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  • 1
    Electronic Resource
    Electronic Resource
    Implementation of finite element nonlinear Galerkin methods using hierarchical bases (1993)
    Springer
    Computational mechanics 11 (1993), S. 384-407 
    Details
    ISSN: 1432-0924
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract The nonlinear Galerkin methods are investigated in the framework of finite element discretization We first describe the theoretical background in relation with multilevel and finite element approximations of attractors. Then on the computational side, we recall the definition of the hierarchical bases and analyze the structure associated to these bases. Finally we present the schemes and report on numerical experiments performed on two-dimensional equations of the Burgers and Navier-Stokes type. Their consistency with the approximation that we make and with the structure of the algorithm is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00350095
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  • 2
    Electronic Resource
    Electronic Resource
    Incremental unknowns for solving partial differential equations (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 255-271 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; 65F35 ; 65N20 ; 35A40 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Incremental unknowns have been proposed in [T] as a method to approximate fractal attractors by using finite difference approximations of evolution equations. In the case of linear elliptic problems, the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization; the method is similar but different from the classical multigrid method. In this article we describe the application of incremental unknowns for solving Laplace equations in dimensions one and two. We provide theoretical results concerning two-level approximations and we report on numerical tests done with multi-level approximations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385779
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  • 3
    Electronic Resource
    Electronic Resource
    Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 271-294 
    Details
    ISSN: 0945-3245
    Keywords: 65M06 ; 65M60
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The IMG algorithm (Inertial Manifold-Multigrid algorithm) which uses the first-order incremental unknowns was introduced in [20]. The IMG algorithm is aimed at numerically implementing inertial manifolds (see e.g. [19]) when finite difference discretizations are used. For that purpose it is necessary to decompose the unknown function into its long wavelength and its short wavelength components; (first-order) Incremental Unknowns (IU) were proposed in [20] as a means to realize this decomposition. Our aim in the present article is to propose and study other forms of incremental unknowns, in particular the Wavelet-like Incremental Unknowns (WIU), so-called because of their oscillatory nature. In this report, we first extend the general convergence results in [20] by proving them under slightly weaker conditions. We then present three sets of incremental unknowns (i.e. the first-order as in [20], the second-order and wavelet-like incremental unknowns). We show that these incremental unknown can be used to construct convergent IMG algorithms. Special stress is put on the wavelet-like incremental unknowns since this set of unknowns has theL 2 orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01388690
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  • 4
    Electronic Resource
    Electronic Resource
    Inertial manifolds, partially space-averaged equations, and the separation of scales in turbulent flows* (1995)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 7 (1995), S. 1791-1793 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Recent theoretical results on the behavior of high-frequency modes in two-dimensional incompressible turbulent flows are reported. These results show that high-frequency modes in the dissipation range are slaved to the low-frequency modes. The physical significance of the results is discussed, as well as its utilization for the numerical simulation of turbulent flows. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.868496
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  • 5
    Electronic Resource
    Electronic Resource
    Approximate inertial manifolds and effective viscosity in turbulent flows (1991)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 3 (1991), S. 898-911 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The recently formulated concept of approximate inertial manifolds is exploited as a means for eliminating systematically the fine structure of the velocity field in two-dimensional flows. The resulting iterative procedure does not invoke any statistical properties of the solutions of Navier–Stokes equations. It leads to a modification of those equations, such that effective viscosity-like terms arise in a natural way. The rigorous mathematical considerations can be related to the corresponding physical concepts and intuition. The result leading to a numerical algorithm, essentially a nonlinear Galerkin method, provides a basis for large eddy simulation in which the subgrid model is derived from the properties of the Navier–Stokes equations, rather than from more or less justifiable ad hoc arguments. Some limited speculations concerning the expected results in three dimensions are also offered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.858212
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  • 6
    Electronic Resource
    Electronic Resource
    A nonlinear galerkin method: The two-level fourier-collocation case (1995)
    Springer
    Journal of scientific computing 10 (1995), S. 371-389 
    Details
    ISSN: 1573-7691
    Keywords: Galerkin ; Fourier ; collocation ; multi-resolution ; Burgers equation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract In this article we study the implementation of the Nonlinear Galerkin method as a multiresolution method when a two-level Fourier-collocation discretization is used. The set of collocation points with an even number of points is considered as the fine grid and decomposed into two coarse grids containing half as many points. Using these two grids we decompose the unknown into the sum of a large scale component containing only low frequency modes and based on one of the coarse grids and a small scale component containing only high frequency modes and based on the other coarse grid. This produces interesting connections between the physical space and the Fourier space representations of the function. A nonlinear Galerkin scheme is then applied to the Burgers equation; finally, implementation issues are discussed showing the advantages of the method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02088956
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  • 7
    Electronic Resource
    Electronic Resource
    Foreword (1995)
    Springer
    Theoretical and computational fluid dynamics 7 (1995), S. 241-241 
    Details
    ISSN: 1432-2250
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00312443
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  • 8
    Electronic Resource
    Electronic Resource
    Dual spaces of stresses and strains, with applications to Hencky plasticity (1983)
    Springer
    Applied mathematics & optimization 10 (1983), S. 1-35 
    Details
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We prove the existence of a dual pairing between admissible stress and displacement fields in the context of Hencky plasticity. We apply this to show (i) that an extremal displacement field exists whenever the safe load condition holds; and (ii) that any extremal stress and displacement fields are related by a “saddle point condition,” expressing in weak form the constitutive hypothesis of Hencky plasticity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01448377
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  • 9
    Electronic Resource
    Electronic Resource
    Stochastic Burgers' equation (1994)
    Springer
    Nonlinear differential equations and applications 1 (1994), S. 389-402 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider a Burgers' equation perturbed by white noise. We prove the existence and uniqueness of the global solution as well as the existence of an invariant measure for the corresponding transition semigroup.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01194987
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  • 10
    Electronic Resource
    Electronic Resource
    A co-area formula with applications to monotone rearrangement and to regularity (1990)
    Springer
    Archive for rational mechanics and analysis 109 (1990), S. 213-238 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00375089
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