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Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis. (English) Zbl 0774.76058

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.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] 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 · Zbl 0652.76063
[2] 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 · Zbl 0562.76086
[3] 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
[4] 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 · Zbl 0497.76041
[5] 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
[6] 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 · Zbl 0678.65083
[7] 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
[8] 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
[9] 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 · Zbl 0545.76130
[10] 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 · Zbl 0699.76099
[11] 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
[12] 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 · Zbl 0707.76093
[13] Dahle, H. K.; Ewing R. E.; Russell, T. F. (to appear): Eulerian-Lagrangian localized adjoint methods for a nonlinear convection-diffusion equation · Zbl 0851.76058
[14] 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
[15] 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 · Zbl 0693.65062
[16] 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 · Zbl 0602.76097
[17] 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 · Zbl 0492.65051
[18] 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 · Zbl 0516.76094
[19] 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.
[20] 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 · Zbl 0607.76103
[21] Ewing, R. E.; Wheeler, M. F. (1980). Galerkin methods for miscible displacement problems in porous media, SIAM J. Numer. Anal. 17, 351-365 · Zbl 0458.76092
[22] Ewing, R. E. (ed) (1983): Research frontiers in applied mathematics, vol. 1. SIAM, Philadelphia
[23] 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
[24] 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
[25] 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
[26] 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. · Zbl 0716.76048
[27] 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. · Zbl 0825.76423
[28] Herrera, I.; Chargoy, L.; Alduncin, G. (1985): Unified formulation of numerical methods. 3. Numerical Methods for PDE’s 1, 241-258 · Zbl 0634.65061
[29] Herrera, I. (1987): The algebraic theory approach for ordinary differential equations: highly accurate finite differences. Numerical Methods for PDE’s 3, 199-218 · Zbl 0657.65102
[30] 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
[31] Johnson, C. (1987): Numerical solutions of partial differential equations by the finite element method. Cambridge University Press, Cambridge · Zbl 0628.65098
[32] Johnson, C. (1989): The characteristic streamline diffusion finite element method. To appear in Proc. from Conf. on Innovative FEM, Rio de, Janeiro, 1989
[33] 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.
[34] Johnson, C.; Pitkaranta, J. (to appear): An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp.
[35] 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 · Zbl 0741.65066
[36] 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
[37] 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 · Zbl 0594.76087
[38] 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.
[39] 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
[40] Thomée, V. (1984): Galerkin finite element methods for parabolic problems. Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer · Zbl 0546.65055
[41] Varoglu, E.; Finn, W. D. L. (1980): Finite elements incorporating characteristics for one-dimensional diffusion-convection equation. J. Comp. Phys. 34, 371-389 · Zbl 0487.76083
[42] 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
[43] 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
[44] Wang, H. (1992): Eulerian-Lagrangian localized adjoint methods: analyses, numerical implementations and their applications. Ph.D. Thesis, Department of Mathematics. University of Wyoming, 1992
[45] Wang, H.; Ewing, R. E.; Russell, T. F. (sumitted): Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis · Zbl 0830.65095
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