scholarly journals Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

2010 ◽  
Vol 44 (2) ◽  
pp. 289-322 ◽  
Author(s):  
Georgios T. Kossioris ◽  
Georgios E. Zouraris
Keyword(s):  
Finite Element ◽  
Parabolic Equation ◽  
White Noise ◽  
Fourth Order ◽  
Space Time ◽  
Order Linear ◽  
Fully Discrete ◽  
Finite Element Approximations ◽  
Discrete Finite Element ◽  
Stochastic Parabolic Equation

scholarly journals Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

2020 ◽  
Vol 54 (6) ◽  
pp. 2199-2227
Author(s):  
Mihály Kovács ◽  
Annika Lang ◽  
Andreas Petersson
Keyword(s):  
Finite Element ◽  
Weak Convergence ◽  
Additive Noise ◽  
Convergence Rates ◽  
Fully Discrete ◽  
Finite Element Approximations ◽  
Negative Order ◽  
Stochastic Wave Equation ◽  
Discrete Finite Element ◽  
Spatial Approximation

The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.

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