Approximations of parabolic variational inequalities. (English) Zbl 0574.65066

Ženíšek, Alexander

Apl. Mat. 30, 11-35 (1985).

This paper considers the problem of finding a function \(u: I\to V\) where \(I=[0,T]\), \(0<T<\infty\) and V is a Hilbert space (satisfying other conditions) such that \((\dot u,w-u)+a(u,w-u)\geq (f,w-u)\) for all w belonging to a closed, convex subset of V and for all t in I except for a set of measure zero. The initial function u(0) is given and a(u,v) is a nonlinear elliptic form having a potential J(v) which is twice G- differentiable. The basic problem is discretized in time by a one-step finite difference method and in space by a finite element method which triangularises the region. The discretization in time are such that I is divided into r equally spaced points and the solution produces \(U^ i\) at \(t=t_ i\), \(i=1,2,...,r\) and \(U^ 0\) is a finite element approximation to u(0). From these abstract functions are constructed: \(U_ r=U^{i-1}+((t-t_{i-1})/\Delta t)(U^ i-U^{i-1}),\quad t\in [t_{i-1},t_ i].\) The form of a(u,v) and further conditions lead to convergence theorems and an error bound providing u(t) is sufficiently smooth.

Reviewer: B.Burrows

MSC:

65K10	Numerical optimization and variational techniques
49J40	Variational inequalities

Keywords:

parabolic variational inequalities; one-step finite difference method; finite element method; convergence; error bound

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