The damped Crank–Nicolson time-marching scheme for the adaptive solution of the Black–Scholes equation

Christian Goll, Rolf Rannacher and Winnifried Wollner

Abstract

ABSTRACT

This paper is concerned with the derivation of a residual-based a posteriori error estimator and mesh-adaptation strategies for the space-time finite element approximation of parabolic problems with irregular data. Typical applications arise in the field of mathematical finance, where the Black-Scholes equation is used for modeling the pricing of European options. A conforming finite element discretization in space is combined with second-order time discretization by a damped Crank-Nicolson scheme for coping with data irregularities in the model. The a posteriori error analysis is developed within the general framework of the dual weighted residual method for sensitivity-based, goal-oriented error estimation and mesh optimization. In particular, the correct form of the dual problem with damping is considered.