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Strong, weak and a posteriori error analysis of the finite element method for parabolic and hyperbolic stochastic equations
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We consider the heat and wave equations driven by additive noise. We
discretize the equations in space by the standard continuous finite
element method and derive various error estimates under sharp regularity assumptions. We use the operator semigroup framework for stochastic PDEs proposed by Da Prato and Zabczyk in the analysis. The first kind of estimate measures the error in the mean square norm and shows the so called strong convergence. Here, appropriate error estimates for the deterministic problem give error estimates for the stochastic equations in a more or less straightforward fashion. In case of the semilinear stochastic heat equation the possibility of truncating the expansion of
the noise is also discussed. In the second type of error analysis the
error is measured in the weak sense of probability measures and implies the so called weak convergence. The analysis in this case is considerably more complicated and uses more advanced results both from infinite dimensional stochastic analysis as well as from functional analysis. Here, the linear stochastic heat equation is discussed. Finally, we consider a posteriori type error estimates for the stochastic heat equation.