| Abstract: |
| We establish a general theory of 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 assumption, 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 $\gamma\in [0,1]$. In the second step, we introduce an auxiliary process and show that both this process and the discrete solutions are uniform unconditionally stable. Finally, we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $\OOO(h^{1+\gamma}+\tau^{1/2})$ and $\OOO(h^{1+\gamma}+\tau^{(1+\gamma)/2})$ for the Galerkin-based Euler and Milstein schemes, respectively. |
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