Transport processes in fractured porous media
This habilitation thesis summarizes author's theoretical work related to development of the Flow123d simulator. This includes especially methods and algorithms for solving Darcy ow problems in saturated and unsaturated fractured porous media. A model with semi-discrete fractures called mixed dimension model is derived at the beginning. Then the abstract model for advection-di usion equation is applied to the Darcy ow. The mixed-hybrid formulation of the Darcy ow mixed dimension problem is presented followed by its discretization using Raviart-Thomas nite elements. An analytical solution to a test single fracture problem is supplied which allows veri cation of the model's implementation. Finally, the BDDC method is applied to obtain a scalable solver of the linear systems arising from the problem's discretization. Subsequently, new developments for the non-conforming mixed meshes are presented. Four methods with common strategy are used to introduce a coupling between equations living on the intersecting nite element meshes of di erent dimension. Further a family of e cient algorithms for computing mesh intersections is presented. Final chapter is devoted to the Richards' equation and modi cation of the mixed-hybrid scheme in order to satisfy discrete maximum principle. This is of particular importance for the Richards' equation where short time steps are often necessary which leads to strong oscillations for the schemes that violate DMP.