In this dissertation we prove the superconvergence of the minimal-dissipation local discontinuous Galerkin method for elliptic problems and construct optimal immersed finite element approximations and discontinuous immersed finite element methods for the Stokes interface problem. In the first part we present an error analysis for the minimal dissipation local discontinuous Galerkin method applied to a model elliptic problem on Cartesian meshes when polynomials of degree at most <i>k</i> and an appropriate approximation of the boundary condition are used. This special approximation allows us to achieve <i>k</i> + 1 order of convergence for both the potential and its gradient in the L² norm. Here we improve on existing estimates for the solution gradient by a factor &#8730;h. In the second part we present discontinuous immersed finite element (IFE) methods for the Stokes interface problem on Cartesian meshes that does not require the mesh to be aligned with the interface. As such, we allow unfitted meshes that are cut by the interface. Thus, elements may contain more than one fluid. On these unfitted meshes we construct an immersed Q₁/Q₀ finite element approximation that depends on the location of the interface. We discuss the basic features of the proposed Q₁/Q₀ IFE basis functions such as the unisolvent property. We present several numerical examples to demonstrate that the proposed IFE approximations applied to solve interface Stokes problems maintain the optimal approximation capability of their standard counterpart applied to solve the homogeneous Stokes problem. Similarly, we also show that discontinuous Galerkin IFE solutions of the Stokes interface problem maintain the optimal convergence rates in both L² and broken H¹ norms. Furthermore, we extend our method to solve the axisymmetric Stokes interface problem with a moving interface and test the proposed method by solving several benchmark problems from the literature. / Ph. D.

This research deals with equivalent dose rate (EDR) in the ground level air caused by gamma radiation. Weighting factor for gamma radiation is equal one, so EDR in the ground air is identical absorbed ionic radiation dose. Highways A6 Kaunas-Zarasai and A11 Panevėžys-Šiauliai-Palanga were investigated. In this research the main parameters determining the level of equivalent dose rate in the ground air by the Lithuanian highways have been defined and analyzed. The objective of this research is to measure EDR in the ground air by the Lithuanian highways and compare to adequate average regional and with reference to modulation values evaluate variation reasons. In the expeditions EDR in the ground air was measured in winter and in summer. Samples of soil and asphalt were examined with gamma spectrometer and analyze presented. There was determined the dependence of EDR in the ground air from soil composition, floral or snow cover and distance to the asphalt cover. Cosmic radiation impact on EDR formation is equal to all locations. Natural origin radionuclides in soil gamma radiation have the main influence to EDR formation in the ground air. Modeling program VISIPLAN was used to define impact of radionuclides radiation in asphalt cover to EDR formation in the roadside zone. Modeling program InterRAS was used to define EDR in the ground air formation from gamma radiation of radionuclide in soil.