Abstract
In this dissertation, we use body-fitting curvilinear coordinates to construct a finite difference scheme, called the Monotone Jump Condition Capturing Scheme (MJCCS), for the elliptic interface problems. The variable coefficients, the solution itself and its normal flux may be discontinuous across the interface. The entries of the coefficient matrix are easily generated via centered difference and average on neighboring mesh points. The resultant matrix of MJCCS is symmetric and positive definite. Most powerful linear solvers such as PCG and multigrid can be applied to invert the matrix. The scheme preserves monotonicity on mild distorted grids. It is proved to achieve second-order accuracy in the sup-norm of global errors. Extensive numerical experiments are performed to demonstrate the second-order convergence rate in the numerical solution and the reconstructed flux. Our scheme automatically captures the jump conditions in the difference formulation without extra enforcement. This is the main advantage of the new scheme. The approximation obtained by our scheme resolves the discontinuities at the interface without numerical smearing at all. Besides, one can apply the scheme to treat more general discontinuities such as multi-phase flows in a straightforward manner. We also extend the idea incorporating with the nonoverlapping domains decomposition method to tackle the domains and interfaces with complicated geometry.