### Electronic Research Archive

2021, Issue 5: 3405-3427. doi: 10.3934/era.2021045
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# An adjoint-based a posteriori analysis of numerical approximation of Richards equation

• Received: 01 November 2020 Revised: 01 April 2021 Published: 24 June 2021
• Primary: 65M60, 65M08; Secondary: 65Z05

• This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.

Citation: Victor Ginting. An adjoint-based a posteriori analysis of numerical approximation of Richards equation[J]. Electronic Research Archive, 2021, 29(5): 3405-3427. doi: 10.3934/era.2021045

### Related Papers:

• This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.

 [1] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-7605-6 [2] Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation. Appl. Numer. Math. (2017) 112: 104-125. [3] Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Advances in Water Resources (2004) 27: 565-581. [4] A posteriori analysis of a space and time discretization of a nonlinear model for the flow in partially saturated porous media. IMA J. Numer. Anal. (2014) 34: 1002-1036. [5] A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. (1990) 26: 1483-1496. [6] Finite volume methods for elliptic PDE's: A new approach. M2AN Math. Model. Numer. Anal. (2002) 36: 307-324. [7] A finite volume element method for a non-linear elliptic problem. Numer. Linear Algebra Appl. (2005) 12: 515-546. [8] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, vol. 2 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9780898718942 [9] Error estimates in $L^2, H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach. Math. Comp. (2000) 69: 103-120. [10] A mass-conservative control volume-finite element method for solving Richards'equation in heterogeneous porous media. BIT (2011) 51: 845-864. [11] (1996) Computational Differential Equations.Cambridge University Press. [12] (1995) Introduction to adaptive methods for differential equations.in Acta Numerica, 1995, Acta Numer., Cambridge Univ. Press. [13] (1995) Introduction to computational methods for differential equations.in Theory and Numerics of Ordinary and Partial Differential Equations (Leicester, 1994), Adv. Numer. Anal., IV, Oxford Univ. Press. [14] A posteriori error bounds and global error control for approximation of ordinary differential equations. SIAM J. Numer. Anal. (1995) 32: 1-48. [15] D. J. Estep, M. G. Larson and R. D. Williams, Estimating the error of numerical solutions of systems of reaction-diffusion equations, Mem. Amer. Math. Soc., 146 (2000), no. 696. doi: 10.1090/memo/0696 [16] The finite volume method for Richards equation. Comput. Geosci. (1999) 3: 259-294. [17] Numerical solution of Richards' equation: A review of advances and challenges. Soil Science Society of America Journal (2017) 81: 1257-1269. [18] Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Science (1958) 85: 228-232. [19] Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality. Acta Numer. (2002) 11: 145-236. [20] Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. (1978) 15: 912-928. [21] G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems, vol. 295 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1995, Translated from the 1992 Russian edition by Guennadi Kontarev and revised by the author. doi: 10.1007/978-94-017-0621-6 [22] Semianalytical solution to Richards' equation for layered porous media. Journal of Irrigation and Drainage Engineering (1998) 124: 290-299. [23] Capillary conduction of liquids through porous mediums. Physics (1931) 1: 318-333. [24] Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils. Water Resour. Res. (1991) 27: 753-762. [25] An analytical solution to Richards' equation for time-varying infiltration. Water Resour. Res. (1991) 27: 763-766.
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