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

Zhihui Liu, Zhonghua Qiao

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)

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(h1+γ+ τ1 / 2) and O(h1+γ+ τ(1+γ)/2) for the Galerkin-based Euler and Milstein schemes, respectively.

Original languageEnglish
Pages (from-to)1-46
Number of pages46
JournalStochastics and Partial Differential Equations: Analysis and Computations
DOIs
Publication statusAccepted/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

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