Exact sequences of conforming finite element spaces with interface constraints for macro polytopal meshes. (15th March 2023) - Devloo, Philippe R.B.; Durán, Omar; Gomes, Sônia M.
Abstract: In this paper we provide guidelines for the construction of new high order conforming finite element exact sequences of subspaces in H 1 ( Ω ), H ( curl, Ω ), H ( div, Ω ), and L 2 ( Ω ) . They are meant for the design of stable and conservative mixed formulations of multiphysics systems combined with advanced numerical strategies. For instance, they support general polytopal meshes and/or allow trace interface constraints. These are appealing attributes for various challenging domains in computational practice, e.g. in hp -adaptivity, meshing of complex regions, multiscale simulations etc. Firstly, we consider standard finite element exact sequences of composite polynomial approximations inside the polytopal subdomains, based on conformal and regular sub-partitions of them. Over a partition of the mesh skeleton (faces and/or edges), we fix a sequence of trace spaces of piecewise polynomials. These trace spaces are supposed to form a hierarchy of lower dimensional exact sequences. Then, the trace-constrained subspaces only keep the non-vanishing trace components constrained by the given (coarser) trace spaces, but the internal components of the original local finite element spaces, having vanishing traces, are all kept. These internal (bubble) components may be richer to different extents when compared with the trace components with respect to internal mesh size, internal polynomial degree, or both. In principle, polynomial degree, mesh resolution and geometry are allowed to vary over different subdomains, but some interface compatibility properties are required. The operators over these trace-constrained exact finite element spaces commuting de Rham diagram are constructed in the context of projection-based interpolants. Some non-conventional examples of these sequences are described, illustrating the use of the specific finite element pairs in H ( div, Ω ) × L 2 ( Ω ) . Highlights: Construction of FE spaces on polytopal meshes with trace interface constraints. Divide-and-conquer approach combined with local bubble enrichment techniques. Focus on advanced stable conservative mixed formulations of multiphysics systems. Attributes for challenging tasks: hp-adaptivity, complex regions, multiscale. Illustrating examples are used for porous media Darcy flows.