Mixed methods on quadrilateral grids are important and applicable for many problems because they maintain the structure of a grid of rectangles while obtaining some of the flexibility of grids of triangles. Although there are many results on this problem, almost of them are obtained by assuming that the grid of quadrilaterals is almost parallelogram. This assumption demands that the shape of quadrilaterals becomes a parallelogram as the mesh size goes to zero. Only a few results have been given without this assumption. In this thesis, we introduce a mixed finite element of the lowest order for general quadrilateral grids, which gives an optimal order of $L^2$ convergence in both the velocity and the divergence. This element is designed so that the H(div)-projection $π_h$ satisfies div $π_h$ = $P_h$ div. A rigorous error estimate is carried out by proving a modified version of the Bramble-Hilbert lemma for a certain subspace of polynomials. Application to Brezzi-Douglas-Marini spaces is also discussed.
In Chapter 4, we suggest a domain decomposition method for the finite element approximations of elliptic problems with anisotropic coefficients in domains consisting of anisotropic shape rectangles. For this kind of problems, the theorems on the traces of functions from Sobolev spaces play an essential role in studying boundary value problems of partial differential equations. These theorems are commonly used for a priori estimates of the stability with respect to boundary conditions, and also play a crucial role in constructing and investigating effective domain decomposition methods. The trace theorem for anisotropic rectangles with anisotropic grids is the main tool to construct domain decomposition preconditioners.