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Approximations of parabolic variational inequalities. (English) Zbl 0574.65066
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
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##### References:
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