Document Type dissertation Author Name Wasyk, Rebecca Dawn Email Address rebecca.wasyk at gmail.com URN etd-120407-095718 Title Numerical Solution of a Transmission Problem with Prefractal Interface Degree PhD Department Mathematical Sciences Advisors Umberto Mosco, Advisor Robert Gilbert, Committee Member Marcus Sarkis, Committee Member Bogdan Vernescu, Department Head Homer Walker, Committee Member Keywords finite element singularities transmission prefractal fractal Date of Presentation/Defense 2007-10-26 Availability unrestricted
Certain physical problems in electrostatics, magnetostatics, and heat transfer give rise to elliptic boundary value problems with transmission conditions on a layer. We focus on a particular problem with a second order transmission condition, representing an infinitely conductive layer. To approximate irregular layers that may naturally arise, a sequence of layers that converge to the fractal von Koch curve is considered. The solution to this transmission problem with a prefractal layer exhibits singularities due to the transmission condition across the layer as well as the reentrant corners introduced in the domain by the prefractal curve. To solve this problem numerically using a finite element method, the mesh must be adjusted to account for these singularities. We exhibit a general algorithm for creating a finite element discretization of the domain that results in linear convergence of the numerical solution to the true solution in a suitable norm.
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