ParaSkel++: a C++ platform for the high-performance, arbitrary-order, 2/3D numerical approximation of PDEs on general polytopal meshes using skeletal Galerkin methods
Résumé
Skeletal Galerkin methods are a vast family of numerical methods for the approximation of PDE-based models that satisfy the following two building principles: (1) the degrees of freedom (DOF) of the method split into (i) skeleton DOF, attached to the geometric entities (vertices, edges, faces) composing the mesh skeleton and common to all cells sharing the geometric entity in question, which prescribe the conformity properties of the underlying discrete functional space, and (ii) bulk DOF (if need be), attached to the interior of the cells, which play no role in the prescription of the conformity properties of the underlying discrete functional space; (2) the global discrete bilinear form of the problem (potentially after linearization, if the problem is nonlinear) writes as the sum over the mesh cells of cell-wise (referred to as local) bilinear contributions. The very structure underpinning skeletal methods grants them the property of being amenable to static condensation, i.e. locally to each cell, bulk DOF can be eliminated in terms of the local skeleton DOF by means of a Schur complement. The final global system to solve thus writes in terms of the skeleton DOF only. The skeletal family encompasses in particular standard FE methods and virtual-like Galerkin methods (VEM, HHO, HDG...). It does not contain (plain vanilla) DG methods. The ParaSkel++ platform offers a high-performance factorized C++ architecture for the implementation of arbitrary-order skeletal methods on general 2/3D polytopal meshes.
