Title page for ETD etd-08252011-160639

Type of Document Dissertation
Author Ben Romdhane, Mohamed
Author's Email Address mbenromd@vt.edu
URN etd-08252011-160639
Title Higher-Degree Immersed Finite Elements for Second-Order Elliptic Interface Problems
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Adjerid, Slimane Committee Chair
Lin, Tao Committee Co-Chair
Hagedorn, George A. Committee Member
Renardy, Yuriko Y. Committee Member
  • Interior Penalty Method
  • Galerkin Method
  • Immersed Finite Elements
  • Interface Problems
Date of Defense 2011-08-01
Availability restricted

A wide range of applications involve interface problems. In most of the cases, mathematical modeling of these interface problems leads to partial differential equations with non-smooth or discontinuous inputs and solutions, especially across material interfaces. Different numerical methods have been developed to solve these kinds of problems and handle the non-smooth behavior of the input data and/or the solution across the interface. The main focus of our work is the immersed finite element method to obtain optimal numerical solutions for interface problems.

In this thesis, we present piecewise quadratic immersed finite element (IFE) spaces that are used with an immersed finite element (IFE) method with interior penalty (IP) for solving two-dimensional second-order elliptic interface problems without requiring the mesh to be aligned with the material interfaces. An analysis of the constructed IFE spaces and their dimensions is presented. Shape functions of Lagrange and hierarchical types are constructed for these spaces, and a proof for the existence is established. The interpolation errors in the proposed piecewise quadratic spaces yield optimal O(h3) and O(h2) convergence rates, respectively, in the L2 and broken H1 norms under mesh refinement. Furthermore, numerical results are presented to validate our theory and show the optimality of our quadratic IFE method.

Our approach in this thesis is, first, to establish a theory for the simplified case of a linear interface. After that, we extend the framework to quadratic interfaces. We, then, describe a general procedure for handling arbitrary interfaces occurring in real physical practical applications and present computational examples showing the optimality of the proposed method. Furthermore, we investigate a general procedure for extending our quadratic IFE spaces to p-th degree and construct hierarchical shape functions for p=3.

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