In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.

在本文中，我们针对具有强单调空间微分算子的拟线性抛物线问题提出了一种经过认证的缩减基数（RB）方法。我们为时空公式提供了基于残差的后验误差估计，并为该方法的认证提供了相应的有效可计算边界。我们介绍了连续时空问题的Petrov-Galerkin有限元离散化方法，并将其用作后验误差控制中的参考。Petrov-Galerkin离散化通过Crank-Nicolson时间行进问题进一步近似。它允许使用POD-Greedy方法构造小尺寸的降基空间，并应用经验插值方法（EIM）来确保有效的离线在线计算程序。在我们的方法中，我们在时间行进框架中计算简化的基础解，而时空范数中的RB近似误差由我们的可计算边界控制。因此，我们将POD-Greedy近似与时空Galerkin方法结合在一起。