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- ItemTopics in Mathematical Fluid Mechanics and Shape Optimization(Technical university of Liberec, 2019-07-08) Stebel, JanThis habilitation thesis is based on the author's contributions to the mathematical theory of incompressible uids and to the shape optimization in uid mechanics. In Chapter 2, an overview of basic equations, theory of weak solutions and nite-element approximation for incompressible uids is given. Chapter 3 is devoted to the formulation of shape optimization problems, to the questions of existence of solutions, approximation and di erentiability. Finally, in Chapter 4 some results obtained by the author are mentioned, namely on the theory of non-Newtonian piezoviscous uids, applied shape optimization and sensitivity analysis.
- ItemTransport processes in fractured porous media(2019-05-13) Březina, JanThis 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.