Reduced Basis Method for Finite Volume Approximation of Evolution Equations on Parametrized Geometries

In this paper we discuss parametrized partial differential equations (PDEs) for parameters that describe the geometry of the underlying problem. One can think of applications in control theory and optimization which depend on time-consuming parameter-studies of such problems. Therefore, we want to reduce the order of complexity of the numerical simulations for such PDEs. Reduced Basis (RB) methods are a means to achieve this goal. These methods have gained popularity over the last few years for model reduction of finite element approximations of elliptic and instationary parabolic equations. We present a RB method for parabolic problems with general geometry parameterization and finite volume (FV) approximations. After a mapping on a reference domain, the parabolic equation leads to a convection-diffusion-reaction equation with anisotropic diffusion tensor. Suitable FV schemes with gradient reconstruction allow to discretize such problems. A model reduction of the resulting numerical scheme can be obtained by an RB technique. We present experimental results, that demonstrate the applicability of the RB method, in particular the computational acceleration.