Darcy’s law is the fundamental equation describing the flow of a fluid through a porous medium. Combined with the principle of mass conservation, it leads to the diffusion Equation (e.g., the groundwater flow equation). In this paper, the Legendre-Petrov-Galerkin method is developed for solving the parabolic problem with Dirichlet boundary conditions based on Darcy’s law-preserving. This problem is transformed into an equivalent first-order system by introducing a flux based on Darcy’s law. Our scheme is based on the Legendre Galerkin method, and the right hand side term is processed using the Legendre/Chebyshev-Gauss-Lobatto points. The time direction is approximated by the Crank-Nicolson method. The algebraic system with a sparse coefficient matrix is obtained by selecting the appropriate basis function. Error estimate of the semi-discrete scheme is given by using Gronwall’s inequality (integral form) and Darcy’s law. Numerical examples show that our scheme has the high-order spectral accuracy and it preserves Darcy’s law.
References
[1]
Hubbert, M.K. (1956) Darcy’s Law and the Field Equations of the Flow of Underground Fluids. Transactions of the AIME, 207, 222-239. https://doi.org/10.2118/749-g
[2]
Durlofsky, L.J. (1994) Accuracy of Mixed and Control Volume Finite Element Approximations to Darcy Velocity and Related Quantities. Water Resources Research, 30, 965-973. https://doi.org/10.1029/94wr00061
[3]
Masud, A. and Hughes, T.J.R. (2002) A Stabilized Mixed Finite Element Method for Darcy Flow. Computer Methods in Applied Mechanics and Engineering, 191, 4341-4370. https://doi.org/10.1016/s0045-7825(02)00371-7
[4]
Chen, Y., Liu, D., Li, Y., Li, Z. and Zhu, Y. (2025) Two-dimensional Modified Darcy Model for Calculating Seepage Problems in Underground Gas Storage Reservoirs. Geoenergy Science and Engineering, 253, Article ID: 213971. https://doi.org/10.1016/j.geoen.2025.213971
[5]
Govindarajan, S.K., Pavan, T.N.V., Devarapu, S.R. and Kumar, G. (2025) Critical Review on Challenges in the Application of Darcy’s Law for Multi-Phase Fluid Flow in Petroleum Reservoirs. Discover Applied Sciences, 7, Article No. 201. https://doi.org/10.1007/s42452-025-06632-y
[6]
Bergamaschi, L., Mantica, S. and Saleri, F. (1994) Mixed Finite Element Approximation of Darcy’s Law in Porous Media. Report CRS4 AppMath-94-20, CRS4.
[7]
Azaïez, M., Belgacem, F.B. and Bernardi, C. (2006) The Mortar Spectral Element Method in Domains of Operators. Part I: The Divergence Operator and Darcy’s Equations. IMA Journal of Numerical Analysis, 26, 131-154. https://doi.org/10.1093/imanum/dri028
[8]
Rui, H. and Pan, H. (2012) A Block-Centered Finite Difference Method for the Darcy-Forchheimer Model. SIAM Journal on Numerical Analysis, 50, 2612-2631. https://doi.org/10.1137/110858239
[9]
Jain, V., Fisser, J., Palha, A. and Gerritsma, M. (2020) A Conservative Hybrid Method for Darcy Flow. In: Sherwin, S.J., Moxey, D., Peiró, J., Vincent, P.E. and Schwab, C., Eds., Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, Springer, 215-227. https://doi.org/10.1007/978-3-030-39647-3_16
[10]
Badia, S. and Codina, R. (2010) Stabilized Continuous and Discontinuous Galerkin Techniques for Darcy Flow. Computer Methods in Applied Mechanics and Engineering, 199, 1654-1667. https://doi.org/10.1016/j.cma.2010.01.015
[11]
Li, H., Ma, H. and Sun, W. (2013) Legendre Spectral Galerkin Method for Electromagnetic Scattering from Large Cavities. SIAM Journal on Numerical Analysis, 51, 353-376. https://doi.org/10.1137/110833853
[12]
Zeng, F., Ma, H. and Liang, D. (2014) Energy-Conserved Splitting Spectral Methods for Two Dimensional Maxwell’s Equations. Journal of Computational and Applied Mathematics, 265, 301-321. https://doi.org/10.1016/j.cam.2013.09.048
[13]
Niu, C., Ma, H. and Liang, D. (2022) Energy-Conserved Splitting Multidomain Legendre-Tau Spectral Method for Two Dimensional Maxwell’s Equations. Journal of Scientific Computing, 90, Article No. 77. https://doi.org/10.1007/s10915-021-01744-0
[14]
Niu, C., Ma, H. and Liang, D. (2023) Legendre-Tau Chebyshev Collocation Spectral Element Method for Maxwell’s Equations with Material Interfaces of Two Dimensional Transverse Magnetic Mode. Computers & Mathematics with Applications, 147, 222-238. https://doi.org/10.1016/j.camwa.2023.07.024
[15]
Shan, W. and Li, H. (2017) The Triangular Spectral Element Method for Stokes Eigenvalues. Mathematics of Computation, 86, 2579-2611. https://doi.org/10.1090/mcom/3173
[16]
Ma, S., Li, H. and Zhang, Z. (2018) Efficient Spectral Methods for Some Singular Eigenvalue Problems. Journal of Scientific Computing, 77, 657-688. https://doi.org/10.1007/s10915-018-0721-y
[17]
An, J., Li, H. and Zhang, Z. (2019) Spectral-Galerkin Approximation and Optimal Error Estimate for Biharmonic Eigenvalue Problems in Circular/Spherical/Elliptical Domains. Numerical Algorithms, 84, 427-455. https://doi.org/10.1007/s11075-019-00760-4
[18]
Jia, L., Li, H. and Zhang, Z. (2020) Numerical Analysis on the Mortar Spectral Element Methods for Schrödinger Eigenvalue Problem with an Inverse Square Potential. Applied Numerical Mathematics, 158, 54-84. https://doi.org/10.1016/j.apnum.2020.06.015
[19]
Wang, Y., Hao, W. and Lin, G. (2018) Two-Level Spectral Methods for Nonlinear Elliptic Equations with Multiple Solutions. SIAM Journal on Scientific Computing, 40, B1180-B1205. https://doi.org/10.1137/17m113767x
[20]
Li, L., Wang, L. and Li, H. (2023) An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions. Journal of Scientific Computing, 95, Article No. 32. https://doi.org/10.1007/s10915-023-02154-0
[21]
Li, X., Tang, T. and Xu, C. (2013) Parallel in Time Algorithm with Spectral-Subdomain Enhancement for Volterra Integral Equations. SIAM Journal on Numerical Analysis, 51, 1735-1756. https://doi.org/10.1137/120876241
[22]
Sheng, C., Wang, Z. and Guo, B. (2014) A Multistep Legendre-Gauss Spectral Collocation Method for Nonlinear Volterra Integral Equations. SIAM Journal on Numerical Analysis, 52, 1953-1980. https://doi.org/10.1137/130915200
[23]
Mohammadi, K.M., Babaei, M., Hajimohammadi, Z. and Parand, K. (2025) A New Numerical Method for Solving Neuro-Cognitive Models via Chebyshev Deep Neural Network (CDNN). Applications of Mathematics, 70, 517-535. https://doi.org/10.21136/am.2025.0082-24
[24]
Wang, Y.P., Akbarpoor Kiasary, S. and Yılmaz, E. (2024) Solving Inverse Nodal Problem with Frozen Argument by Using Second Chebyshev Wavelet Method. Applications of Mathematics, 69, 339-354. https://doi.org/10.21136/am.2024.0038-21
[25]
Elbarbary, E.M.E. (2007) Efficient Chebyshev-Petrov-Galerkin Method for Solving Second-Order Equations. Journal of Scientific Computing, 34, 113-126. https://doi.org/10.1007/s10915-007-9161-9
[26]
Ismail, M.S. (2008) Numerical Solution of Complex Modified Korteweg-de Vries Equation by Petrov-Galerkin Method. Applied Mathematics and Computation, 202, 520-531. https://doi.org/10.1016/j.amc.2008.02.033
[27]
Hu, J., Huang, X., Shen, J. and Yang, H. (2022) A Fast Petrov-Galerkin Spectral Method for the Multidimensional Boltzmann Equation Using Mapped Chebyshev Functions. SIAM Journal on Scientific Computing, 44, A1497-A1524. https://doi.org/10.1137/21m1420721
[28]
Leng, Y., Tian, X., Demkowicz, L., Gomez, H. and Foster, J.T. (2022) A Petrov-Galerkin Method for Nonlocal Convection-Dominated Diffusion Problems. Journal of Computational Physics, 452, Article ID: 110919. https://doi.org/10.1016/j.jcp.2021.110919
[29]
Vaziri Astaneh, A., Keith, B. and Demkowicz, L. (2018) On Perfectly Matched Layers for Discontinuous Petrov-Galerkin Methods. Computational Mechanics, 63, 1131-1145. https://doi.org/10.1007/s00466-018-1640-3
[30]
Ma, H. and Sun, W. (2000) A Legendre-Petrov-Galerkin and Chebyshev Collocation Method for Third-Order Differential Equations. SIAM Journal on Numerical Analysis, 38, 1425-1438. https://doi.org/10.1137/s0036142999361505
[31]
Boffi, D., Brezzi, F. and Fortin, M. (2013) Mixed Finite Element Methods and Applications: Volume 44 of Springer Series in Computational Mathematics. Springer.
[32]
Cockburn, B., Karniadakis, G.E. and Shu, C. (2000) The Development of Discontinuous Galerkin Methods. In: Cockburn, B., Karniadakis, G.E. and Shu, C.W., Eds., Discontinuous Galerkin Methods, Springer, 3-50. https://doi.org/10.1007/978-3-642-59721-3_1
[33]
Cockburn, B. (2003) Discontinuous Galerkin Methods. ZAMM—Journal of Applied Mathematics and Mechanics/Zeitschrift für AngewandteMathematik und Mechanik, 83, 731-754. https://doi.org/10.1002/zamm.200310088
[34]
Zhang, H., Wu, B. and Meng, X. (2022) Analysis of the Local Discontinuous Galerkin Method with Generalized Fluxes for One-Dimensional Nonlinear Convection-Diffusion Systems. Science China Mathematics, 66, 2641-2664. https://doi.org/10.1007/s11425-022-2035-y
[35]
Dolejší, V. and Feistauer, M. (2015) Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow, Volume 48 of Springer Series in Computational Mathematics. Springer.
[36]
Liang, D. and Du, C. (2014) The Efficient S-DDM Scheme and Its Analysis for Solving Parabolic Equations. Journal of Computational Physics, 272, 46-69. https://doi.org/10.1016/j.jcp.2014.04.015
[37]
Zhou, Z. and Liang, D. (2016) The Mass-Preserving S-DDM Scheme for Two-Dimensional Parabolic Equations. Communications in Computational Physics, 19, 411-441. https://doi.org/10.4208/cicp.070814.190615a
[38]
Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (2006) Spectral Methods: Fundamentals in Single Domains. Springer.
[39]
Shen, J., Tang, T. and Wang, L.L. (2011) Spectral Methods: Algorithms, Analysis and Applications, Volume 41 of Springer Series in Computational Mathematics. Spring-er.
[40]
Shen, J. (1994) Efficient Spectral-Galerkin Method I. Direct Solvers of Second-and Fourth-Order Equations Using Legendre Polynomials. SIAM Journal on Scientific Computing, 15, 1489-1505. https://doi.org/10.1137/0915089
[41]
Bernardi, C. and Maday, Y. (1997) Spectral Methods. In: Ciarlet, P.G. and Lions, J.L., Eds., Handbook of Numerical Analysis, Elsevier, 209-485. https://doi.org/10.1016/s1570-8659(97)80003-8
[42]
Guo, B.Y. (1998) Spectral Methods and Their Applications. World Scientific.
[43]
Hu, Y.Q., Qin, Y.H. and Fan, Y.K. (2021) Legendre Spectral Collocation Least Squares Method for the Parabolic Problem. Journal of Guilin University of Electronic Technology, 41, 55-60. (In Chinese)
