Certified Reduced Basis Methods for Nonaffine Linear Time–Varying Partial Differential Equations

We present reduced basis approximations and associated a posteriori error bounds for nonaffine linear time-varying parabolic partial differential equations. We employ the Empirical Interpolation Method in order to construct “affine” coefficient-function approximations of the “nonaffine” parametrized functions. To this end, we extend previous work on time-invariant functions to time-varying functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We present an efficient offline-online computational procedure for the calculation of the reduced basis approximation and associated error bound. Numerical results are presented to confirm and test our approach.