Abstract
In this article we analyze an immersed interface finite volume method for second order elliptic and   parabolic interface problems. We show the optimal convergence of the elliptic interface problem in L^2 and energy norms.
 For the parabolic interface problem, we prove the optimal order in L^2 and energy norms for piecewise constant and variable diffusion coefficients respectively. Furthermore, for the elliptic interface problem, we demonstrate super convergence at element nodes when the diffusion coefficient  is a piecewise constant.  Numerical examples  are also provided to confirm the optimal error estimates.
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