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  • 1
    Electronic Resource
    Electronic Resource
    Fourier analysis of the Eulerian-Lagrangian least squares collocation method (1990)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 11 (1990), S. 427-444 
    Details
    ISSN: 0271-2091
    Keywords: Fourier analysis ; Eulerian-Lagrangian ; Least squares ; Transport ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A Fourier analysis was performed in order to study the numerical characteristics of the effective Eulerian-Lagrangian least squares collocation (ELLESCO) method. As applied to the transport equation, ELLESCO requires a C1-continuous trial space and has two degrees of freedom per node. Two coupled discrete equations are generated for a typical interior node for a one-dimensional problem. Each degree of freedom is expanded separately in a Fourier series and is substituted into the discrete equations to form a homogeneous matrix equation. The required singularity of the system matrix leads to a ‘physical’ amplification factor that characterizes the numerical propagation of the initial conditions and a ‘computational’ one that can affect stability.Unconditional stability for time-stepping weights greater than or equal to 0-5 is demonstrated. With advection only, ELLESCO accurately propagates spatial wavelengths down to 2Δx. As the dimensionless dispersion number becomes large, implicit formulations accurately propagate the phase, but the higher-wave-number components are underdamped. At large dispersion numbers, phase errors combined with underdamping cause oscillations in Crank-Nicolson solutions. These effects lead to limits on the temporal discretization when dispersion is present. Increases in the number of collocation points per element improve the spectral behaviour of ELLESCO.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650110406
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  • 2
    Electronic Resource
    Electronic Resource
    A brief note on upwind collocation (1983)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 3 (1983), S. 307-313 
    Details
    ISSN: 0271-2091
    Keywords: Collocation ; Finite Element Method ; Upwind Schemes ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Upwind collocation on Hermite cubics is compared to orthogonal collocation with respect to effective diffusion. The one-dimensional constant coefficient advection-diffusion equation is employed to this end. The effective diffusion coefficient is determined exactly and is found to be dependent on the nodal solution values. The effective diffusion coefficients of three other upwinding schemes are also presented. Upwind collocation is found to have an effective diffusion coefficient like other upwinding schemes plus an extra term which may enhance or reduce the non-advective flux, depending on the problem solution and point location within the domain.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650030309
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A least-squares method for solving the mixed form of the groundwater flow equations (1992)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 14 (1992), S. 729-751 
    Details
    ISSN: 0271-2091
    Keywords: Least squares ; Mixed formulation ; Collocation ; Groundwater ; Vertically averaged flow ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The mixed form of the areal groundwater flow equations is solved with a least-squares finite element procedure (LESFEM). Hydraulic head and x- and y-directed fluxes are state variables. Physical parameters and state variables are approximated using a bilinear basis. Grid refinements and irregular domain boundaries are implemented on rectangular meshes.Residuals are constructed at collocation points for conservation of mass and Darcy's law. Boundary condition residuals are constructed at discrete points along the boundary. The residuals are weighted, squared and summed. A set of algebraic equations is formed by taking the derivatives of the weighted sum of the squares of the residuals with respect to each unknown parameter in the approximation for the state variable and setting them to zero.Proper choice of a potential scaling parameter and residual weights is essential for the effective application of the algorithm. Test problem results demonstrate that the method is effective for both transient and steady state cases.The LESFEM algorithm generates a C°-continuous velocity field. The continuous velocity field and the rectangular mesh simplify the implementation of algorithms that require tracking. In addition, rectangular meshes simplify mesh and boundary generation.
    Additional Material: 12 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650140606
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  • 4
    Electronic Resource
    Electronic Resource
    A weighted least squares method for first-order hyperbolic systems (1995)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 20 (1995), S. 191-212 
    Details
    ISSN: 0271-2091
    Keywords: Weighted least squares ; Hyperbolic system of equations ; Shallow water equations ; Newton-Raphson method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The paper presents a generalization of the classical L2-norm weighted least squares method for the numerical solution of a first-order hyperbolic system. This alternative least squares method consists of the minimization of the weighted sum of the L2 residuals for each equation of the system. The order of accuracy of global conservation of each equation of the system is shown to be inversely proportional to the weight associated with the equation. The optimal relative weights between the equations are then determined in order to satisfy global conservation of the energy of the physical system.As an application of the algorithm, the shallow water equations on an irregular domain are first discretized in time and then solved using Laplace modification and the proposed least squares method.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650200302
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  • 5
    Electronic Resource
    Electronic Resource
    Analytical integration formulae for linear isoparametric finite elements (1984)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 20 (1984), S. 1153-1163 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Exact and approximate analytical expressions can be derived for integrals arising in finite element methods, employing isoparametric linear quadrilaterals in two space dimensions with bilinear basis functions. The formulae associated with rectangular elements, arbitrarily oriented in space, can be shown to be a special case. The proposed method provides considerable savings in computational effort, in comparison with a numerical method that employs Gaussian quadrature procedures. In addition, the method, when applied to a quadrilateral inscribable in a circle, can be shown to produce better accuracy than the associated (2 × 2) Gaussian quadrature formulae.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620200613
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    An iterative penalty method for the least squares solution of boundary value problems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 257-281 
    Details
    ISSN: 0749-159X
    Keywords: boundary value problems ; collocation least squares method ; augmented Lagrangian method ; Uzawa's algorithm ; preconditioned conjugate gradient ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This article is concerned with iterative techniques for linear systems of equations arising from a least squares formulation of boundary value problems. In its classical form, the solution of the least squares method is obtained by solving the traditional normal equation. However, for nonsmooth boundary conditions or in the case of refinement at a selected set of interior points, the matrix associated with the normal equation tends to be ill-conditioned. In this case, the least squares method may be formulated as a Powell multiplier method and the equations solved iteratively. Therein we use and compare two different iterative algorithms. The first algorithm is the preconditioned conjugate gradient method applied to the normal equation, while the second is a new algorithm based on the Powell method and formulated on the stabilized dual problem. The two algorithms are first compared on a one-dimensional problem with poorly conditioned matrices. Results show that, for such problems, the new algorithm gives more accurate results. The new algorithm is then applied to a two-dimensional steady state diffusion problem and a boundary layer problem. A comparison between the least squares method of Bramble and Schatz and the new algorithm demonstrates the ability of the new method to give highly accurate results on the boundary, or at a set of given interior collocation points without the deterioration of the condition number of the matrix. Conditions for convergence of the proposed algorithm are discussed. © 1997 John Wiley & Sons, Inc.
    Additional Material: 15 Ill.
    Type of Medium: Electronic Resource
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    Paper (German National Licenses)
  • 7
    Electronic Resource
    Electronic Resource
    Solution of the advective-dispersive transport equation using a least squares collocation, Eulerian-Lagrangian method (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 227-240 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Numerical solution of the advective-dispersive transport equation is difficult when advection dominates. Difficulties arise because of the first-order spatial derivatives which can be elminated by a local coordinate transformation to the characteristic lines of the first order hyperbolic portion of the equation. The resulting differential equation is discretized using a finite difference in time and finite elements in space employing cubic Hermite basis functions. The residuals at individual collocation points are then computed. The sum of the squares of the residuals is minimized to form the necessary set of algebraic equations. The method has performed well in one-dimensional test problems.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690050306
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  • 8
    Electronic Resource
    Electronic Resource
    Least squares collocation solution of differential equations on irregularly shaped domains using orthogonal meshes (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 347-361 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The least squares collocation (LESCO) method has been formulated to solve differential equations defined over irregular domains using a more convenient orthogonal computational mesh. The LESCO method is described in detail for second-order boundary value problems and applied to the time-dependent diffusion and advection-diffusion equations defined over two-dimensional irregular domains. Particular attention is given to the proper procedure for applying boundary conditions. Accuracy, convergence, and consistency are examined. For cubic elements with arbitrary location of collocation points, the convergence rate is between 3rd and 4th order. The major advantages of this method are reduced input data requirements, a more robust procedure for forming the equations, positive definite matrices, and flexibility in distrbuting errors.
    Additional Material: 13 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690050406
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