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In this paper, we propose a certified reduced basis (RB) method forquasilinear parabolic problems. The method is based on a space-time variationalformulation. We provide a residual-based a-posteriori error bound on aspace-time level and the corresponding efficiently computable estimator for thecertification of the method. We use the Empirical Interpolation method (EIM) toguarantee the efficient offline-online computational procedure. The error ofthe EIM method is then rigorously incorporated into the certificationprocedure. The Petrov-Galerkin finite element discretization allows to benefitfrom the Crank-Nicolson interpretation of the discrete problem and to use aPOD-Greedy approach to construct the reduced-basis spaces of small dimensions.It computes the reduced basis solution in a time-marching framework while theRB approximation error in a space-time norm is controlled by the estimator.Therefore the proposed method incorporates a POD-Greedy approximation into aspace-time certification.

A space-time certified reduced basis method for quasilinear parabolic partial differential equations