The aim of this paper is to investigate the finite element methods for pricing the American put option on bonds. Based on a new variational inequality equation for the option pricing problems, both semidiscrete and fully discretized finite element approximation schemesare established. It is proved that the finite element methods are stable and convergent under L2 and HI norms.

In this paper, we investigate the finite element methods for a second type variational inequality problem. The linear finite element approximation scheme including its numerical integration modification form is proposed by a new approach. We show the unique existence and stability of finite element solutions. In particular, some abstract error estimates in H1 and L2 norms are established which imply the optimal convergence rates in order and regularity. Finally, we also give an error estimate in L1 norm.

In this paper, we present a general error analysis framework for the finite volume element (FVE) approximation to the Ritz-Volterra projection, the Sobolev equations and parabolic integro-differential equations. The main idea in our paper is to consider the FVE methods as perturbations of standard finite element methods which enables us to derive the optimal L2 and H1 norm error estimates, and the L1 and W1 1 norm error estimates by means of the time dependent Green functions. Our discussions also include elliptic and parabolic problems as the special cases.

A highly accurate derivative recovery formula is presented for the k-order finite element approximations to the two-point boundary value problems. This formula possesses the O (hk+1) order of superconvergence on the whole domain in L1 norm and O (h2k) order of ultraconvergence at the mesh points, and also the lowest regularity requirement for the exact solutions. Numerical experiments are given to verify the high accuracy of our formula.

The object of this paper is to investigate the uperconvergence properties of finite element approximations to parabolic and hyperbolic integral-differential equations. The quasi projection technique introduced earlier by Douglas, etc. is developed to derive the O (h2r) order knot superconvergence in a single pace variable, and to show the optimal order negative norm estimates in case of several space variables.