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  • 1
    Electronic Resource
    Electronic Resource
    Analysis of mixed finite element methods on locally refined grids (1992)
    Springer
    Numerische Mathematik 63 (1992), S. 183-194 
    Details
    ISSN: 0945-3245
    Keywords: 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the mixed finite element method for locally refined triangulations. A local projection operator $$\hat \Pi _h $$ is defined to satisfy the commutation property that is required in the general theory of mixed methods. Our results can be applied to every known space of arbitrary order over rectangles or triangles. Optimal-order error estimates and superconvergence for the flux along the Gauss lines are established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385855
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  • 2
    Electronic Resource
    Electronic Resource
    Determination of a source term in a linear parabolic partial differential equation (1976)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 27 (1976), S. 393-401 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenfassung Wir betrachten eine lineare parabolische Differentialgleichung, deren Koeffizienten zeitunabhängig sind. Die Bestimmung der unbekannten zeitunabhängigen Wärmequelle aus dem seitlichen Wärmefluss, zusätzlich zu den üblichen Randbedingungen ist kein wohlbestimmtes Problem im Sinne von Hadamard. Wir betrachten die stetige Abhängigkeit der Lösungen von diesen Daten und zeigen, dass die asymptotische Abhängigkeit höchstens logarithmisch ist, wenn die Norm der Daten gegen Null strebt.
    Notes: Abstract The determination of a spatially dependent source term in a linear parabolic differential equation whose coefficients depend only on the space variable from the specification of the heat flux at one of the boundaries in addition to the usual boundary-initial conditions is a not wellposed problem in the sense of Hadamard. Continuous dependence upon the data is studied and it is demonstrated that the asymptotic dependence as the norm of the data tends to zero is no worse than logarithmic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01590512
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  • 3
    Electronic Resource
    Electronic Resource
    Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis (1993)
    Springer
    Computational mechanics 12 (1993), S. 97-121 
    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 In this paper, we develop Eulerian-Lagrangian localized adjoint methods (ELLAM) to solve the initial-boundary value problems for linear advection or advection-reaction equations. In contrast to many methods for advection-type problems, our ELLAM scheme naturally incorporates the inflow boundary conditions into its formulations and does not need an artificial outflow boundary condition. It does conserve mass. Moreover, optimal-order error estimates for ELLAM have been obtained. In contrast, many methods have only suboptimal-order estimates when applied to solve these problems. Furthermore, our ELLAM scheme provides a systematic approach to treat the interface problems of advection-type equations and can be naturally combined with domain decomposition and local refinement techniques to solve these problems. Numerical results in one and two dimensions are presented and discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00370489
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  • 4
    Electronic Resource
    Electronic Resource
    Numerical well model for non-Darcy flow through isotropic porous media[*]This work was supported in part by the EPA under grant # R 825207-01-1, by the State of Texas under ARP/ATP grant # 010366-168 and by a gift grant from Mobil Oil Corp. (1999)
    Springer
    Computational geosciences 3 (1999), S. 185-204 
    Details
    ISSN: 1573-1499
    Keywords: Forchheimer flow ; well models ; porous media ; finite element ; mixed finite element ; 65c20 ; 65n30 ; 65m60
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Computer Science
    Notes: Abstract Numerical simulation of fluid flow in a hydrocarbon reservoir has to account for the presence of wells. The pressure of a grid cell containing a well is different from the average pressure in that cell and different from the bottom-hole pressure for the well [17]. This paper presents a study of grid pressures obtained from the simulation of single phase flow through an isotropic porous medium using different numerical methods. Well equations are proposed for Darcy flow with Galerkin finite elements and mixed finite elements. Furthermore, high velocity (non-Darcy) flow well equations are developed for cell-centered finite difference, Galerkin finite element and mixed finite element techniques.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1011543412675
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  • 5
    Electronic Resource
    Electronic Resource
    The analysis of multigrid algorithms for cell centered finite difference methods (1996)
    Springer
    Advances in computational mathematics 5 (1996), S. 15-29 
    Details
    ISSN: 1572-9044
    Keywords: Primary 65N30 ; Secondary 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we examine multigrid algorithms for cell centered finite difference approximations of second order elliptic boundary value problems. The cell centered application gives rise to one of the simplest non-variational multigrid algorithms. We shall provide an analysis which guarantees that the W-cycle and variable V-cycle multigrid algorithms converge with a rate of iterative convergence which can be bounded independently of the number of multilevel spaces. In contrast, the natural variational multigrid algorithm converges much more slowly.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02124733
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  • 6
    Electronic Resource
    Electronic Resource
    Preconditioned conjugate gradient methods for large-scale fluid flow applications (1989)
    Springer
    BIT 29 (1989), S. 850-866 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The simulation of large-scale fluid flow applications often requires the efficient solution of extremely large nonsymmetric linear and nonlinear sparse systems of equations arising from the discretization of systems of partial differential equations. While preconditioned conjugate gradient methods work well for symmetric, positive-definite matrices, other methods are necessary to treat large, nonsymmetric matrices. The applications may also involve highly localized phenomena which can be addressed via local and adaptive grid refinement techniques. These local refinement methods usually cause non-standard grid connections which destroy the bandedness of the matrices and the associated ease of solution and vectorization of the algorithms. The use of preconditioned conjugate gradient or conjugate-gradient-like iterative methods in large-scale reservoir simulation applications is briefly surveyed. Then, some block preconditioning methods for adaptive grid refinement via domain decomposition techniques are presented and compared. These techniques are being used efficiently in existing large-scale simulation codes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932749
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  • 7
    Electronic Resource
    Electronic Resource
    From Single-Phase to Compositional Flow: Applicability of Mixed Finite Elements (1997)
    Springer
    Transport in porous media 27 (1997), S. 225-242 
    Details
    ISSN: 1573-1634
    Keywords: fractional flow ; mixed methods ; finite elements ; multiphase flow ; porous media.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Technology
    Notes: Abstract In this paper we discuss the formulation of the governing equations that describe flow of fluids in porous media. Various types of fluid flow, ranging from single-phase flow to compositional flow, are considered. It is shown that all the differential equations governing these types of flow can be effectively rewritten in a fractional flow formulation; i.e., in terms of a global pressure and saturation (or saturations), and that mixed finite element methods can be accurately exploited to solve the pressure equation. Numerical results are presented to see the performance of the mixed methods for the flow equations in three space dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1006507816183
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  • 8
    Electronic Resource
    Electronic Resource
    Simulation of multiphase flows in porous media (1991)
    Springer
    Transport in porous media 6 (1991), S. 479-499 
    Details
    ISSN: 1573-1634
    Keywords: Multiphase flow ; numerical simulation ; viscous fingering
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Technology
    Notes: Abstract The ability to numerically simulate single phase and multiphase flow of fluids in porous media is extremely important in developing an understanding of the complex phenomena governing the flow. The flow is complicated by the presence of heterogeneities in the reservoir and by phenomena such as diffusion, dispersion, and viscous fingering. These effects must be modeled by terms in coupled systems of nonlinear partial differential equations which form the basis of the simulator. The simulator must be able to handle both single and multiphase flows and the transition regimes between the two. A discussion of some of the aspects of modeling dispersion and viscous fingering is presented along with directions for future work. The partial differential equation models are convection-dominated and contain important local effects. An operator-splitting technique is used to address these different effects accurately. Convection is treated by time stepping along the characteristics of the associated pure convection problem, and diffusion is modeled via a Galerkin method for single phase flow and a Petrov-Galerkin technique for multiphase regimes. ELLAM (Eulerian-Lagrangian Localized Adjoint Methods) are discussed to effectively treat the advection-dominated processes. Accurate approximations of the fluid velocities needed in the Eulerian-Lagrangian time-stepping procedure are obtained by mixed finite element methods. Adaptive local grid refinement techniques are then indicated to resolve important local phenomena around wells and large heterogeneities or to resolve the moving internal boundary layers which often govern the mass transfer between phases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00137846
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  • 9
    Electronic Resource
    Electronic Resource
    A coupled non-linear hyperbolic-sobolev system (1977)
    Springer
    Annali di matematica pura ed applicata 114 (1977), S. 331-349 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A boundary-initial value problem for a quasilinear hyperbolic system in one space variable is coupled to a boundary-initial value problem for a quasilinear equation of Sobolev type in two space variables of the form Mut(x, t)+L(t) u (x, t)=f(x, t, u(x, t)) where M and L(t) are second order elliptic spacial operators. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the equation of Sobolev type. Such a coupling can arise in the consideration of oil flowing in a fissured medium and out of that medium via a pipe. Barenblatt, Zheltov, and Kochina[2] have modeled flow in a fissured medium via a special case of the above equation. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, some applications of results of R. Showalter and the contraction mapping theorem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02413794
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  • 10
    Electronic Resource
    Electronic Resource
    Application of conjugate-gradient-like methods to a hyperbolic problem in porous-media flow (1987)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 7 (1987), S. 551-566 
    Details
    ISSN: 0271-2091
    Keywords: Generalized Conjugate Residual Method ; Nested Factorization ; Buckley-Leverett Equation ; Preconditioned Conjugate Gradients ; 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: This paper presents the application of a preconditioned conjugate-gradient-like method to a non-self-adjoint problem of interest in underground flow simulation. The method furnishes a reliable iterative solution scheme for the non-symmetric matrices arising at each iteration of the non-linear time-stepping scheme. The method employs a generalized conjugate residual scheme with nested factorization as a preconditioner. Model runs demonstrate significant computational savings over direct sparse matrix solvers.
    Additional Material: 12 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650070603
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