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We study a two-scale coupled system consisting of a macroscopic elliptic equation and a microscopic parabolic equation. This system models the interplay between a gas and liquid close to equilibrium within a porous medium with distributed microstructures. We use formal homogenization arguments to derive the target system. We start by proving well-posedness and inverse estimates for the two-scale system. We follow up by proposing a Galerkin scheme which is continuous in time and discrete in space, for which we obtain well-posedness, a priori error estimates and convergence rates. Finally, we propose a numerical error reduction strategy by refining the grid based on residual error estimators.

5 TITLE: A priori analysis as a part of teacher's lesson planning AUTHOR: Mgr. Hana Nováková DEPARTMENT: Department of Mathematics and Mathematical Education SUPERVISOR: Prof. RNDr. Jarmila Novotná, CSc. This thesis focuses on a priori analysis as a part of teacher's lesson planning. The theoretical background consists of the Theory of didactical situations in Mathematics (TDSM). In TDSM, the a priori analysis is seen as one of the teacher's tools that he/she has when planning a lesson. The goal of the thesis is to analyse differences between a priori analysis as described in TDSM and the reality in teacher's practice, to compare lesson plans of experienced teachers with those of students and demonstrate the significance and application of a priori analysis in teacher's and researcher's practice. The thesis consists of three parts, theoretical, experimental and applicational. In the theoretical part, the main concepts of TDSM linked with a priori analysis are explained and the issue of teacher's lesson planning is presented. The experimental part starts with a pre-experiment. Its results contributed to precise the structure of a priori analysis for further use. During the main experiment, the lesson plans of experienced teachers were compared with those of pre-service teachers. The differences and the...

In the first part of this thesis we are concerned with the constitutive the- ory for incompressible fluids characterized by a continuous monotone rela- tion between the velocity gradient and the Cauchy stress. We, in particular, investigate a class of activated fluids that behave as the Euler fluid prior activation, and as the Navier-Stokes or power-law fluid once the activation takes place. We develop a large-data existence analysis for both steady and unsteady three-dimensional flows of such fluids subject either to the no-slip boundary condition or to a range of slip-type boundary conditions, including free-slip, Navier's slip, and stick-slip. In the second part we show that the W−1,q norm is localizable provided that the functional in question vanishes on locally supported functions which constitute a partition of unity. This represents a key tool for establishing local a posteriori efficiency for partial differential equations in divergence form with residuals in W−1,q . In the third part we provide a novel analysis for the pressure convection- diffusion (PCD) preconditioner. We first develop a theory for the precon- ditioner considered as an operator in infinite-dimensional spaces. We then provide a methodology for constructing discrete PCD operators for a broad class of pressure discretizations. The...