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Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise. (English) Zbl 1476.65243

Summary: We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov assumptions, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the \(L_\omega^p L_t^\infty \dot{H}^{1+\gamma}\)-norm and a temporal Hölder regularity under the \(L_\omega^p L_x^2\)-norm for the solution of the proposed equation with an \(\dot{H}^{1+\gamma}\)-valued initial datum for \(\gamma \in [0,1]\). Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates \({\mathscr{O}}(h^{1+\gamma}+\tau^{1/2})\) and \({\mathscr{O}}(h^{1+\gamma}+\tau^{(1+\gamma)/2})\) for the Galerkin-based Euler and Milstein schemes, respectively.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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