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Error estimation techniques [1, 2] are widely used as the solution verification in the numerical simulations. One can employ either a posteriori error estimation based on known quantities in the solution or a priori error estimation by controlling the error parameters in unknown quantities. This study proposes a posteriori criteria for finite element approximation of the problems in flow through porous media. We introduce a technique to verify the numerical solution of Darcy and Darcy–Brinkman models and their modifications. We show that for given boundary conditions of all kinematically admissible vector fields, the Darcy and the Darcy–Brinkman velocities have minimum total dissipation. Our proposed dissipation is a parameter for accuracy assessment and grid convergence studies. This solution verification technique is not only mathematical method such as error measurements in energy or other standard norms, but also has a firm physical basis. Moreover, our assessment is applicable for adaptive (h-version) finite element approximations and for the domain contains the nonsmooth or polluted solution [3]. To support our theory, the proposed dissipation is used to verify the numerical solution of some problems in the flow through porous media. REFERENCES [1] Ainsworth, M., Oden, J.T. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering. 1997, 142:1–88. [2] Roache, P.J. Verification and Validation in Computational Science and Engineering. Hermosa Publishers, New Mexico; 1998. [3] Babuška, I., Oh, H.S. Pollution problem of the p- and hp-versions of the finite element method. Communications in Applied Numerical Methods. 1987, 3:553–561.

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On assessing accuracy of numerical solutions of porous media models

Error estimation techniques [1, 2] are widely used as the solution verification in the numerical simulations. One can employ either a posteriori error estimation based on known quantities in the solution or a priori error estimation by controlling the error parameters in unknown quantities. This study proposes a posteriori criteria for finite element approximation of the problems in flow through porous media. We introduce a technique to verify the numerical solution of Darcy and Darcy–Brinkman models and their modifications. We show that for given boundary conditions of all kinematically admissible vector fields, the Darcy and the Darcy–Brinkman velocities have minimum total dissipation. Our proposed dissipation is a parameter for accuracy assessment and grid convergence studies. This solution verification technique is not only mathematical method such as error measurements in energy or other standard norms, but also has a firm physical basis. Moreover, our assessment is applicable for adaptive (h-version) finite element approximations and for the domain contains the nonsmooth or polluted solution [3]. To support our theory, the proposed dissipation is used to verify the numerical solution of some problems in the flow through porous media. REFERENCES [1] Ainsworth, M., Oden, J.T. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering. 1997, 142:1–88. [2] Roache, P.J. Verification and Validation in Computational Science and Engineering. Hermosa Publishers, New Mexico; 1998. [3] Babuška, I., Oh, H.S. Pollution problem of the p- and hp-versions of the finite element method. Communications in Applied Numerical Methods. 1987, 3:553–561.