Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus. (English) Zbl 1462.60089
Summary: In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on \(\mathbb{T}^2\). First we prove that the convergence rate for stochastic 2D heat equation is of order \(\alpha-\delta\) in Besov space \(\mathcal{C}^{-\alpha}\) for \(\alpha\in (0,1)\) and \(\delta>0\) arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order \(\alpha-\delta\) in \(\mathcal{C}^{-\alpha}\) for \(\alpha\in (0,2/9)\) and \(\delta>0\) arbitrarily small.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 82C28 | Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics |
| 60H40 | White noise theory |
Keywords:
stochastic Allen-Cahn equations; convergence rate; Galerkin projection; Besov space; white noiseReferences:
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