Finite Element Approximation of the Linear Stochastic Wave Equation with Additive Noise
Abstract
Semidiscrete finite element approximation of the linear stochastic wave equation (LSWE) with additive noise is studied in a semigroup framework. Optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions. These are used to prove strong convergence estimates for the stochastic problem. The theory presented here applies to multidimensional domains and spatially correlated noise. Numerical examples illustrate the theory.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
E. J. Allen, Modeling with Itô Stochastic Differential Equations, Math. Model. Theory Appl. 22, Springer, Dordrecht, The Netherlands, 2007.
2.
E. J. Allen, S. J. Novosel, and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep., 64 (1998), pp. 117–142.
3.
G. A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal., 13 (1976), pp. 564–576.
4.
P. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2007.
5.
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.
6.
Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J. Numer. Anal., 40 (2002), pp. 1421–1445.
7.
T. Dupont, $L\sp{2}$-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal., 10 (1973), pp. 880–889.
8.
M. Geissert, M. Kovács, and S. Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, BIT, 49 (2009), pp. 343–356.
9.
I. Gyöngy, Approximations of stochastic partial differential equations, in Stochastic Partial Differential Equations and Applications (Trento, 2002), Lect. Notes Pure Appl. Math. 227, Dekker, New York, 2002, pp. 287–307.
10.
M. Kovács, S. Larsson, and F. Lindgren, Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Noise, preprint 2009:37, Department of Mathematical Sciences, Chalmers University of Technology, Göteborg, Sweden, 2009.
11.
M. Kovács, S. Larsson, and F. Lindgren, Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise, Numer. Algorithms, 53 (2010), pp. 309–320.
12.
A. Martin, S. M. Prigarin, and G. Winkler, Exact and fast numerical algorithms for the stochastic wave equation, Int. J. Comput. Math., 80 (2003), pp. 1535–1541.
13.
C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math. 1905, Springer, Berlin, 2007.
14.
L. Quer-Sardanyons and M. Sanz-Solé, Space semi-discretisations for a stochastic wave equation, Potential Anal., 24 (2006), pp. 303–332.
15.
J. Rauch, On convergence of the finite element method for the wave equation, SIAM J. Numer. Anal., 22 (1985), pp. 245–249.
16.
C. Roth, Weak approximations of solutions of a first order hyperbolic stochastic partial differential equation, Monte Carlo Methods Appl., 13 (2007), pp. 117–133.
17.
C. Schwab and R. A. Todor, Karhunen-Loève approximation of random fields by generalized fast multipole methods, J. Comput. Phys., 217 (2006), pp. 100–122.
18.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer-Verlag, Berlin, 2006.
19.
J. B. Walsh, An introduction to stochastic partial differential equations, in École d'été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math. 1180, Springer, Berlin, 1986, pp. 265–439.
20.
J. B. Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006), pp. 991–1018.
21.
Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise, BIT, 44 (2004), pp. 829–847.
22.
Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), pp. 1363–1384.
Information & Authors
Information
Published In
Copyright
Copyright © 2010 Society for Industrial and Applied Mathematics.
History
Submitted: 28 September 2009
Accepted: 8 February 2010
Published online: 23 April 2010
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Strong Convergence Rates for a Full Discretization of Stochastic Wave Equation with Nonlinear DampingJournal of Scientific Computing, Vol. 102, No. 2 | 3 January 2025
- Weak convergence rates for temporal numerical approximations of the semilinear stochastic wave equation with multiplicative noiseNumerische Mathematik, Vol. 156, No. 6 | 10 October 2024
- Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial dataScience China Mathematics, Vol. 67, No. 12 | 27 March 2024
- Temporal approximation of stochastic evolution equations with irregular nonlinearitiesJournal of Evolution Equations, Vol. 24, No. 2 | 9 May 2024
- Novel structure-preserving schemes for stochastic Klein–Gordon–Schrödinger equations with additive noiseJournal of Computational Physics, Vol. 500 | 1 Mar 2024
- Strong optimal error estimates of discontinuous Galerkin method for multiplicative noise driving nonlinear SPDEsNumerical Methods for Partial Differential Equations, Vol. 39, No. 3 | 23 November 2022
- Discontinuous Galerkin for the wave equation: a simplified a priori error analysisInternational Journal of Computer Mathematics, Vol. 100, No. 3 | 3 November 2022
- Newton's method for stochastic semilinear wave equations driven by multiplicative time‐space noiseMathematische Nachrichten, Vol. 296, No. 2 | 28 November 2022
- Finite Element Approximations of a Class of Nonlinear Stochastic Wave Equations with Multiplicative NoiseJournal of Scientific Computing, Vol. 91, No. 2 | 3 April 2022
- Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noiseJournal of Computational Physics, Vol. 451 | 1 Feb 2022
- Infinite-Dimensional Stochastic Hamiltonian SystemsSymplectic Integration of Stochastic Hamiltonian Systems | 16 November 2022
- Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative NoiseApplied Mathematics & Optimization, Vol. 84, No. S2 | 6 November 2021
- Strong Convergence of a Verlet Integrator for the Semilinear Stochastic Wave EquationSIAM Journal on Numerical Analysis, Vol. 59, No. 4 | 15 July 2021
- A local discontinuous Galerkin method for nonlinear parabolic SPDEsESAIM: Mathematical Modelling and Numerical Analysis, Vol. 55 | 26 February 2021
- A full-discrete exponential Euler approximation of the invariant measure for parabolic stochastic partial differential equationsApplied Numerical Mathematics, Vol. 157 | 1 Nov 2020
- Energy estimates and model order reduction for stochastic bilinear systemsInternational Journal of Control, Vol. 93, No. 8 | 8 November 2018
- A new type of singular perturbation approximation for stochastic bilinear systemsMathematics of Control, Signals, and Systems, Vol. 32, No. 2 | 25 May 2020
- Newton’s method for nonlinear stochastic wave equationsForum Mathematicum, Vol. 32, No. 3 | 11 February 2020
- A stochastic telegraph equation from the six-vertex modelThe Annals of Probability, Vol. 47, No. 6 | 1 Nov 2019
- Error estimates of finite element method for semilinear stochastic strongly damped wave equationIMA Journal of Numerical Analysis, Vol. 39, No. 3 | 5 June 2018
- Efficient Stochastic Galerkin Methods for Maxwell’s Equations with Random InputsJournal of Scientific Computing, Vol. 80, No. 1 | 7 March 2019
- Online Identification of Nonlinear Stochastic Spatiotemporal System With Multiplicative Noise by Robust Optimal Control-Based Kernel Learning MethodIEEE Transactions on Neural Networks and Learning Systems, Vol. 30, No. 2 | 1 Feb 2019
- Regularity analysis of metamaterial Maxwell’s equations with random coefficients and initial conditionsComputer Methods in Applied Mechanics and Engineering, Vol. 335 | 1 Jun 2018
- Enhancing the Order of the Milstein Scheme for Stochastic Partial Differential Equations with Commutative NoiseSIAM Journal on Numerical Analysis, Vol. 56, No. 4 | 14 August 2018
- Weak convergence of Galerkin approximations of stochastic partial differential equations driven by additive Lévy noiseMathematics and Computers in Simulation, Vol. 143 | 1 Jan 2018
- Regularity of weak solutions for the stochastic wave equation with additive noise2017 36th Chinese Control Conference (CCC) | 1 Jul 2017
- Nonlinear stochastic evolution equations of second order with dampingStochastics and Partial Differential Equations: Analysis and Computations, Vol. 5, No. 1 | 11 October 2016
- An accelerated exponential time integrator for semi-linear stochastic strongly damped wave equation with additive noiseJournal of Mathematical Analysis and Applications, Vol. 447, No. 2 | 1 Mar 2017
- Approximating Stochastic Evolution Equations with Additive White and Rough NoisesSIAM Journal on Numerical Analysis, Vol. 55, No. 4 | 17 August 2017
- Numerical methods for stochastic differential equationsNumerical Methods for Stochastic Partial Differential Equations with White Noise | 2 September 2017
- Existence and convergence of G alerkin approximation for second order hyperbolic equations with memory termNumerical Methods for Partial Differential Equations, Vol. 32, No. 2 | 19 August 2015
- Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative NoiseSIAM Journal on Numerical Analysis, Vol. 54, No. 2 | 12 April 2016
- An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave EquationJournal of Scientific Computing, Vol. 64, No. 1 | 12 October 2014
- Stochastic exponential integrator for finite element spatial discretization of stochastic elastic equationComputers & Mathematics with Applications, Vol. 69, No. 8 | 1 Apr 2015
- A fully discrete approximation of the one-dimensional stochastic wave equationIMA Journal of Numerical Analysis, Vol. 13 | 11 March 2015
- Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Lévy NoiseSIAM/ASA Journal on Uncertainty Quantification, Vol. 3, No. 1 | 8 December 2015
- Identification of partially known non‐linear stochastic spatio‐temporal dynamical systems by using a novel partially linear Kernel methodIET Control Theory & Applications, Vol. 9, No. 1 | 1 January 2015
- Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational InequalitiesSIAM Journal on Numerical Analysis, Vol. 52, No. 3 | 29 May 2014
- Higher Order Strong Approximations of Semilinear Stochastic Wave Equation with Additive Space-time White NoiseSIAM Journal on Scientific Computing, Vol. 36, No. 6 | 11 November 2014
- Full‐discrete finite element method for the stochastic elastic equation driven by additive noiseNumerical Methods for Partial Differential Equations, Vol. 29, No. 6 | 14 April 2013
- Identification of non‐linear stochastic spatiotemporal dynamical systemsIET Control Theory & Applications, Vol. 7, No. 17 | 1 November 2013
- Weak Convergence of Finite Element Method for Stochastic Elastic Equation Driven By Additive NoiseJournal of Scientific Computing, Vol. 56, No. 3 | 24 January 2013
- Nonconforming finite element method for stochastic Stokes equationsApplied Mathematical Modelling, Vol. 37, No. 8 | 1 Apr 2013
- Multilevel Monte Carlo method for parabolic stochastic partial differential equationsBIT Numerical Mathematics, Vol. 53, No. 1 | 18 September 2012
- A Trigonometric Method for the Linear Stochastic Wave EquationSIAM Journal on Numerical Analysis, Vol. 51, No. 1 | 24 January 2013
- Multilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEsMultiscale Modeling & Simulation, Vol. 11, No. 4 | 22 October 2013
- Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemesBIT Numerical Mathematics, Vol. 13 | 8 November 2012
- A posteriori analysis of finite element discretizations of stochastic partial differential delay equationsJournal of Difference Equations and Applications, Vol. 18, No. 10 | 1 Oct 2012
- Weak convergence of finite element approximations of linear stochastic evolution equations with additive noiseBIT Numerical Mathematics, Vol. 52, No. 1 | 2 August 2011
- Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficientsNumerische Mathematik, Vol. 119, No. 1 | 10 April 2011
- Sparse tensor discretizations of high-dimensional parametric and stochastic PDEsActa Numerica, Vol. 20 | 28 April 2011
- Spatial approximation of stochastic convolutionsJournal of Computational and Applied Mathematics, Vol. 235, No. 12 | 1 Apr 2011
- Weak approximation of the stochastic wave equationJournal of Computational and Applied Mathematics, Vol. 235, No. 1 | 1 Nov 2010
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].