Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs dri
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Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure Aurelien Junior Noupelah1 · Antoine Tambue2,3 Received: 31 December 2019 / Accepted: 1 November 2020 / © The Author(s) 2020
Abstract In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter H > 12 and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order O(h2 + Δt) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise. Keywords Stochastic parabolic partial differential equations · Fractional Brownian motion · Finite element method · Errors estimate · Finite element methods · Timestepping methods · Poisson random measure Mathematics Subject Classification (2010) 65C30 · 74S05 · 74S60
Antoine Tambue
[email protected]; [email protected]
Extended author information available on the last page of the article.
Numerical Algorithms
1 Introduction We analyse the strong numerical approximation of an SPDE defines in Λ ⊂ Rd , d ∈ {1, 2, 3} with initial value and boundary conditions (Dirichlet, Neumann, Robin boundary conditions or mixed Dirichlet and Neumann). In Hilbert space, our model equation can be formulated as the following parabolic SPDE dX(t) + AX(t)dt = F (X(t))dt + φ(t)dB H (t) + χ z0 N(dz, dt) (1) X(0) = X0 , in Hilbert space H = L2 (Λ), with z0 ∈ χ, where χ is the mark set defined by χ := H \ {0}. Let B (Γ ) be the smallest σ -algebra containing all open sets of Γ . Let (χ, B (χ), ν) be a σ -finite measurable space and ν (with ν ≡ 0) a L´evy measure on B (χ) such that ν({0}) = 0 and min(z2 , 1)ν(dz) < ∞. (2) χ
Let N(dz, dt) be the H-valued Poisson distributed σ -finite measure on the product σ -algebra B (χ) and B (R+ ) with intensity ν(dz)dt, where dt is the Lebesgue mea dt) stands for the compensated sure on B (R+ ). In our model problem (1), N(dz, Poisson random measure defined by (dz, dt) := N(dz, dt) − ν(dz)dt. N
(3)
We denote by
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