Implicit boundary integral methods
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Boundary integral methods (BIMs) solve constant coefficient, linear partial differential equations (PDEs) which have been formulated as integral equations. Implicit BIMs (IBIMs) transform these boundary integrals in a level set framework, where the boundaries are described implicitly as the zero level set of a Lipschitz function. The advantage of IBIMs is that they can work on a fixed Cartesian grid without having to parametrize the boundaries. This dissertation extends the IBIM model and develops algorithms for problems in two application areas. The first part of this dissertation considers nonlinear interface dynamics driven by bulk diffusion, which involves solving Dirichlet Laplace Problems for multiply connected regions and propagating the interface according to the solutions of the PDE at each time instant. We develop an algorithm that inherits the advantages of both level set methods (LSMs) and BIMs to simulate the nonlocal front propagation problem with possible topological changes. Simulation results in both 2D and 3D are provided to demonstrate the effectiveness of the algorithm. The second part considers wave scattering problems in unbounded domains. To obtain solutions at eigenfrequencies, boundary integral formulations use a combination of double and single layer potentials to cover the null space of the single layer integral operator. However, the double layer potential leads to a hypersingular integral in Neumann problems. Traditional schemes involve an interpretation of the integral as its Hadamard's Finite Part or a complicated process of element kernel regularization. In this thesis, we introduce an extrapolatory implicit boundary integral method (EIBIM) that evaluates the natural definition of the BIM. It is able to solve the Helmholtz problems at eigenfrequencies and requires no extra complication in different dimensions. We illustrate numerical results in both 2D and 3D for various boundary shapes, which are implicitly described by level set functions.