A time-parallel framework for coupling finite element and lattice Boltzmann methods

In this work, we propose a new numerical procedure for the simulation of time-dependent problems based on the coupling between the finite element method (FEM) and the lattice Boltzmann method. The procedure exploits the Parareal paradigm to efficiently couple the two numerical methods, allowing independent grid size and time-step size. The motivations behind this approach are wide-ranging. In particular, one technique may be more efficient or physically more appropriate or less memory consuming than the other depending on the target of the simulation and/or on the sub-region of the computational domain. Furthermore, the coupling with FEM may circumvent some difficulties inherent to lattice Boltzmann discretization, for some domains with complex boundaries, or for some kind of boundary conditions. The theoretical and numerical framework is presented for the time-dependent heat equation in order to describe and validate numerically the methodology in a simple situation.