## 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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