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 $L_{\omega }^p{L}_t^{\mathrm{\infty }}{\dot{H}}^{1+\gamma }$-norm and a temporal H"older 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 $O(h^{1+\gamma }+{\tau }^{1\mathrm{/}2})$ and $O(h^{1+\gamma }+{\tau }^{(1+\gamma )\mathrm{/}2})$ for the Galerkin-based Euler and Milstein schemes, respectively.