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  • 1
    Electronic Resource
    Electronic Resource
    Superconvergence analysis of approximate boundary-flux calculations (1992)
    Springer
    Numerische Mathematik 63 (1992), S. 483-501 
    Details
    ISSN: 0945-3245
    Keywords: 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Certain projection post-processing techniques have been proposed for computing the boundary flux for two-dimensional problems (e.g., see Carey, et al. [5]). In a series of numerical experiments on elliptic problems they observed that these post-processing formulas for approximate fluxes were almost (O(h 2)-accurate for linear triangular elements. In this paper we prove that the computed boundary flux isO(h 2 ln 1/h)-accurate in the maximum norm for the partial method of [5]. If the solutionuφH 3(Ω) then the boundary flux error isO(h 3/2) in theL 2-norm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385871
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  • 2
    Electronic Resource
    Electronic Resource
    Convergence analysis of least-squares mixed finite elements (1993)
    Springer
    Computing 51 (1993), S. 111-123 
    Details
    ISSN: 1436-5057
    Keywords: 65N30 ; 65N15 ; 65L10 ; Least-squares ; mixed finite element method ; superconvergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Methode der kleinsten Fehlerquadrate wird bei gemischten finiten Elementen für die Differentialgleichungen erster Ordnung angewandt, die den linearen elliptischen 1-D Randwertaufgaben zweiter Ordnung entsprechen. Es werden theoretische Untersuchungen vorgestellt und Fehlerabschätzungen vorgenommen. Diese sind mit den früther von Carey and Shen [5] veröffentlichten numerischen Analysen konsistent. Die LBB Bedingung ist nicht notwendig und es werden Abschätzungen für verschiedene Grade der Austatz-Polynome vorgenommen. Superkonvergente Abschätzungen werden ebenso vorgestellt.
    Notes: Abstract The least-squares finite element method for first order systems corresponding to second order linear two-point boundary value problems is considered. A theoretical analysis and error estimates are developed and the estimates are seen to be consistent with our previous numerical studies in Carey and Shen [5]. The method is not subject to the LBB condition and we consider, in particular, the estimates when the polynomial degree differs for the corresponding variables. Superconvergence estimates are also developed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243846
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  • 3
    Electronic Resource
    Electronic Resource
    Least-squares finite element methods for compressible Euler equations (1990)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 10 (1990), S. 557-568 
    Details
    ISSN: 0271-2091
    Keywords: Compressible Euler equations ; Finite element ; Least-squares method ; Shock resolution ; 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 method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L2-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L2-method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.
    Additional Material: 9 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650100504
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  • 4
    Electronic Resource
    Electronic Resource
    Periodic viscous flow: A bench-mark problem (1990)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 11 (1990), S. 87-97 
    Details
    ISSN: 0271-2091
    Keywords: Periodic ; Unsteady ; Viscous flow ; Finite elements ; 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 double-transform technique provides a semi-analytic solution in the form of a series expansion for unsteady axisymmetric Stokes flow in the entrance region of a semi-infinite rigid cylindrical tube. This in turn offers an appropriate bench-mark problem for evaluating the quality of numerical approximations. To illustrate this, periodic axial flow in a circular cylinder is considered. Some aspects of the bench-mark problem that are of interest include the reverse flow in the wall layers, the accuracy of the approximate method in different flow regimes and the mesh grading. This bench-mark problem and the numerical study provide some insight into practical issues pertinent to the approximate solution of unsteady and periodic flows.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650110106
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  • 5
    Electronic Resource
    Electronic Resource
    A least squares finite element method for viscoelastic fluid flow problems (1993)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 17 (1993), S. 943-953 
    Details
    ISSN: 0271-2091
    Keywords: Viscoelastic ; Least squares ; Finite elements ; 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: Viscoelastic flows remain a demanding class of problems for approximate analysis, particularly at increasing Weissenberg numbers. Part of the difficulty stems from the convective behavior and in the treatment of the stress field as a primary unknown. This latter aspect has led to the use of higher-order piecewise approximations for the stress approximation spaces in recent finite element research. The computational complexity of the discretized problem is increased significantly by this approach but at present it appears the most viable technique for solving these problems. Motivated by recent success in treating mixed systems and convective problems, we formulate here a least squares finite element method for the viscoelastic flow problem. Numerical experiments are conducted to test the method and examine its strengths and limitations. Some difficulties and open issues are identified through the numerical experiments. We consider the use of high degree elements (p refinement) to improve performance and accuracy.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650171103
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  • 6
    Electronic Resource
    Electronic Resource
    A perturbation analysis and finite element approximate model for free surface flow over curved beds (1991)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 31 (1991), S. 493-507 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A perturbation analysis is developed for 1-D shallow water flow over a curved bed for applications such as spillways. The perturbation approach leads to a new formulation of the problem with associated weak integral statement and approximation using finite elements. The flow may exhibit a hydraulic jump in the downstream regime. An artificial dissipation technique is introduced to stabilize the non-linear problem and suppress numerical oscillations. Numerical results demonstrate the performance of the model and compare it with the steep-slope shallow water formulation corresponding to the model with zero curvature.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620310306
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  • 7
    Electronic Resource
    Electronic Resource
    Viscous flow and transport with moving free and reactive surfaces (1990)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 30 (1990), S. 1181-1194 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Transport of a chemical species through a viscous fluid with reaction at an interface is modelled by a quasi-static process in which the transient behaviour is dominated by motion of the reactive surface. The reaction is accompanied by a molar volume change which leads to motion of the free surface. A decoupled stream function-vorticity formulation is introduced in conjunction with a moving finite element method. This also permits convenient treatment of the free and reactive surface boundary conditions. The model is applied to the growth of oxide films on silicon surfaces and reveals the effect of surface curvature on film growth.
    Additional Material: 10 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620300607
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  • 8
    Electronic Resource
    Electronic Resource
    Mesh generation/refinement using fractal concepts and iterated function systems (1992)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 33 (1992), S. 287-305 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A novel method of mesh generation is proposed which is based on the use of fractal concepts to derive contractive, affine transformations. The transformations are constructed in such a manner that the attractors of the resulting maps are a union of the points, lines and surfaces in the domain. In particular, the mesh nodes may be generated recursively as a sequence of points which are obtained by applying the transformations to a coarse background mesh constructed from the given boundary data. A Delaunay triangulation or similar edge connection approach can then be performed on the resulting set of nodes in order to generate the mesh. Local refinement of an existing mesh can also be performed using the procedure. The method is easily extended to three dimensions, in which case the Delaunay triangulation is replaced by an analogous 3-D tesselation.
    Additional Material: 16 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620330205
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  • 9
    Electronic Resource
    Electronic Resource
    Shape optimization using adaptive shape refinement (1993)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 36 (1993), S. 2435-2451 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A large part of the computational effort in shape optimization problems is expended in the numerical computation of the gradients for sensitivity information. This effort increases dramatically with an increase in the number of variables used to represent the shape. An adaptation of the gradient projection algorithm for shape optimization problems is described here along with a method to reduce the intermediate size of the optimization problem by allowing adaptive refinement of the shape. The method is demonstrated with a simple representative test case.
    Additional Material: 9 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620361408
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  • 10
    Electronic Resource
    Electronic Resource
    Stream function vorticity solution using high-p element-by-element techniques (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 387-395 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A p-type finite-element scheme is introduced for Navier-Stokes problems in a fully coupled stream function/vorticity formulation. The resulting coupled non-linear system is solved using a Newton iteration with incremental continuation in the Reynold's number. A biconjugate gradient (BCG) scheme in an element-by-element setting is applied to solve the linearized system at each Newton iteration. Pseudo-element block preconditioners produce a practical PBCG method, even for ill-conditioned systems arising from high-p elements in the case of Stokes flow. The problem at higher Reynolds numbers further limits p.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090504
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