Title page for ETD etd-04032017-132806

Type of Document Master's Thesis
Author's Email Address ruchi91@vt.edu
URN etd-04032017-132806
Title A Linear Immersed Finite Element Space Defined by Actual Interface Curve on Triangular Meshes
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Tao Lin Committee Chair
Christopher Beattie Committee Member
Slimane Adjerid Committee Member
  • Elliptic Interface Problems
  • Immersed Finite Element
  • Interpolation Error Analysis
  • Interface Independent Mesh
  • Linear Finite Element
Date of Defense 2017-03-21
Availability unrestricted
In this thesis, we develop the a new immersed finite element(IFE) space formed by piecewise linear polynomials defined on sub-elements cut by the actual interface curve for solving elliptic interface problems on interface independent meshes. A group of geometric identities and estimates on interface elements are derived. Based on these geometric identities and estimates, we establish a multi-point Taylor expansion of the true solutions and show the estimates for the second order terms in the expansion. Then, we construct the local IFE spaces by imposing the weak jump conditions and nodal value conditions on the piecewise polynomials. The unisolvence of the IFE shape functions is proven by the invertibility of the well-known Sherman-Morrison system. Furthermore we derive a group of fundamental identities about the IFE shape functions, which show that the two polynomial components in an IFE shape function are highly related. Finally we employ these fundamental identities and the multi-point Taylor expansion to derive the estimates for IFE interpolation errors in L2 and semi-H1 norms.
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