| We study and numerically solve elliptic differential equations in the presence of interfaces where the solution is not smooth. We use uniform Cartesian grids and do not require the interfaces to be aligned with the grid. We develop a one-dimensional theory for the new Explicit Jump Immersed Interface Method (EJIIM), which culminates in a proof of second-order convergence for piecewise-constant coefficients for single-point interfaces. The proof is interesting in not requiring the numerical scheme to satisfy a discrete maximum principle, the usual means by which such results are proved, and in providing error bounds that are independent of the geometry and the contrast in the coefficients. EJIIM works by focusing on the jumps in the solutions and their derivatives, rather than on finding coefficients of new finite difference schemes, like the original Immersed Interface Method (IIM). In our formulation, the jump conditions for many different problems all turn out to depend only on limits from one side of the interface. This view allows the use of fast solvers for the resulting large sparse systems, and easy incorporation of multiple interfaces. The one-dimensional corrections to finite difference schemes in the presence of discontinuities could prove useful far beyond their scope of EJIIM. We move on to prove second-order convergence for singular source problems in two dimensions, and find bounds on the coefficients of the scheme for elliptic equations with discontinuous coefficients. New jump conditions for irregular domain problems are found, and the Liouville transformation is extended to discontinuous coefficients. Numerical examples demonstrate the second-order behavior of EJIIM and improved performance in the presence of large contrasts in the coefficients on a variety of boundary value problems and interface problems including multiply connected domains, and improved smooth dependence of the solutions for smoothly varying interface locations. Finally, we present a method to recover perturbations of a circular interface from measurements taken at the boundary of a circular domain, by linearizing about the circular interface and using Fourier series. |
