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. / Master of Science / Interface problems occur in many mathematical models in science and engineering that are posed on domains consisting of multiple materials. In general, materials in a modeling domain have different physical or chemical properties; thus, the transmission behaviors across the interface between different materials must be considered. Partial differential equations (PDEs) are often employed in these models and their coefficients are usually discontinuous across the material interface. This leads to the so called interface problems for the involved PDEs whose solutions are usually not smooth across the interface, and this non-smoothness is an obstacle for mathematical analysis and numerical computation. In this thesis, we present a new immersed finite element (IFE) space for efficiently solving a class of interface problems on interface independent meshes. The new IFE space is formed by piecewise linear polynomials defined on sub-elements cut by the actual interface. We present the construction procedure for this IFE space and establish fundamental properties for its shape functions. Furthermore, we prove that the proposed IFE space has the optimal approximation capability.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/79946 |
| Date | 17 April 2017 |
| Creators | Guo, Ruchi |
| Contributors | Mathematics, Lin, Tao, Beattie, Christopher A., Adjerid, Slimane |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Thesis, Text |
| Format | application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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