Section: Research Program

Structure-preserving discretizations and discrete element methods

Our second important activity is the design of numerical methods for the devised partial differential equation 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 non-matching interfaces, and we develop them in response to a high demand from our industrial partners. 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 developed schemes, we study the existence, uniqueness, and stability questions, and derive a priori convergence estimates. Our special interest is in higher-order methods, almost nonexisting these days even on the theoretical side. Albeit their use in practice may not be immediate, we believe that they represent the future generation of numerical methods for industrial simulations. These developments are and will be undertaken by Alexandre Ern and Laurent Monasse from ENPC who shall enlarge SERENA in the joint Inria–ENPC team shortly.