In this dissertation, we explore applications of partial differential equations with discontinuous coefficients. We consider the nonconforming immersed finite element methods (IFE) for modeling and simulating these partial differential equations. A one-dimensional second-order parabolic initial-boundary value problem with discontinuous coefficients is studied. We propose an extension of the immersed finite element method to a high-order immersed finite element method for solving one-dimensional parabolic interface problems. In addition, we introduce a nonconforming immersed finite element method to solve the two-dimensional parabolic problem with a moving interface. In the nonconforming IFE framework, the degrees of freedom are determined by the average integral value over the element edges. The continuity of the nonconforming IFE framework is in the weak sense in comparison the continuity of the conforming IFE framework. Numerical experiments are provided to demonstrate the features and the robustness of these methods. We introduce a class of lowest-order nonconforming immersed finite element methods for solving two-dimensional Stokes interface problem. On triangular meshes, the Crouzeix-Raviart element is used for velocity approximation, and piecewise constant for pressure. On rectangular meshes, the Rannacher-Turek rotated $Q_1$-$Q_0$ finite element is used. We also consider a new mixed immersed finite element method for the Stokes interface problem on an unfitted mesh. The proposed IFE space uses conforming linear elements for one velocity component and nonconforming linear elements for the other component. The new vector-valued IFE functions are constructed to approximate the interface jump conditions. Basic properties including the unisolvency and the partition of unity of these new IFE methods are discussed. Numerical approximations are observed to converge optimally. Lastly, we apply each class of the new immersed finite element methods to solve the unsteady Stokes interface problem. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. A comparison of the degrees of freedom and number of elements are presented for each method. Numerical experiments are provided to demonstrate the features of these methods.
| Identifer | oai:union.ndltd.org:MSSTATE/oai:scholarsjunction.msstate.edu:td-6113 |
| Date | 30 April 2021 |
| Creators | Jones, Derrick T. |
| Publisher | Scholars Junction |
| Source Sets | Mississippi State University |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses and Dissertations |
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