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 $L_\omega^p L_t^\infty \dot H^{1+\gamma}$-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 $O(h^{1+\gamma}+\tau^{1/2})$ and $O(h^{1+\gamma}+\tau^{(1+\gamma)/2})$ for the Galerkin-based Euler and Milstein schemes, respectively.