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Optimal upper and lower error estimates for strong full-discrete numericalapproximations of the stochastic heat equation driven by space-time white noiseare obtained. In particular, we establish the optimality of strong convergencerates for full-discrete approximations of stochastic Allen-Cahn equations withspace-time white noise which have recently been obtained in [Becker, S., Gess,B., Jentzen, A., and Kloeden, P. E., Strong convergence rates for explicitspace-time discrete numerical approximations of stochastic Allen-Cahnequations. arXiv:1711.02423 (2017)].

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