Inf-Sup Theory for the Biot equations: analysis and discretisation
A talk in the BI.discrete Workshop series by
Christian Kreuzer from Dortmund
| Abstract: | The quasi-static Biot equations in poroelasticity describe the flow of a Newtonian fluid inside an elastic porous medium. The main unknowns are the displacement of the elastic medium and the pressure of the fluid. In addition the two auxiliary variables of the total pressure and the fluid content play a central role in our analysis.
We shall present a new analysis establishing existence of a unique solution based on the Banach-Necas inf-sup theory. Compared to results in the Literature, this establishes an isomorphism between data and solution and thus requires only minimal regularity of the data.
Guided by the inf-sup theory, we propose then a new finite element discretisation of the equations resorting to the the backward Euler scheme in time and discretising all variables in space by conforming Lagrange finite elements on simplicial meshes.
We establish the well-posedness, the stability and the quasi-optimality of the discretisation. All constants involved in our analysis are robust with respect to the material parameters. Within the CRC this talk is associated to the project(s): A7, B3, B7 |