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In this paper we develop a numerical method for pricing American options under regime-switching model, whose solutions are option buyer indifference prices. The problem is formulated as a nonlinear complementarity problem. We apply interior penalty method to approximate the differential complementarity problem, which results in a system of one degenerate parabolic equation and one ordinary differential equation, weakly coupled by nonlinear exponential term. We formulate the problems in variational form in appropriate spaces and prove comparison principle. Then, as a consequence, we derive estimates about the convergence rate of the interior penalty method. We discretize the penalized problem by fully implicit finite difference scheme and prove that the early exercise constraint is strictly satisfied. We establish comparison principle, uniqueness and boundedness of the numerical solution. To solve the nonlinear system of algebraic equations we use Newton method and partial Newton method. We present numerical results that confirm the theoretical statements. To improve the computational efficiency, we unfold the two-grid idea, combining both iteration processes – Newton method and partial Newton method. We illustrate the efficiency of this approach. [ABSTRACT FROM AUTHOR]