A posteriori error analysis for parabolic variational inequalities

Kyoung-Sook Moon; Ricardo H. Nochetto; Tobias von Petersdorff; Chen-song Zhang

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 3, page 485-511
  • ISSN: 0764-583X

Abstract

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Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain Ω ⊂ ℝ d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L2(0,T;H1(Ω)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the non-contact region, and for the time error estimator. Numerical results for d=1,2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step τ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.

How to cite

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Moon, Kyoung-Sook, et al. "A posteriori error analysis for parabolic variational inequalities." ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 485-511. <http://eudml.org/doc/250072>.

@article{Moon2007,
abstract = { Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain $\Omega\subset\mathbb\{R\}^d$ with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L2(0,T;H1(Ω)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the non-contact region, and for the time error estimator. Numerical results for d=1,2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step τ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions. },
author = {Moon, Kyoung-Sook, Nochetto, Ricardo H., von Petersdorff, Tobias, Zhang, Chen-song},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori error analysis; finite element method; variational inequality; American option pricing.; backward Euler method; numerical results; a posteriori error analysis; variational inequality; American option pricing},
language = {eng},
month = {8},
number = {3},
pages = {485-511},
publisher = {EDP Sciences},
title = {A posteriori error analysis for parabolic variational inequalities},
url = {http://eudml.org/doc/250072},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Moon, Kyoung-Sook
AU - Nochetto, Ricardo H.
AU - von Petersdorff, Tobias
AU - Zhang, Chen-song
TI - A posteriori error analysis for parabolic variational inequalities
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/8//
PB - EDP Sciences
VL - 41
IS - 3
SP - 485
EP - 511
AB - Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain $\Omega\subset\mathbb{R}^d$ with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L2(0,T;H1(Ω)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the non-contact region, and for the time error estimator. Numerical results for d=1,2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step τ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.
LA - eng
KW - A posteriori error analysis; finite element method; variational inequality; American option pricing.; backward Euler method; numerical results; a posteriori error analysis; variational inequality; American option pricing
UR - http://eudml.org/doc/250072
ER -

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