A posteriorierror analysis for parabolic variational inequalities

Abstract

Abstract.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 L 2 (0, T ; H 1 (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.Mathematics Subject Classification. 58E35, 65N15, 65N30.

A posterioriestimates for the Cahn–Hilliard equation with obstacle free energy

“…A short review on a posteriori estimates for elliptic obstacle problems is given in [10]. A posteriori estimates for parabolic variational inequalities were derived in [24] by extending the ideas of [27]. We also refer to work in optimal control theory, where very recently an error estimator for a control problem with side constraints involving PDEs and inequality constraints has been introduced in [20,21].…”

An efficient and reliable residual-type a posteriori error estimator for the Signorini problem

“…For obstacle problems different choices ofλ m have been proposed in [12,[24][25][26]31]. We refer to [35] for a discussion about residual-type a posteriori error estimators for obstacle and contact problems.…”

“…We refer to [35] for a discussion about residual-type a posteriori error estimators for obstacle and contact problems. In this section we define aλ m which depends on the discrete solution and data and reflects the properties of λ as, e.g., in [12,24]. We call it quasi-discrete contact force density.…”

“…In order to derive and analyze the error estimator, we adopt the framework presented in [31] and used in, e.g., [12,[24][25][26] for obstacle problems. A key ingredient of this approach is the so-called Galerkin functional.…”

“…In [12,24,26] the approximate Lagrange multiplier is constructed from the discrete solution and given data with help of the partition of unity given by the canonical basis function. Notice that, generically, the support of the Lagrange multiplier is d-dimensional for obstacle problems, while it is (d − 1)-dimensional for Signorini problems.…”