This dissertation aims to study finite element methods for two-dimensional stationary second order interface model problems on homogeneous and heterogeneous media, under discretizations non-fitted with the interface. We propose novel numerical methods for the model interface problems, proving optimal convergence of the errors and robust error estimates with respect to the curvature of the interface and the physical parameters of the problems. In the first part of the dissertation, we consider the Poisson and Stokes interface problems on homogeneous media. In both cases, we design correction terms relying on the interface jump conditions of the problem for each unknown variable, which are then added to the right-hand side of the system. This technique is developed for arbitrary polynomial order of the finite element subspace, by means of higher order derivative jumps of the solution derived from the problem. We provide pointwise error analyses of the methods obtaining optimal convergence of the errors. The second part assesses the classical elliptic interface model problem on heterogeneous media: Poisson equations with discontinuous constant coefficients. In particular, we examine the case in which the contrast of the diffusion coefficients is high. We present an a priori analysis of the regularity in order to develop error estimates for the numerical methods consistent with the behavior of the solution. Furthermore, we propose and analyze two finite element methods, following distinct approaches, for the high contrast Poisson interface problem. The first method follows the philosophy of Immersed Finite Element methods, incorporating the jump conditions of the problem strongly in the finite element subspace. Optimal error estimates independent of the contrast of the diffusion coefficients are proved. A detailed analysis incorporating the curvature of the interface and the regularity of the solution is presented, guaranteeing robustness of the method. The second numerical approximation is based on a Nitsche finite element method, with flux stabilization terms. This approach imposes the jumps conditions of the problem weakly, in the form of penalty terms with support on the interface that are added to the system. Optimal error estimates independent of the high-contrast are demonstrated, including a novel error estimate for the flux of the solution. Finally, numerical evidence of the properties of the finite element approximations are displayed for each problem, corroborating the theoretical findings.
Sanchez, Manuel A.,
"Finite Element Methods for Interface Problems Using Unfitted Meshes: Design and Analysis"
(2016).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0JD4V58
