Section: Research Program

Structure-preserving discretizations and discrete element methods

We consequently design numerical methods for the devised model. Traditionally, we have worked in the context of finite element, finite volume, mixed finite element, and discontinuous Galerkin methods. Novel classes of schemes enable the use of general polygonal and polyhedral meshes with nonmatching interfaces, and we develop them in response to a high demand from our industrial partners (namely EDF and IFP Energies Nouvelles). Our requirement is to derive structure-preserving methods, i.e., methods that mimic at the discrete level fundamental properties of the underlying PDEs, such as conservation principles and preservation of invariants. Here, the theoretical questions are closely linked to differential geometry for the lowest-order schemes. For the schemes we develop, we study existence, uniqueness, and stability questions, and derive a priori convergence estimates. Our special interest is in higher-order methods like the hybrid high-order method, which have recently begun to receive significant attention. Even though their use in practice may not be immediate, we believe that they represent the future generation of numerical methods for industrial simulations.