A Posteriori Error Bounds for Reduced Basis Approximations of Nonaffine and Nonlinear Parabolic Partial Differential Equations

We present a posteriori error bounds for reduced basis approximations of parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct “affine” coefficient-function approximations of the “nonaffine” (or nonlinear) parametrized functions. 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 show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an efficient offline-online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. We also introduce a new sampling approach to generate the collateral reduced basis space for functions with a nonlinear dependence on the field variable. Numerical results are presented to confirm and test our approach.