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>frac {1}{2} 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 mathcal {O}(h^{2}+varDelta t) 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.
Highlights
We analyse the strong numerical approximation of an stochastic partial differential equation (SPDE) defines in2 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
We provide the strong convergence of the exponential scheme [21, 34] for (
Summary
2 3 with initial value and boundary conditions (Dirichlet, Neumann, Robin boundary conditions or mixed Dirichlet and Neumann). We are interested in considering an fBm with values in a Hilbert space and giving the definition of the corresponding stochastic integral. If the linear operator is given by (34), the following optimal regularity results in space and time hold (28). Note that the exponential integrator scheme (49) is an explicit stable scheme when the SPDE (5) is driven by its linear part as the linear implicit method, while the stochastic exponential Rosenbrock scheme (SERS) (50) is very stable when (5) is driven by its linear or nonlinear part. When dealing with SERS, the strong convergence proof will make use of the following assumption, used in [26, 27]
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