Weak Galerkin finite element methods for semilinear Klein-Gordon equation on polygonal meshes. (English) Zbl 07862450
Summary: The article presents the development of the weak Galerkin finite element method (WG-FEM) for semilinear hyperbolic problems. Semidiscrete error estimate in \(L^2\)-norm as well as \(H^1\)-norm have been executed for the weak Galerkin space \((\mathbf{P}_k(\mathcal{K}), \mathbf{P}_k(\partial\mathcal{K}), [\mathbf{P}_{k-1}(\mathcal{K})]^2)\), where \(k \geq 1\) is an integer. For a fully discrete scheme, we employ the Newmark scheme for temporal discretization. Finally, a few numerical results are provided to validate theoretical results.
MSC:
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
semilinear hyperbolic equation; weak Galerkin method; semi-discrete scheme; fully discrete scheme; newmark schemes; optimal error estimatesReferences:
| [1] | Ablowitz, MJ; Kruskal, MD; Ladik, JF, Solitary wave collisions, SIAM J Appl Math, 36, 428-437, 1979 · Zbl 0408.65075 · doi:10.1137/0136033 |
| [2] | Adak, D.; Natarajan, E.; Kumar, S., Virtual element method for semilinear hyperbolic problems on polygonal meshes, Int J Comput Math, 96, 971-991, 2019 · Zbl 1499.65478 · doi:10.1080/00207160.2018.1475651 |
| [3] | Adams, R.; Fournier, J., Sobolev spaces, 2003, Amsterdam: Academic Press, Amsterdam · Zbl 1098.46001 |
| [4] | Ashyralyev, A.; Sirma, A., A note on the numerical solution of the semilinear Schrödinger equation, Nonlinear Anal Theory Methods Appl, 71, 12, 2507-2516, 2009 · Zbl 1239.65053 · doi:10.1016/j.na.2009.05.048 |
| [5] | Chen, CM; Larsson, S.; Zhang, NY, Error estimates of optimal order for finite element methods with interpolated coefficients for the nonlinear heat equation, IMA J Numer Anal, 9, 4, 507-524, 1989 · Zbl 0686.65077 · doi:10.1093/imanum/9.4.507 |
| [6] | Chen, L.; Wang, J.; Ye, X., A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J Sci Comput, 59, 496-511, 2014 · Zbl 1307.65153 · doi:10.1007/s10915-013-9771-3 |
| [7] | Chrysafinos, K.; Hou, LS, Error estimates for the semidiscrete finite element method approximations of linear and semilinear parabolic equations, SIAM J Numer Anal, 40, 282-306, 2002 · Zbl 1020.65068 · doi:10.1137/S0036142900377991 |
| [8] | Deka, B.; Kumar, N., A systematic study on weak Galerkin finite element method for second order parabolic problems, Numer Methods PDE, 39, 2444-2474, 2023 · Zbl 07777013 · doi:10.1002/num.22973 |
| [9] | Deka, B.; Roy, P., Weak Galerkin finite element methods for parabolic interface problems with nonhomogeneous jump conditions, Numer Funct Anal Optim, 40, 259-279, 2019 · Zbl 1420.65099 · doi:10.1080/01630563.2018.1549074 |
| [10] | Feistauer, M.; Sobotikova, V., Finite element approximation of nonlinear elliptic problems with discontinuous coefficients, RAIRO Model Math Anal Numer, 24, 4, 457-500, 1990 · Zbl 0712.65097 · doi:10.1051/m2an/1990240404571 |
| [11] | Han, H.; Zhang, Z., Split local absorbing conditions for one-dimensional nonlinear Klein-Gordon equation on unbounded domain, J Comput Phys, 227, 8992-9004, 2008 · Zbl 1152.65092 · doi:10.1016/j.jcp.2008.07.006 |
| [12] | Hu, X.; Mu, L.; Ye, X., A weak Galerkin finite element method for the Navier-Stokes equations, J Comput Appl Math, 362, 614-625, 2019 · Zbl 1422.65389 · doi:10.1016/j.cam.2018.08.022 |
| [13] | Huang, Y.; Li, J.; Li, D., Developing weak Galerkin finite element methods for the wave equation, Numer Methods Partial Differ Equ, 33, 868-884, 2017 · Zbl 1377.65129 · doi:10.1002/num.22127 |
| [14] | Irk, D.; Kirli, E.; Gorgulu, MZ, A high order accurate numerical solution of the Klein-Gordon equation, Appl Math Inf Sci, 16, 331-339, 2022 |
| [15] | Kim, S.; Lim, H., High order schemes for acoustic waveform simulation, Appl Numer Math, 57, 402-414, 2007 · Zbl 1113.65087 · doi:10.1016/j.apnum.2006.05.003 |
| [16] | Kirby RC, Kieu, TT (2013) Galerkin finite element methods for nonlinear Klein-Gordon equations. Math Comput 189(4) |
| [17] | Lehrenfeld C (2010) Hybrid discontinuous Galerkin methods for solving incompressible flow problems. Diploma Thesis, MathCCES/IGPM, RWTH Aachen |
| [18] | Li, J.; Chen, Y-T, Computational partial differential equations using MATLAB, 2008, London: Chapman and Hall/CRC Press, London · doi:10.1201/9781420089059 |
| [19] | Li, B.; Sun, W., Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media, SIAM J Numer Anal, 51, 1959-1977, 2013 · Zbl 1311.76067 · doi:10.1137/120871821 |
| [20] | Li, QH; Wang, J., Weak Galerkin finite element methods for parabolic equations, Numer Methods Partial Differ Equ, 29, 2004-2024, 2013 · Zbl 1307.65133 · doi:10.1002/num.21786 |
| [21] | Li, H.; Mu, L.; Ye, X., Interior energy error estimates for the weak Galerkin finite element method, Numer Math, 139, 447-478, 2018 · Zbl 1447.65154 · doi:10.1007/s00211-017-0940-4 |
| [22] | Li, D.; Nie, Y.; Wang, C., Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions, Comput Math Appl, 78, 905-928, 2019 · Zbl 1442.65380 · doi:10.1016/j.camwa.2019.03.010 |
| [23] | Li, D.; Wang, C.; Wang, J., Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions, Appl Numer Math, 150, 396-417, 2020 · Zbl 1434.65266 · doi:10.1016/j.apnum.2019.10.013 |
| [24] | Li, D.; Wang, C.; Wang, J., A primal-dual finite element method for transport equations in nondivergence form, J Comput Appl Math, 412, 2022 · Zbl 1486.65256 · doi:10.1016/j.cam.2022.114313 |
| [25] | Lin, G.; Liu, J.; Sadre-Marandi, F., A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods, J Comput Appl Math, 273, 346-362, 2015 · Zbl 1295.65114 · doi:10.1016/j.cam.2014.06.024 |
| [26] | Lin, R.; Ye, X.; Zhang, S.; Zhu, P., A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J Numer Anal, 56, 1482-1497, 2018 · Zbl 1448.65239 · doi:10.1137/17M1152528 |
| [27] | Lipo, D.; Paye, KR; Popivanov, NI, On the degenerate hyperbolic Goursat problem for linear and nonlinear equation of Tricomi type, Nonlinear Anal Theory Methods Appl, 108, 29-56, 2014 · Zbl 1297.35145 · doi:10.1016/j.na.2014.05.009 |
| [28] | Liu, Y.; Nie, Y., A priori and a posteriori error estimates of the weak Galerkin finite element method for parabolic problems, Comput Math Appl, 99, 73-83, 2021 · Zbl 1524.65571 · doi:10.1016/j.camwa.2021.08.002 |
| [29] | Liu X, Li J, Chen Z (2018a) A weak Galerkin finite element method for the Navier-Stokes equations. J Comput Appl Math 333:442-457 · Zbl 1395.76040 |
| [30] | Liu J, Tavener S, Wang Z (2018b) Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes. SIAM J Sci Comput 40:B1229-B1252 · Zbl 1404.65232 |
| [31] | Liu, J.; Tavener, S.; Wang, Z., Penalty-free any-order weak Galerkin FEMs for elliptic problems on quadrilateral meshes, J Sci Comput, 83, 47, 2020 · Zbl 1440.65219 · doi:10.1007/s10915-020-01239-4 |
| [32] | Liu, Y.; Guan, Z.; Nie, Y., Unconditionally optimal error estimates of a linearized weak Galerkin finite element method for semilinear parabolic equations, Adv Comput Math, 48, 47, 2022 · Zbl 1492.65321 · doi:10.1007/s10444-022-09961-3 |
| [33] | Mu, L.; Chen, Z., A new WENO weak Galerkin finite element method for time dependent hyperbolic equations, Appl Numer Math, 159, 106-124, 2021 · Zbl 1459.65187 · doi:10.1016/j.apnum.2020.09.002 |
| [34] | Mu L, Wang J, Ye X (2015a) Weak Galerkin finite element methods on polytopal meshes. Int J Numer Anal Model 12:31-53 · Zbl 1332.65172 |
| [35] | Mu L, Wang J, Xiu Y, Zhang S (2015b) A weak Galerkin finite element method for the Maxwell equations. J Sci Comput 65:363-386 · Zbl 1327.65220 |
| [36] | Mu, L.; Wang, J.; Ye, X., A least-squares-based weak Galerkin finite element method for second order elliptic equations, SIAM J Sci Comput, 39, A1531-A1557, 2017 · Zbl 1404.65233 · doi:10.1137/16M1083244 |
| [37] | Shields, S.; Li, J.; Machorro, E., Weak Galerkin methods for time-dependent Maxwell’s equations, Comput Math Appl, 74, 2106-2124, 2017 · Zbl 1397.65192 · doi:10.1016/j.camwa.2017.07.047 |
| [38] | Sinha, R.; Deka, B., Finite element methods for semilinear elliptic and parabolic interface problems, Appl Numer Math, 59, 1870-1883, 2009 · Zbl 1172.65064 · doi:10.1016/j.apnum.2009.02.001 |
| [39] | Sun, S.; Huang, Z.; Wang, C., Weak Galerkin finite element method for a class of quasilinear elliptic problems, Appl Math Lett, 79, 67-72, 2018 · Zbl 1466.65202 · doi:10.1016/j.aml.2017.11.017 |
| [40] | Thomée, V., Galerkin finite element methods for parabolic problems, 2007, Berlin: Springer Science & Business Media, Berlin |
| [41] | Wang, C.; Wang, J., A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form, Math Comput, 87, 515-545, 2018 · Zbl 1380.65390 · doi:10.1090/mcom/3220 |
| [42] | Wang, J.; Ye, X., A weak Galerkin finite element method for second order elliptic problems, J Comput Appl Math, 241, 103-115, 2013 · Zbl 1261.65121 · doi:10.1016/j.cam.2012.10.003 |
| [43] | Wang, J.; Ye, X., A weak Galerkin mixed finite element method for second order elliptic problems, Math Comput, 83, 2101-2126, 2014 · Zbl 1308.65202 · doi:10.1090/S0025-5718-2014-02852-4 |
| [44] | Wang, J.; Wang, R.; Zhai, Q.; Zhang, R., A systematic study on weak Galerkin finite element methods for second order elliptic problems, J Sci Comput, 74, 1369-1396, 2018 · Zbl 1398.65311 · doi:10.1007/s10915-017-0496-6 |
| [45] | Wang, X.; Gao, F.; Sun, Z., Weak Galerkin finite element method for viscoelastic wave equations, J Comput Appl Math, 375, 2020 · Zbl 1435.65168 · doi:10.1016/j.cam.2020.112816 |
| [46] | Wheeler, MF, A priori \(L^2\) error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J Numer Anal, 10, 723-759, 1973 · doi:10.1137/0710062 |
| [47] | Yang L (2006) Numerical studies of the Klein-Gordon-Schrödinger equations. Master’s Thesis, National University Singapore, Singapore |
| [48] | Yang, H., High-order energy and linear momentum conserving methods for the Klein-Gordon equation, Mathematics, 6, 10, 200, 2018 · Zbl 1515.65261 · doi:10.3390/math6100200 |
| [49] | Yin, F.; Tian, T.; Song, J.; Zhu, M., Spectral methods using Legendre wavelets for nonlinear Klein/Sine-Gordon equations, J Comput Appl Math, 275, 321-334, 2015 · Zbl 1334.65175 · doi:10.1016/j.cam.2014.07.014 |
| [50] | Zenisek, A., The finite element method for nonlinear elliptic equations with discontinuous coefficients, Numer Math, 58, 51-77, 1990 · Zbl 0709.65081 · doi:10.1007/BF01385610 |
| [51] | Zhang, T., A posteriori error analysis for the weak Galerkin method for solving elliptic problems, Int J Comput Methods, 15, 1850075, 2018 · Zbl 1404.65285 · doi:10.1142/S0219876218500755 |
| [52] | Zhang, T.; Chen, Y., An analysis of the weak Galerkin finite element method for convection-diffusion equation, Appl Math Comput, 346, 612-621, 2019 · Zbl 1429.65281 |
| [53] | Zhang, H.; Zou, Y.; Xu, Y.; Zhai, Q.; Yue, H., Weak Galerkin finite element method for second order parabolic equations, Int J Numer Anal Model, 13, 525-544, 2016 · Zbl 1362.65110 |
| [54] | Zhou, S.; Gao, F.; Li, B.; Sun, Z., Weak Galerkin finite element method with second-order accuracy in time for parabolic problems, Appl Math Lett, 90, 118-123, 2019 · Zbl 1410.65387 · doi:10.1016/j.aml.2018.10.023 |
| [55] | Zlamal, M., A finite element solution of the nonlinear heat equation, RAIRO Anal Numer, 14, 203-216, 1980 · Zbl 0442.65103 · doi:10.1051/m2an/1980140202031 |
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