Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise

摘要

We consider a semilinear parabolic PDE driven by additive noise. The equation is discretized in space by a standard piecewise linear finite element method. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing the asymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is smooth enough. For example, if the covariance operator is given by the Gauss kernel, then the number of terms to be kept is the quasi-logarithm of the number of terms in the original expansion. Then one can reduce the size of the corresponding linear algebra problem enormously and hence reduce the computational complexity, which is a key issue when stochastic problems are simulated. Keywords Finite element - Semilinear parabolic equation - Wiener process - Error estimate - Stochastic partial differential equation - Truncation Mathematics Subject Classifications (2000) 65N30 - 60H35 - 60H15 - 35R60 M. Kovács was partially supported by the Swedish Research Council (VR). S. Larsson was partially supported by the Swedish Foundation for Strategic Research through GMMC, the Gothenburg Mathematical Modeling Centre and the Swedish Research Council (VR).