Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrödinger equation on the unit sphere.

show abstract

“…We end this section by observing that several strategies for proving weak rates of convergence of numerical solutions to SPDEs in the literature could be extended to the present setting or in the case of numerical discretizations of nonlinear stochastic wave equations on the sphere, see for instance [13,24,34,5,20,16,23] and references therein. This could be subject of future research.…”

Section: Convergence Analysismentioning

confidence: 99%

Numerical approximation and simulation of the stochastic wave equation on the sphere

Cohen¹,

Lang²

2021

Preprint

Self Cite

0

View full text Add to dashboard Cite

Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrödinger equation on the unit sphere.

show abstract

“…Besides global Lipschitz continuity, no further regularity assumptions are imposed on the nonlinearity F and noise G. Now, our goal is to show pathwise uniform convergence of contractive time discretisation schemes for such irregular nonlinearities and rough initial data, focusing on the hyperbolic setting. It has been extensively studied in recent years (see [1,2,4,7,[10][11][12][14][15][16][17]19,20,26,29,32,[36][37][38][39]42,53] and references therein). When passing to the parabolic setting (i.e.…”

Section: Introductionmentioning

confidence: 99%

Temporal approximation of stochastic evolution equations with irregular nonlinearities

Klioba,

Veraar

2024

J. Evol. Equ.

0

View full text Add to dashboard Cite

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error$$\begin{aligned} \textrm{E}_{k}^{\infty } {:}{=}\Big (\mathbb {E}\sup _{j\in \{0, \ldots , N_k\}} \Vert U(t_j) - U^j\Vert _X^p\Big )^{1/p} \rightarrow 0\quad (k \rightarrow 0), \end{aligned}$$ E k ∞ : = ( E sup j ∈ { 0 , … , N k } ‖ U ( t j ) - U j ‖ X p ) 1 / p → 0 ( k → 0 ) , where $$p \in [2,\infty )$$ p ∈ [ 2 , ∞ ) , U is the mild solution, $$U^j$$ U j is obtained from a time discretisation scheme, k is the step size, and $$N_k = T/k$$ N k = T / k for final time $$T>0$$ T > 0 . This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error$$\begin{aligned} \textrm{E}_k {:}{=}\bigg (\sup _{j\in \{0,\ldots ,N_k\}}\mathbb {E}\Vert U(t_j) - U^{j}\Vert _X^p\bigg )^{1/p}, \end{aligned}$$ E k : = ( sup j ∈ { 0 , … , N k } E ‖ U ( t j ) - U j ‖ X p ) 1 / p , which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

show abstract

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

Cited by 9 publications

References 25 publications

Numerical approximation and simulation of the stochastic wave equation on the sphere

Numerical approximation and simulation of the stochastic wave equation on the sphere

Numerical approximation and simulation of the stochastic wave equation on the sphere

Temporal approximation of stochastic evolution equations with irregular nonlinearities

Contact Info

Product

Resources

About