Marien-Lorenzo Hanot: Thursday, 23rd March at 11:00
We are interested in the discretization of advanced differential complexes. That is to say, complexes presenting higher regularity or additional algebraic constraints compared to the De Rham complex.
This type of complex appears naturally in the discretization of many systems of differential equations. For example, the Stokes complex uses the same operators as the De Rham complex. Still, it requires an increased regularity, or the Div-Div complex appears in biharmonic equations and requires the use of fields with values in symmetric or traceless matrices.
The principle of polytopal methods is to use discrete functions not belonging to a subset of the continuous functions but are composed of a collection of polynomials defined on objects of any dimension of the mesh (on edges, faces, cells…).
This allows using very generic meshes, in our case composed of arbitrary contractible polytopes, while keeping the computability of discrete functions.
The objective is to present the construction of a family of discrete 3-dimensional Div-Div complexes for arbitrary polynomial degrees. These complexes are consistent on polynomial functions, which is the basis for obtaining an optimal convergence of the schemes built on them. Moreover, they preserve the algebraic structure of the continuous complex, in the sense that the cohomology of the discrete complex is isomorphic to that of the continuous.