Numerical homogenization of a nonlinearly coupled elliptic-parabolic system, reduced basis method, and application to nuclear waste storage (Q2857743)

In the present paper, a simple model of radionuclide transport in porous media is considered. This yields a convection-diffusion equation, where the coefficients are determined through the Darcy law by solving a nonhomogeneous elliptic equation. The resulting nonlinearly coupled elliptic-parabolic system has coefficients depending on a small scale, which do not satisfy a uniform \(L^\infty\)-estimate. The authors show the existence and uniqueness of the weak solution.NEWLINENEWLINEFurthermore, the goal is to find a coarse-grained effective model and the corresponding efficient numerical method. The main computational effort is devoted to the evaluation of the coefficients of the effective equations since the homogenized diffusion matrix depend on the space variable. In order to reduce the computational costs of the so-called cell problems, the reduced basis method is applied to a family of elliptic equations. A specific treatment is required because of the non-affine dependence of the elliptic operator on the parameters. The numerical method relies on a suitable parametrization, which allows the use of the fast Fourier transformation to construct efficiently stiffness matrices. Numerical simulations demonstrate the efficiency and convergence of the proposed scheme.