Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative &amp; additive noise

Stochastic Processes and their Applications

2020

Self Cite

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretise the SPDE in space by the finite element method and propose a new scheme in time appropriate for such equations, called stochastic Rosenbrock-Type scheme, which is based on the local linearisation of the semi-discrete problem obtained after space discretisation. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise. Our convergence rates are in agreement with results in the literature. Numerical experiments to sustain our theoretical results are provided.

Section: Novel Fully Discrete Scheme and Main Resultsmentioning

confidence: 99%

“…We analyze the strong convergence of the new fully discrete scheme toward the exact solution in the L 2 -norm. The challenge here is that the resolvent of the operator S m h,∆t (ω) appearing in the numerical scheme (41) is not constant as it changes at each time step. Furthermore the operator S m h,∆t (ω) is a random operator.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

Mukam

Stochastic Processes and their Applications

2020

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“…Theorem 7 Let X(t m ) and X h m be respectively the mild solution (3) and the numerical approximation given by (35) at t m = m∆t. Let Assumption 1, Assumption 2, Assumption 3 and Assumption 4 be fulfilled.…”

Section: Resultsmentioning

confidence: 99%

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

Mukam

2017

J Sci Comput

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In this paper, we consider the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme (SERS) based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square L 2 norm. Numerical experiments to sustain theoretical results are provided.

“…where Γ (α) is the Gamma function (see [10]). Under condition (35), it is well known (see, e.g., [8]) that the linear operator −A given by (34) generates an analytic semigroup S(t) ≡ e −tA . Following [8,21], we characterize the domain of the operator A r/2 denoted by D(A r/2 ), r ∈ {1, 2} with the following equivalence of norms, useful in our convergence proofs…”

Section: Numerical Schemesmentioning

confidence: 99%

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

Noupelah

2020

Numer Algor

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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 $H>\frac {1}{2}$ H > 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 $\mathcal {O}(h^{2}+\varDelta t)$ O ( h 2 + Δ 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.