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Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise

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 $L_ω^p L_t^\infty \dot H^{1+γ}$-norm and a temporal Hölder regularity under the $L_ω^p L_x^2$-norm for the solution of the proposed equation with an $\dot H^{1+γ}$-valued initial datum for $γ\in [0,1]$. Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $O(h^{1+γ}+τ^{1/2})$ and $O(h^{1+γ}+τ^{(1+γ)/2})$ for the Galerkin-based Euler and Milstein schemes, respectively.