Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

Strong convergence rates of an explicit scheme for stochastic Cahn-Hilliard equation with additive noise

“…Comparing (1.4) with the sharp temporal Hölder continuity result in Theorem 2.6, one can easily observe, for γ ∈ ( d 2 , 2], the rate of convergence is in accordance with the temporal Hölder continuity of the mild solution. For γ ∈ [2,4], the rate of the convergence can reach 1 and higher than the Hölder continuity of the mild solution. It must be emphasized that the derivation of (1.4) is not an easy task and requires a variety of delicate error estimates, which are elaborated in subsection 3.3.…”

“…As indicated in [2], the fully discrete exponential Euler and fully discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. Later, some explicit modified Euler-type schemes have been proposed in [3,4,5,9,19,28] to numerically solve the stochastic Allen-Cahn equations. Based on a spectral Galerkin spatial approximation of (1.1), given by dX N (t) + A(AX N (t) + P N F (X N (t)))dt = P N dW (t), t ∈ (0, T ]; X N (0) = P N X 0 ,…”

Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

“…This shows that for all N ∈ N we have that N 3 realizations of standard normal random variables are used to calculate one realization of X N 2 ,N T . Combining (6) with the computational effort N 3 illustrates that for all ε ∈ (0, 1 /6) we have that the approximation scheme in (2)-( 3) converges with the overall rate 1 /6 − ε with respect to the number of used independent standard normal random variables.…”

“…Next we would like to point out that the numerical approximation scheme (3) has been proposed in Hutzenthaler et al [21] and has there been referred to as a nonlinearity-truncated approximation scheme (cf. [21, (3) in Section 1] and, e.g., [17][18][19][20]24,25,35,36,38,40] for further research articles on explicit approximation schemes for stochastic differential equations with superlinearly growing nonlinearities).…”

“…We also would like to point out that the strong convergence rates established in Theorem 1.1 can, in general, not essentially be improved. More formally, [3,Corollary 2.7] proves in the case where 3 i=0 |a i | = 0 and ξ = 0 in the framework of Theorem 1.1 that there exist real numbers c, C ∈ (0, ∞) such that for all M, N ∈ N we have that…”