Strong convergence of the linear implicit Euler method for the finite element discretization of semilinear SPDEs driven by multiplicative or additive noise

Applied Numerical Mathematics

2020

Self Cite

This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than the autonomous ones when modeling real world phenomena. Such equations find applications in many fields such as transport in porous media, quantum fields theory, electromagnetism and nuclear physics. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet well understood. Here, a non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the time depending, not necessarily self-adjoint linear operators with the corresponding semigroup and provide the strong convergence result of the fully discrete scheme toward the mild solution. The results indicate how the converge order depends on the regularity of the initial solution and the noise. Additionally, for additive noise we achieve convergence order in time approximately 1 under less regularity assumptions on the nonlinear drift term than required in the current literature, even in the autonomous case.Numerical simulations motivated from realistic porous media flow are provided to illustrate our theoretical finding.We consider numerical approximation of the following non-autonomous SPDE defined in Λ ⊂on the Hilbert space L 2 (Λ, R), T > 0 is the final time, F and B are nonlinear functions and X 0 is the initial data, which is random. The family of linear operators A(t) are unbounded, not necessarily self-adjoint, and for all s ∈ [0, T ], −A(s) is the generator of an analytic semigroup S s (t) =: e −tA(s) , t ≥ 0. The noise W (t) is a Q−Wiener process defined in a filtered probability space (Ω, F , P, {F t } t≥0 ). The filtration is assumed to fulfill the usual conditions (see e.g. [38,Definition 2.1.11]). Note that the noise can be represented as follows (see e.g. [38,37])where q i , e i , i ∈ N d are respectively the eigenvalues and the eigenfunctions of the covariance operator Q, and β i , i ∈ N, are independent and identically distributed standard Brownian motions. Precise assumptions on F , B, X 0 and A(t) will be given in the next section to ensure the existence of the unique mild solution X of (1). In many situations it is hard to exhibit explicit solutions of SPDEs. Therefore, numerical algorithms are good tools to provide realistic approximations. Strong approximations of autonomous SPDEs with constant linear self-adjoint operator A(t) = A are widely investigated in the literature, see e.g. [47,46,20,17,25,48] and references therein. When we turn our attention to the case of semilinear SPDEs, still with constant operator A(t) = A, but not necessary self-adjoint, the list of references becomes remarkably short, see e.g. [24,31]. Note that modelling real world phenomena with time dependent linear operator is more realistic than modelling with time independent lin...

Section: Preparatory Resultsmentioning

confidence: 99%

Strong convergence of the linear implicit Euler method for the finite element discretization of semilinear non-autonomous SPDEs driven by multiplicative or additive noise

Mukam

Applied Numerical Mathematics

2020

Self Cite

“…For any 0 ≤ ρ ≤ 1, 0 ≤ γ ≤ 2 and 0 ≤ υ ≤ H with H ∈ 1 2 , 1 , if the linear operator is given by (34), there exists a positive constant C such that for all…”

Section: Lemmamentioning

confidence: 99%

“…For time discretization, we will first update the implicit linear for finite element method and not necessarily self-adjoint. We also provide the strong convergence of the exponential scheme [21,34] for (H ∈ ( 1 2 , 1]). Note that this scheme is an explicit stable scheme, where the implementation is based on the computation of matrix exponential functions [21].…”

Section: Introductionmentioning

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

Self Cite

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.

“…Linear implicit Euler method has been investigated in the literature, see e.g. [18,23,40,44] and the references therein. The resolvent operator (I + ∆tA h ) −1 plays a key role to stabilise the linear implicit Euler method, where A h is the discrete form of A after space discretization.…”

Section: Introductionmentioning

confidence: 99%

“…For such equations, both linear implicit Euler method [18,23,40,44] and exponential integrators [11,22,43] behave like the standard explicit Euler method (see Section 2.3) and therefore lose their good stabilities properties. For such problems in the deterministic context, Exponential Rosenbrock-type [10,39] methods and Rosenbrock-type [28,29,39] methods were proved to be efficient.…”

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

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.