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 language | English |
---|---|
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