The Immersed Interface Method in Three Space Dimensions and Applied to Elliptic Inverse Problems

The immersed interface method (IIM) by Li and LeVeque has been used successfully for a wide variety of problems. We extended it to nonlinear one-dimensional elliptic and parabolic problems and applied it to traac ow. When applying the IIM to inverse problems in electrical impedance tomography, we found unstabilities that prohibited its use. Recent improvements by Li have overcome the problems (to some extent) for the special case of piecewise constant conductivities. We have solved some particular inverse problems successfully. We propose a generalization of Li's improvement for general conductivities and an extension of the IIM to three space dimensions. Applications of the IIM to inverse problems in medical imaging will be investigated.