## 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 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.

Original language | English |
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Pages (from-to) | 1-46 |

Number of pages | 46 |

Journal | Stochastics and Partial Differential Equations: Analysis and Computations |

DOIs | |

Publication status | Accepted/In press - 3 Aug 2020 |

## Keywords

- Euler scheme
- Galerkin finite element method
- Milstein scheme
- Monotone stochastic partial differential equation
- Stochastic Allen–Cahn equation

## ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics