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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. 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 L2-, L∞-norm error estimates and an H1-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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solution of the biharmonic equation using Adini nonconforming finite element, and report new results of the asymptotic error expansions of the interpolation error functionals and nonconforming remainder. These expansions are used to develop two extrapolation formulas and a series of sharp error estimates. Finally, the numerical results have verified the extrapolation theory. [ABSTRACT FROM AUTHOR]