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 LωpLt∞H˙1+γ-norm and a temporal Hölder regularity under the LωpLx2-norm for the solution of the proposed equation with an H˙ ^1+γ-valued initial datum for γ∈ [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.