Non-conforming Mixed Finite Element Methods for Diffusion Equation

In this dissertation, we consider new approaches to the construction of meshes, discretization, and preconditioning of the resulting algebraic systems for the diffusion equation with discontinuous coefficients. In the first part, we discuss mixed finite element approximations of the diffusion equation on general polyhedral meshes. We introduce a non-conforming approximation method for the flux vector functions, and propose a benchmark problem which allows us to analyze its accuracy in the case of 3D diffusion equation with non-homogeneous boundary conditions on domains with oblique parallel layers. In the second part, we propose a two-stage preconditioning method for the algebraic system resulting from the application of the introduced method to the diffusion equation on the prismatic meshes. We provide the description of the recommended implementation and show the results of numerical experiments used to compare its performance with some well-known preconditioners. In the third part, we consider application of non-conforming meshes on rectangular domains with oblique parallel or curved concentric layers. We describe possible choices of such meshes for each case, and introduce benchmark problems used to compare the accuracy of finite element methods on conforming and non-conforming meshes. The results of numerical experiments are provided.