several authors to eliminate the inaccuracies and oscillations obtained with Galerkin methods when applied to diffusion-convection problems at high Péclet numbers: the difficulty is to select the appropriate test space for a given trial space. We investigate here choices of test space which either exactly or approximately symmetrize the associated bilinear form and so retain the optimal character of the approximate solution. This is the key to high accuracy and superconvergence, and optimal recovery techniques are proposed to obtain the maximum information from the approximations. Examples are given to show how the position and thickness of boundary layers may be estimated with relatively coarse meshes.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620151004
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  • 6
    Electronic Resource
    Electronic Resource
    Stability of finite difference approximations to a diffusion-convection equation (1980)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 15 (1980), S. 677-683 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620150505
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    Paper (German National Licenses)
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