Convergence Rate for Galerkin Approximation of the Stochastic Allen—Cahn Equations on 2D Torus

Abstract

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on T 2 . First we prove that the convergence rate for stochastic 2D heat equation is of order α − δ in Besov space C −α for α ∈ (0, 1) and δ > 0 arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order α − δ in C −α for α ∈ (0, 2/9) and δ > 0 arbitrarily small.

Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations

“…Considering the solutions as elements of distributional spaces is necessary in higher dimensional versions of the stochastic Allen-Cahn equations, also known as the dynamical Φ 4 d models, which for this reason have to be renormalised. The dependence of the rate of convergence of approximations on the choice of the Besov exponent is observed in [MZ20], where the authors consider the spatial semidiscretisation of Φ 4 2 and bound the B −θ ∞,∞ norm of the error by n −θ+ε for any ε > 0, under the constraint θ ∈ (0, 2/9). In [HM18,ZZ18] the convergence of spatial semidiscretisations of Φ 4…”