Type of Document Dissertation Author Camp, Brian David URN etd-12042003-220747 Title A Class of Immersed Finite Element Spaces and Their Application to Forward and Inverse Interface Problems Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Lin, Tao Committee Chair Adjerid, Slimane Committee Member Beattie, Christopher A. Committee Member Borggaard, Jeffrey T. Committee Member Rogers, Robert C. Committee Member Keywords

- mixed equation error
- immersed finite elements
- elliptic interface problem
- least squares
- inverse problems
Date of Defense 2003-11-19 Availability unrestricted AbstractA class of immersed finite element (IFE) spaces is developed for solving elliptic boundary value problems that have interfaces. IFE spaces are finite element approximation spaces which are based upon meshes that can be independent of interfaces in the domain. Three different quadratic IFE spaces and their related biquadratic IFE spaces are introduced here for the purposes of solving both forward and inverse elliptic interface problems in 1D and 2D. These different spaces are constructed by (i) using a hierarchical approach, (ii) imposing extra continuity requirements or (iii) using a local refinement technique. The interpolation properties of each space are tested against appropriate testing functions in 1D and 2D. The IFE spaces are also used to approximate the solution of a forward elliptic interface problem using the Galerkin finite element method and the mixed least squares finite element method. Finally, one appropriate space is selected to solve an inverse interface problem using either an output least squares approach or the least squares with mixed equation error method.

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