Strong Convergence of a Fully Discrete Finite Element Method for a Class of Semilinear Stochastic Partial Differential Equations with Multiplicative Noise

2021 ◽  
Vol 39 (4) ◽  
pp. 591-616
Author(s):  
global sci
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Strong Convergence ◽  
Stochastic Partial Differential Equations ◽  
Multiplicative Noise ◽  
Partial Differential ◽  
Fully Discrete ◽  
Discrete Finite Element

scholarly journals A finite element method for time fractional partial differential equations

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
Neville Ford ◽  
Jingyu Xiao ◽  
Yubin Yan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fully Discrete ◽  
Convergence Error ◽  
Time Stepping ◽  
Element Method

AbstractIn this paper, we consider the finite element method for time fractional partial differential equations. The existence and uniqueness of the solutions are proved by using the Lax-Milgram Lemma. A time stepping method is introduced based on a quadrature formula approach. The fully discrete scheme is considered by using a finite element method and optimal convergence error estimates are obtained. The numerical examples at the end of the paper show that the experimental results are consistent with our theoretical results.

scholarly journals Strong approximation of monotone stochastic partial differential equations driven by white noise

2019 ◽  
Vol 40 (2) ◽  
pp. 1074-1093 ◽  
Author(s):  
Zhihui Liu ◽  
Zhonghua Qiao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rate ◽  
Strong Convergence ◽  
White Noise ◽  
Stochastic Partial Differential Equations ◽  
Partial Differential ◽  
Fully Discrete ◽  
Backward Euler ◽  
Strong Convergence Rate

Abstract We establish an optimal strong convergence rate of a fully discrete numerical scheme for second-order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen–Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretise this random equation. By the monotonicity assumption, in combination with the factorisation method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme is obtained. Several numerical experiments are carried out to verify the theoretical result.

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