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.

我们建立了由乘法无穷维维纳过程驱动的具有单调漂移的二阶抛物线型随机偏微分方程数值近似的最佳强误差估计的一般理论。该方程通过Galerkin方法在空间上离散，并且通过漂移隐式Euler和Milstein方案在时间上离散。通过单调和Lyapunov假设，我们使用变分和半群方法来导出\（L_ \ omega ^ p L_t ^ \ infty \ dot {H} ^ {1+ \ gamma} \）- norm下的空间Sobolev正则性和在（（L_ \ omega ^ p L_x ^ 2 \）-范数下的时间Hölder正则性，用于具有（（dot）{H} ^ {1+ \ gamma} \）值的初始基准的拟议方程的解对于\（\ gamma \ in [0,1] \）。然后，我们充分利用随机演算中方程和工具的单调性，得出尖锐的强收敛速度\（{\ mathscr {O}}（h ^ {1+ \ gamma} + \ tau ^ {1/2} ）\）和\（{\ mathscr {O}}（h ^ {1+ \ gamma} + \ tau ^ {（1+ \ gamma / 2/2））\）分别用于基于Galerkin的Euler和Milstein方案。