Error Analysis of the Immersed Interface Method for Elliptic Problems with an Interfac

The immersed interface method has been applied to a lot of interface problems. Error analysis of the immersed interface method for some interface problems with Dirichlet boundary conditions has shown that the immersed interface method gives second-order accurate numerical solutions. But the error analysis of the immersed interface method for problems with Neumann boundary conditions is still missing. During my PhD study, we have worked on error analysis of the immersed interface method for the Poisson interface problems with a Neumann boundary condition and the Stokes equations with an interface. For the Poisson interface equation with a constant coefficient, we use the method-based on the results from Dr. Beale et al. With the discrete Poincare-Neumann inequality, we show that the numerical solution is second-order accurate. Next we consider the Poisson interface problem with a Neumann boundary condition and a piece-wise smooth coefficient. For this problem, an additional condition is needed to guarantee the problem well-posed. We use the method from the maximum principle and related theorems. A comparison function is constructed for the error estimation. From the result of the maximum principle and related theorems, it turns out that the immersed interface method is second-order accurate for the problems in both one-dimensional and two-dimensional spaces. After that, we consider a general elliptic problem with an interface and Neumann boundary conditions. But this problem is well-posed such that no additional condition is needed. We also use the method from the maximum principle and related theorems to show that the second-order accurate solution can be given by the immersed interface method. Finally we consider the static Stokes equations with an interface. We use the three Poisson-equations approach to decouple velocity and pressure in the Stokes equations. To preserve divergence-free of velocity, a Neumann boundary condition for pressure can be derived. Then our problem becomes three Poisson equations with an interface. We apply the result for the Poisson equation to show that the numerical solution of pressure is second-order accurate. With the result of analysis for pressure, we show that the numerical solutions of velocities are also second-order accurate.