| In this dissertation, our focus is on the most important and best known class of elliptic boundary value problems, namely the second order diffusion equation. We discuss the mixed method and hybridization techniques. Different mixed and mixed hybrid formulations of the problem are presented.; The finite element approximation of problems which are set in the mixed form involves vector valued function spaces which are subspaces of H(div, O). Given a simplicial triangulation of the computational domain O, the lowest order Raviart-Thomas finite element space is considered as an example of a finite element space which requires the minimal number of degrees of freedom. Each degree of freedom represents constant normal trace of the flux vector function on an interface between two neighboring simplexes.; In this dissertation, we construct the lowest order finite element spaces on general polygonal/polyhedral triangulations. In particular, the new numerical scheme is applied to logically rectangular and logically prismatic meshes. The results of numerical experiments confirm that the proposed method is no worse than the existing numerical methods based on the Piola transformation. The main advantage of the new method is that it works on meshes for which standard discretizations cannot be applied; for instance, on pyramidal cells in 3D, and nonconvex, and even degenerating, quadrilaterals in 2D.; Nonoverlapping domain decomposition consists of splitting the original computational domain into several nonintersecting subdomains. In each of the subdomains, a local triangulation is introduced, which may result in nonmatching meshes on the interfaces between subdomains. In this dissertation, two approaches to discretize the problem are discussed, and numerical results for one of them are presented. |
