Extension Operators and Finite Elements for Fractal Boundary Value Problems

The dissertation is organized into two main parts. The first part considers fractal extension operators. Although extension operators are available for general subsets of Euclidean domains or metric spaces, our extension operator is unique in that it utilizes both the iterative nature of the fractal and finite element approximations to construct the operator. The resulting operator is especially well suited for future numerical work on domains with prefractal boundaries. In the dissertation we prove the existence of a linear extension operator, Π from the space of Hölder continuous functions on a fractal set S to the space of Hölder continuous functions on a larger domain Ω. Moreover this same extension operator maps functions of finite energy on the fractal to H1 functions on the larger domain Ω. In the second part, we consider boundary value problems in domains with fractal boundaries. First we consider the Sierpinski prefractal and how we might apply the technique of singular homogenization to thin layers constructed on the prefractal. We will also discuss numerical approximation in domains with fractal boundaries and introduce a finite element mesh developed for studying problems in domains with prefractal Koch boundaries. This mesh exploits the self-similarity of the Koch curve for arbitrary rational values of α and its construction is crucial for future numerical study of problems in domains with prefractal Koch curve boundaries. We also show a technique for mesh refinement so that singularities in the domain can be handled and present sample numerical results for the transmission problem