Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise

Abstract

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.