Strong convergence of linear implicit virtual element methods for the nonlinear stochastic parabolic equation with multiplicative noise (original) (raw)

References

  1. Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013). https://doi.org/10.1016/j.camwa.2013.05.015
    Article MathSciNet Google Scholar
  2. Antonietti, P.F., Beirão da Veiga, L., Scacchi, S., Verani, M.: A \(C^1\) virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 34–56 (2016) https://doi.org/10.1137/15M1008117
  3. Adak, D., Natarajan, S.: Virtual element methods for nonlocal parabolic problems on general type of meshes. Adv. Comput. Math. 46(5), 74–29 (2020). https://doi.org/10.1007/s10444-020-09811-0
    Article MathSciNet Google Scholar
  4. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013) https://doi.org/10.1142/S0218202512500492
  5. Beirão da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013) https://doi.org/10.1137/120874746
  6. Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016) https://doi.org/10.1142/S0218202516500160
  7. Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The virtual element method. Acta Numer. 32, 123–202 (2023) https://doi.org/10.1017/S0962492922000095
  8. Bréhier, C.-E., Cohen, D., Ulander, J.: Analysis of a positivity-preserving splitting scheme for some semilinear stochastic heat equations. ESAIM Math. Model. Numer. Anal. 58(4), 1317–1346 (2024). https://doi.org/10.1051/m2an/2024032
    Article MathSciNet Google Scholar
  9. Brenner, S.C., Guan, Q., Sung, L.-Y.: Some estimates for virtual element methods. Comput. Methods Appl. Math. 17(4), 553–574 (2017). https://doi.org/10.1515/cmam-2017-0008
    Article MathSciNet Google Scholar
  10. Becker, S., Jentzen, A.: Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Stoch. Process. Appl. 129(1), 28–69 (2019). https://doi.org/10.1016/j.spa.2018.02.008
    Article MathSciNet Google Scholar
  11. Beirão da Veiga, L., Lovadina, C., Vacca, G.: Virtual elements for the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 56(3), 1210–1242 (2018) https://doi.org/10.1137/17M1132811
  12. Bauzet, C., Nabet, F., Schmitz, K., Zimmermann, A.: Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise. ESAIM Math. Model. Numer. Anal. 57(2), 745–783 (2023). https://doi.org/10.1051/m2an/2022087
    Article MathSciNet Google Scholar
  13. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Texts in Appl. Math., 15. Springer, Berlin (2008). https://doi.org/10.1007/978-0-387-75934-0
  14. Buckwar, E., Winkler, R.: Multistep methods for SDEs and their application to problems with small noise. SIAM J. Numer. Anal. 44(2), 779–803 (2006). https://doi.org/10.1137/040602857
    Article MathSciNet Google Scholar
  15. Cangiani, A., Chatzipantelidis, P., Diwan, G., Georgoulis, E.H.: Virtual element method for quasilinear elliptic problems. IMA J. Numer. Anal. 40(4), 2450–2472 (2020). https://doi.org/10.1093/imanum/drz035
    Article MathSciNet Google Scholar
  16. Cangiani, A., Gyrya, V., Manzini, G.: The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54(6), 3411–3435 (2016). https://doi.org/10.1137/15M1049531
    Article MathSciNet Google Scholar
  17. Chen, C., Hong, J.: Symplectic Runge-Kutta semidiscretization for stochastic Schrödinger equation. SIAM J. Numer. Anal. 54(4), 2569–2593 (2016). https://doi.org/10.1137/151005208
    Article MathSciNet Google Scholar
  18. Chen, L., Huang, J.: Some error analysis on virtual element methods. Calcolo 55(1), 5–23 (2018). https://doi.org/10.1007/s10092-018-0249-4
    Article MathSciNet Google Scholar
  19. Cui, J., Hong, J.: Strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one-sided Lipschitz coefficient. SIAM J. Numer. Anal. 57(4), 1815–1841 (2019). https://doi.org/10.1137/18M1215554
    Article MathSciNet Google Scholar
  20. Chen, L., Huang, X.: Nonconforming virtual element method for \(2m\)th order partial differential equations in \({R}^n\). Math. Comp. 89(324), 1711–1744 (2020). https://doi.org/10.1090/mcom/3498
    Article MathSciNet Google Scholar
  21. Chen, C., Hong, J., Ji, L.: Mean-square convergence of a symplectic local discontinuous Galerkin method applied to stochastic linear Schrödinger equation. IMA J. Numer. Anal. 37(2), 1041–1065 (2017). https://doi.org/10.1093/imanum/drw023
    Article MathSciNet Google Scholar
  22. Chow, P.-L.: Stochastic Partial Differential Equations, 2nd edn. Advances in Applied Mathematics. CRC Press, Boca Raton, FL (2015)
    Google Scholar
  23. Cui, J., Hong, J., Sun, L.: Strong convergence of full discretization for stochastic Cahn-Hilliard equation driven by additive noise. SIAM J. Numer. Anal. 59(6), 2866–2899 (2021). https://doi.org/10.1137/20M1382131
    Article MathSciNet Google Scholar
  24. Chen, L., Wang, F.: A divergence free weak virtual element method for the Stokes problem on polytopal meshes. J. Sci. Comput. 78(2), 864–886 (2019). https://doi.org/10.1007/s10915-018-0796-5
    Article MathSciNet Google Scholar
  25. Dedner, A., Hodson, A.: A higher order nonconforming virtual element method for the Cahn-Hilliard equation. J. Sci. Comput. 101(3), 81 (2024). https://doi.org/10.1007/s10915-024-02721-z
    Article MathSciNet Google Scholar
  26. Feng, F., Han, W., Huang, J.: Virtual element method for an elliptic hemivariational inequality with applications to contact mechanics. J. Sci. Comput. 81(3), 2388–2412 (2019). https://doi.org/10.1007/s10915-019-01090-2
    Article MathSciNet Google Scholar
  27. Feng, X., Li, Y., Zhang, Y.: Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise. SIAM J. Numer. Anal. 55(1), 194–216 (2017). https://doi.org/10.1137/15M1022124
    Article MathSciNet Google Scholar
  28. Feng, X., Li, Y., Zhang, Y.: A fully discrete mixed finite element method for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise. J. Sci. Comput. 83(1), 23–24 (2020). https://doi.org/10.1007/s10915-020-01202-3
    Article MathSciNet Google Scholar
  29. Feng, X., Li, Y., Zhang, Y.: Strong convergence of a fully discrete finite element method for a class of semilinear stochastic partial differential equations with multiplicative noise. J. Comput. Math. 39(4), 574–598 (2021). https://doi.org/10.4208/jcm.2003-m2019-0250
    Article MathSciNet Google Scholar
  30. Feng, X., Prohl, A., Vo, L.: Optimally convergent mixed finite element methods for the stochastic Stokes equations. IMA J. Numer. Anal. 41(3), 2280–2310 (2021). https://doi.org/10.1093/imanum/drab006
    Article MathSciNet Google Scholar
  31. Feng, F., Yu, Y.: A modified interior penalty virtual element method for fourth-order singular perturbation problems. J. Sci. Comput. 101(1), 21–32 (2024). https://doi.org/10.1007/s10915-024-02665-4
    Article MathSciNet Google Scholar
  32. Gao, X., Qin, Y., Li, J.: Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise. Adv. Comput. Math. 50(3), 46–48 (2024). https://doi.org/10.1007/s10444-024-10122-x
    Article MathSciNet Google Scholar
  33. Huang, C., Shen, J.: Stability and convergence analysis of a fully discrete semi-implicit scheme for stochastic Allen-Cahn equations with multiplicative noise. Math. Comp. 92(344), 2685–2713 (2023). https://doi.org/10.1090/mcom/3846
    Article MathSciNet Google Scholar
  34. Jentzen, A., Pušnik, P.: Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. IMA J. Numer. Anal. 40(2), 1005–1050 (2020). https://doi.org/10.1093/imanum/drz009
    Article MathSciNet Google Scholar
  35. Kovács, M., Lang, A., Petersson, A.: Approximation of SPDE covariance operators by finite elements: a semigroup approach. IMA J. Numer. Anal. 43(3), 1324–1357 (2022). https://doi.org/10.1093/imanum/drac020
    Article MathSciNet Google Scholar
  36. Kruse, R.: Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise. IMA J. Numer. Anal. 34(1), 217–251 (2014). https://doi.org/10.1093/imanum/drs055
    Article MathSciNet Google Scholar
  37. Kruse, R., Weiske, R.: The BDF2-Maruyama method for the stochastic Allen-Cahn equation with multiplicative noise. J. Comput. Appl. Math. 419, 114634–13 (2023) https://doi.org/10.1016/j.cam.2022.114634
  38. Liu, X., Chen, Z.: The nonconforming virtual element method for the Navier-Stokes equations. Adv. Comput. Math. 45(1), 51–74 (2019). https://doi.org/10.1007/s10444-018-9602-z
    Article MathSciNet Google Scholar
  39. Liu, X., He, Z., Chen, Z.: A fully discrete virtual element scheme for the Cahn-Hilliard equation in mixed form. Comput. Phys. Commun. 246, 106870–11 (2020) https://doi.org/10.1016/j.cpc.2019.106870
  40. Liu, Z., Qiao, Z.: Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise. Stoch. Partial Differ. Equ. Anal. Comput. 9(3), 559–602 (2021). https://doi.org/10.1007/s40072-020-00179-2
    Article MathSciNet Google Scholar
  41. Li, Y., Shu, C.-W., Tang, S.: A discontinuous Galerkin method for stochastic conservation laws. SIAM J. Sci. Comput. 42(1), 54–86 (2020). https://doi.org/10.1137/19M125710X
    Article MathSciNet Google Scholar
  42. Li, Y., Shu, C.-W., Tang, S.: An ultra-weak discontinuous Galerkin method with implicit-explicit time-marching for generalized stochastic KdV equations. J. Sci. Comput. 82(3), 61–36 (2020). https://doi.org/10.1007/s10915-020-01162-8
    Article MathSciNet Google Scholar
  43. Li, Y., Shu, C.-W., Tang, S.: A local discontinuous Galerkin method for nonlinear parabolic SPDEs. ESAIM Math. Model. Numer. Anal. 55, 187–223 (2021) https://doi.org/10.1051/m2an/2020026
  44. Li, M., Zhao, J., Huang, C., Chen, S.: Nonconforming virtual element method for the time fractional reaction-subdiffusion equation with non-smooth data. J. Sci. Comput. 81(3), 1823–1859 (2019). https://doi.org/10.1007/s10915-019-01064-4
    Article MathSciNet Google Scholar
  45. Li, M., Zhao, J., Wang, N., Chen, S.: Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: a unified framework. Comput. Methods Appl. Mech. Engrg. 380, 113793–27 (2021) https://doi.org/10.1016/j.cma.2021.113793
  46. Meng, J., Mei, L.: A mixed virtual element method for the vibration problem of clamped Kirchhoff plate. Adv. Comput. Math. 46(5), 68–18 (2020). https://doi.org/10.1007/s10444-020-09810-1
    Article MathSciNet Google Scholar
  47. Majee, A.K., Prohl, A.: Optimal strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise. Comput. Methods Appl. Math. 18(2), 297–311 (2018). https://doi.org/10.1515/cmam-2017-0023
    Article MathSciNet Google Scholar
  48. Mora, D., Rivera, G.: A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations. IMA J. Numer. Anal. 40(1), 322–357 (2020). https://doi.org/10.1093/imanum/dry063
    Article MathSciNet Google Scholar
  49. Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25(8), 1421–1445 (2015). https://doi.org/10.1142/S0218202515500372
    Article MathSciNet Google Scholar
  50. Meng, J., Wang, G., Mei, L.: Mixed virtual element method for the Helmholtz transmission eigenvalue problem on polytopal meshes. IMA J. Numer. Anal. 43(3), 1685–1717 (2023). https://doi.org/10.1093/imanum/drac019
    Article MathSciNet Google Scholar
  51. Qi, R., Wang, X.: Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise. J. Sci. Comput. 80(2), 1171–1194 (2019). https://doi.org/10.1007/s10915-019-00973-8
    Article MathSciNet Google Scholar
  52. Qi, X., Zhang, Y., Xu, C.: An efficient approximation to the stochastic Allen-Cahn equation with random diffusion coefficient field and multiplicative noise. Adv. Comput. Math. 49(5), 73–24 (2023). https://doi.org/10.1007/s10444-023-10072-w
    Article MathSciNet Google Scholar
  53. Vacca, G., Beirão da Veiga, L.: Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Diff. Equ. 31(6), 2110–2134 (2015) https://doi.org/10.1002/num.21982
  54. Wang, X.: Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise. IMA J. Numer. Anal. 37(2), 965–984 (2017). https://doi.org/10.1093/imanum/drw016
    Article MathSciNet Google Scholar
  55. Yang, X., Zhao, W., Zhao, W.: Strong optimal error estimates of discontinuous Galerkin method for multiplicative noise driving nonlinear SPDEs. Numer. Methods Partial Diff. Equ. 39(3), 2073–2095 (2023). https://doi.org/10.1002/num.22958
    Article MathSciNet Google Scholar
  56. Zhang, B., Zhao, J., Yang, Y., Chen, S.: The nonconforming virtual element method for elasticity problems. J. Comput. Phys. 378, 394–410 (2019) https://doi.org/10.1016/j.jcp.2018.11.004

Download references