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Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis

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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.

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References

  • Allen, III, M. B.; Behie, G. A. I.; Trangenstein, J. A. (1988): Multiphase flow in porous media. Lecture Notes in Engineering. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Barrett, J. W.; Morton, K. W. (1984): Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems. Comp. Meth. Appl. Mech. Eng. 45, 97–122

    Google Scholar 

  • Bouloutas, E. T.; Celia, M. A. (1988): An analysis of a class of Petrov-Galerkin and optimal test functions methods. In: Celia, M. A. (ed): Proceedings of the Seventh International Conference on Computational Methods in Water Resources, 15–20

  • Brooks, A.; Hughes, T. J. R. (1982): Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng. 32, 199–295

    Google Scholar 

  • Celia, M. A.; Herrera, I.; Bouloutas, E. T. (1989): Adjoint Petrov-Galerkin methods for multi-dimensional flow problems. Proceedings of the Seventh International Conference on Finite Element Methods in Flow Problems, Huntsville, Alabama, 953–958

  • Celia, M. A.; Herrera, I.; Bouloutas, E. T.; Kinder, J. S. (1989): A new numerical approach for the advective-diffusive transport equation. Numerical Methods for PDE's 5, 203–226

    Google Scholar 

  • Celia, M. A.; Russell, T. F.; Herrera, I.; Ewing, R. E. (1990): An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation. Advances in Water Resources 13, 187–206

    Google Scholar 

  • Celia, M. A.; Zisman, S. (1990): An Eulerian-Lagrangian localized adjoint method for reactive transport in groundwater. Computational Methods in Subsurface Hydrology, Proceedings of the Eighth International Conference on Computational Methods in Water Resources, Venice, Italy, 383–392

  • Chavent, G.; Cohen, G.; Jaffre, J. (1984): Discontinuous upwinding and mixed finite elements for two-phase flows in reservoir simulation. Comp. Meth. Appl. Mech. Eng. 47, 93–118

    Google Scholar 

  • Dahle, H. K., Espedal, M. S.; Ewing, R. E. (1988): Characteristic Petrov-Galerkin subdomain methods for convection diffusion problems. In: Wheeler, M. F. (ed): Numerical simulation in oil recovery, IMA vol. 11, 77–88. Berlin: Springer

    Google Scholar 

  • Dahle, H. K. (1988): Adaptive characteristic operator splitting techniques for convection-dominated diffusion problems in one and two space dimensions, Ph.D. Thesis, Department of Mathematics, University of Bergen, Norway

  • Dahle, H. K.; Espedal, M. S.; Ewing, R. E.; Sævareid, O. (1990): Characteristic adaptive sub-domain methods for reservoir flow problems. Numerical Methods for PDE's, 279–309

  • Dahle, H. K.; Ewing R. E.; Russell, T. F. (to appear): Eulerian-Lagrangian localized adjoint methods for a nonlinear convection-diffusion equation

  • Dawson, C. N.; Dupont, T. F.; Wheeler, M. F. (1989): The rate of convergence of the modified method of characteristics for linear advection equation in one dimension. In: Diaz, J. C. (ed): Mathematics for large scale computing. New York: Marcel Dekker, 115–126

    Google Scholar 

  • Dawson, C. N.; Russell, T. F.; Wheeler, M. F. (1989): Some improved error estimates for the modified method of characteristics, SIAM J. Numer. Anal. 26, 1487–1512

    Google Scholar 

  • Demkowicz, L.; Oden, J. T. (1986): An adaptive characteristic Petro-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables. Comp. Meth. Appl. Mech. Eng. 55, 3–87

    Google Scholar 

  • Douglas, J. Jr.; Russell, T. F. (1982): Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885

    Google Scholar 

  • Douglas, J. Jr.; Ewing, R. E.; Wheeler, M. F. (1983): The approximation of the pressure by a mixed method in the simulation of miscible displacement. R.A.I.R.O. Analyse Numerique 17, 17–33

    Google Scholar 

  • Ericksson, K.; Johnson, C. (1990): Adaptive streamline diffusion finite element methods for convection-diffusion problems. Technical report 1990: 18, Chalmers University of Technology. To appear in SIAM J. Numer. Anal.

  • Espedal, M. S.; Ewing, R. E. (1987): Characteristic Petrov-Galerkin subdomain methods for two-phase immiscible flow. Comp. Meth. Appl. Mech. Eng. 64, 113–135

    Google Scholar 

  • Ewing, R. E.; Wheeler, M. F. (1980). Galerkin methods for miscible displacement problems in porous media, SIAM J. Numer. Anal. 17, 351–365

    Google Scholar 

  • Ewing, R. E. (ed) (1983): Research frontiers in applied mathematics, vol. 1. SIAM, Philadelphia

    Google Scholar 

  • Ewing, R. E. (1990): Operator splitting and Eulerian-Lagrangian localized adjoint methods for multiphase flow. The Mathematics of Finite Elements and Applications VII (MAFELAP, 1990), (Whiteman, I. ed), Academic Press, Inc., San Diego, CA, 1991, 215–232

    Google Scholar 

  • Ewing, R. E.; Wang, H. (1991): Eulerian-Lagrangian localized adjoint methods for linear advection equations. Proceedings of International Conference on Computational Engineering Science. Melbourne, Australia 1991, 245–250

  • Ewing, R. E.; Celia M. A. (1992): Numerical methods for reactive transport and biodegradation. In: Russel, Ewing, Brebbia, Gray, Pinder (eds): Computational Methods in Water Resources IX. Vol. I: Numerical Methods in Water Resources. Computational Mechanics Publications and Elsevier Applied Science, London and New York, 51–58

    Google Scholar 

  • Hansbo, P. (1990): The characteristic streamline diffusion methods for convection-diffusion problems. Technical report 1990, Chalmers University of Technology. To appear in Comp. Meth. Appl. Meth. Eng.

  • Hansbo, P. (1991): The characteristic streamline diffusion method for the incompressible Navier-Stokes equations. Technical report 1991-14, Chalmers University of Technology. To appear in Comp. Meth. Appl. Meth. Eng.

  • Herrera, I.; Chargoy, L.; Alduncin, G. (1985): Unified formulation of numerical methods. 3. Numerical Methods for PDE's 1, 241–258

    Google Scholar 

  • Herrera, I. (1987): The algebraic theory approach for ordinary differential equations: highly accurate finite differences. Numerical Methods for PDE's 3, 199–218

    Google Scholar 

  • Hughes, T. J. R.; Brooks, A. (1982): A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions. Applications to the streamline-upwind procedure. In: Gallagher (ed): Finite Elements in Fluids, vol. 4, Wiley

  • Johnson, C. (1987): Numerical solutions of partial differential equations by the finite element method. Cambridge University Press, Cambridge

    Google Scholar 

  • Johnson, C. (1989): The characteristic streamline diffusion finite element method. To appear in Proc. from Conf. on Innovative FEM, Rio de, Janeiro, 1989

  • Johnson, C. (1990): A new approach to algorithm for convection problems which are based on exact transport + projection. Technical report 1990-24. Chalmers University of Technology. To appear in Comp. Meth. Appl. Meth. Eng.

  • Johnson, C.; Pitkaranta, J. (to appear): An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp.

  • Lin, T.; Sochacki, J.; Ewing, R. E.; George, J. (1991): Some grid refinement schemes for hyperbolic equations with piecewise constant coefficients. Math. Comp. 56, 61–86

    Google Scholar 

  • Russell, T. F.; Wheeler, M. F. (1983): Finite element and finite difference methods for continuous flow in porous media. In: Ewing, R. E. (ed): Mathematics of reservoir simulation. Frontiers in Applied Math. Philadelphia, Phennsylvania 35–106

  • Russell, T. F. (1985): Time-stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal. 22, 970–1013

    Google Scholar 

  • Russell, T. F. (1989): Eulerian-Lagrangian localized adjoint methods for advection-dominated problems. In: Griffiths, D. F.; Watson, G. A. (eds): Numerical analysis proceedings of the 13th Dundee Conference on Numerical Analysis, Pitmann Research Notes in Mathematics Series, vol. 228, 206–228. Longman Scientific & Technical, Harlow, U.K.

    Google Scholar 

  • Russell, T. F.; Trujilo, R. V. (1990): Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions. Computational Methods in Surface Hydrology, Proceedings of the Eighth International Conference on Computational Methods in Water Resources, Venice, Italy, 357–363

  • Thomée, V. (1984): Galerkin finite element methods for parabolic problems. Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Varoglu, E.; Finn, W. D. L. (1980): Finite elements incorporating characteristics for one-dimensional diffusion-convection equation. J. Comp. Phys. 34, 371–389

    Google Scholar 

  • Wang, H.; Ewing, R. E.; Russell, T. F. (1992): ELLAM for variable-coefficient convection-diffusion problems arising in groundwater applications. In: Russell, Ewing, Brebbia, Gray, Pinder (eds): Computational Methods in Resources IX. Vol. I: Numerical Methods in Water Resources. London and New York: Elsevier. Computational Mechanics Publications and Applied Science, 25–31

    Google Scholar 

  • Wang, H.; Lin, T.; Ewing, R. E. (1992): ELLAM with domain decomposition and local refinement techniques for advectionreaction problems with discontinuous coefficients. In: Russell, Ewing, Brebbia, Gray, Pinder (eds): Computational Methods in Water Resources IX. Vol. I: Numerical Methods in Water Resources. London and New York: Elsevier. Computational Mechanics Publications and Applied Science, 17–24

    Google Scholar 

  • Wang, H. (1992): Eulerian-Lagrangian localized adjoint methods: analyses, numerical implementations and their applications. Ph.D. Thesis, Department of Mathematics. University of Wyoming, 1992

  • Wang, H.; Ewing, R. E.; Russell, T. F. (sumitted): Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis

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Authors and Affiliations

  1. Department of Mathematics, University of Wyoming, 82071, Laramie, WY, USA

    Richard E. Ewing & Hong Wang

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  1. Richard E. Ewing
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  2. Hong Wang
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Communicated by S. N. Atluri, December 1, 1992

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Ewing, R.E., Wang, H. Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis. Computational Mechanics 12, 97–121 (1993). https://doi.org/10.1007/BF00370489

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