Immersed interface method for three dimensional interface problems and applications

This thesis describes a maximum principle preserving scheme and a fast algorithm for three-dimensional elliptic interface problems, in which the partial differential equations have discontinuities and singularities in the coefficients and the solutions. Such problems arise in many physical applications.; The immersed interface method (IIM) was developed in [46] and is designed for elliptic equations having discontinuous coefficients and singular source terms. This method is second order accurate and has been applied to many problems in one or two dimensions. In this thesis, we first pursue the extension of the IIM method to three dimensions. Then based on the IIM method, we present a maximum principle preserving scheme for arbitrary coefficients in three dimensions using direct finite difference discretization. The new scheme satisfies the sign property that guarantees the discrete maximum principle. The sign property is enforced through a constrained quadratic optimization problem. The Successive Overrelaxation method (SOR) or the Algebraic Multigrid method (AMG) can then be used to solve the resulting system of linear equations. Numerical experiments confirm the expected second order accuracy.; We also present a second order fast algorithm for three-dimensional elliptic equations with piecewise constant coefficients. Before applying the IIM method, we precondition the differential equation. In order to take advantage of existing fast Poisson solvers, an intermediate unknown function, the jump in the normal derivative of the solution across the interface, is introduced. Then the Generalized Minimal Residual method (GMRES) is employed to solve the Schur complement system derived from the discretization. Numerical experiments show that the fast algorithm is very efficient. Especially, the number of iterations in solving the Schur complement system is independent of the mesh size.; We then investigate some applications of the fast algorithm. We develop an embedding technique to solve interior or exterior Poisson equations with Dirichlet or Neumann boundary conditions. Then we investigate how to use the fast algorithm to solve an inverse interface problem.