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American options; variational inequality; finite element methods; optimal and superconvergent estimates; interpolation postprocessing; a posteriori error estimators
In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. {\it 39} (2001), 834--857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique, we present sharp $L^2$-, $L^{\infty }$-norm error estimates and an $H^1$-norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be used to generate an efficient a posteriori error estimator.
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