Previous Articles    

STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE

Xiaobing Feng1, Yukun Li2, Yi Zhang3   

  1. 1 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA;
    2 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA;
    3 Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402, USA
  • Received:2018-06-07 Revised:2020-02-13 Published:2021-08-06
  • Contact: Yukun Li,Email:yukun.li@ucf.edu
  • Supported by:
    The work of the first author was partially supported by the NSF grant DMS-1318486. The work of the second author was partially supported by the startup grant from University of Central Florida.

Xiaobing Feng, Yukun Li, Yi Zhang. STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE[J]. Journal of Computational Mathematics, 2021, 39(4): 574-598.

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition. These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations (SODEs) under “minimum assumptions” were studied. As a result, the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established, which are difficult and delicate. A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal. Finally, stability estimates for the second and higher order moments of the L2-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence. Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.

CLC Number: 

[1] R. Bank and H. Yserentant, On the H1-stability of the L2-projection onto finite element spaces, Numer. Math., 126(2014), 361-381.
[2] M. Beccari, M. Hutzenthaler, A. Jentzen, R. Kurniawan, F. Lindner and D. Salimova, Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities, 2019.
[3] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer, 2008.
[4] J. Brandts, A. Hannukainen, S. Korotov and M. Křȋžek, On angle conditions in the finite element method, SeMA Journal, 56(2011), 81-95.
[5] C-E Bréhier, and L. Goudenège, Analysis of some splitting schemes for the stochastic Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 24(2019), 4169-4190.
[6] K. Burrage and J. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal., 16(1979), 46-57.
[7] J. Butcher, A stability property of implicit Runge-Kutta methods, BIT, 15(1975), 358-361.
[8] P. Ciarlet, The finite element method for elliptic problems, Classics in Appl. Math., 40(2002), 1-511.
[9] X. Feng and Y. Li, Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow, IMA J. Numer. Anal., 35(2015), 1622-1651.
[10] X. Feng, Y. Li and A. Prohl, Finite element approximations of the stochastic mean curvature flow of planar curves of graphs, Stoch. PDEs: Analysis and Computations, 2(2014), 54-83.
[11] X. Feng, Y. Li and Y. Zhang, Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise, SIAM J. Numer. Anal., 55(2017), 194-216.
[12] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94(2003), 33-65.
[13] G. Dahlquist, Error analysis for a class of methods for stiff non-linear initial value problems, SIAM J. Numer. Anal. (1976), 60-72.
[14] K. Dekker, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, 2(1984).
[15] B. Gess, Strong solutions for stochastic partial differential equations of gradient type, J. Funct. Anal., 263(2012), 2355-2383.
[16] I. Gyöngy and A. Millet, On discretization schemes for stochastic evolution equations, Potential analysis, 23(2005), 99-134.
[17] I. Gyöngy, S. Sabanis and D. Šiška, Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stochastics and Partial Differential Equations: Analysis and Computations, 4(2016), 225-245.
[18] D. Higham, X. Mao and A. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40(2002), 1041-1063.
[19] M. Hutzenthaler, A. Jentzen and P. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467(2010), 1563-1576.
[20] M. Hutzenthaler, A. Jentzen and P. Kloeden, Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations, Ann. Appl. Probab., 23:5(2013), 1913- 1966.
[21] K. Itô, 109. stochastic integral, Proceedings of the Imperial Academy, 20(1944), 519-524.
[22] A. Jentzen and P. Pušnik, Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities, IMA J. Numer. Anal., (2020), 1-38.
[23] M. Kovács, S. Larsson and F. Lindgren, On the backward Euler approximation of the stochastic Allen-Cahn equation, J. Appl. Probab., 52(2015), 323-338.
[24] M. Kovács, S. Larsson and F. Lindgren, On the discretisation in time of the stochastic Allen-Cahn equation, Mathematische Nachrichten, 291(2018), 966995.
[25] M. Krızek and Q. Lin, On diagonal dominance of stiffness matrices in 3D, East-West J. Numer. Math, 3(1995), 59-69.
[26] Y. Li, Numerical Methods for Deterministic and Stochastic Phase Field Models of Phase Transition and Related Geometric Flows, Ph.D. thesis, The University of Tennessee, Knoxville, 2015.
[27] Z. Liu and Z. Qiao, Strong approximation of monotone stochastic partial differential equations driven by white noise, IMA J. Numer. Anal., 2019.
[28] A. Majee and A. Prohl, Optimal Strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise, Comput. Methods Appl. Math., 18(2018), 297-311.
[29] X. Mao, Stochastic differential equations and applications, 2nd Edition, Elsevier, 2007.
[30] P. Kloeden and E. Platen, Numerical Methods for Stochastic Differential Equations, Springer, 1991.
[31] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, 1992.
[32] A. Majee and A. Prohl, A. Prohl, Strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise, Comput. Methods Appl. Math., (2018), 297-311.
[33] A. Stuart and A. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1998.
[34] J. Xu, Y. Li, S. Wu and A. Bousquet, On the stability and accuracy of partially and fully implicit schemes for phase field modeling, Comput. Methods in Appl. Mech. Eng., 345(2019), 826-853.
[35] J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp., 68(1999), 1429-1446.
[1] Xiaocui Li, Xu You. MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 130-146.
[2] Tingting Qin, Chengjian Zhang. A GENERAL CLASS OF ONE-STEP APPROXIMATION FOR INDEX-1 STOCHASTIC DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS [J]. Journal of Computational Mathematics, 2019, 37(2): 151-169.
[3] Charles-Edouard Bréhier, Martin Hairer, Andrew M. Stuart. WEAK ERROR ESTIMATES FOR TRAJECTORIES OF SPDEs UNDER SPECTRAL GALERKIN DISCRETIZATION [J]. Journal of Computational Mathematics, 2018, 36(2): 159-182.
[4] Christoph Reisinger, Zhenru Wang. ANALYSIS OF MULTI-INDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE [J]. Journal of Computational Mathematics, 2018, 36(2): 202-236.
[5] Rikard Anton, David Cohen. EXPONENTIAL INTEGRATORS FOR STOCHASTIC SCHRÖDINGER EQUATIONS DRIVEN BY ITÔ NOISE [J]. Journal of Computational Mathematics, 2018, 36(2): 276-309.
[6] Xiaocui Li, Xiaoyuan Yang. ERROR ESTIMATES OF FINITE ELEMENT METHODS FOR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2017, 35(3): 346-362.
[7] Xiaoyuan Yang, Xiaocui Li, Ruisheng Qi, Yinghan Zhang. FULL-DISCRETE FINITE ELEMENT METHOD FOR STOCHASTIC HYPERBOLIC EQUATION [J]. Journal of Computational Mathematics, 2015, 33(5): 533-556.
Viewed
Full text


Abstract