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On finite element approximation in the \(L^{\infty}\)-norm of variational inequalities. (English) Zbl 0574.65064
Using a discrete maximum principle and the properties of the so called ”M-functions” it is proved that the solution \(u_ h\) of the discrete variational inequality: \(a_ h(u_ h,v_ h-u_ h)\geq (f,v_ h-u_ h),\) \(u_ h\leq r_ h\psi\), \(v_ h\leq r_ h\psi\), \(\forall v_ h\in V_ h\) approximates the solution u of the continuous problem: \(a(u,v- u)\geq (f,v-u),\) \(u\leq \psi\), \(v\leq \psi\), \(\forall v\in V\) in the \(L^{\infty}\)-norm as follows: \(\| u-u_ h\|_{\infty}\leq Ch^{\nu}| \log h|^{\mu}\) where \(0\leq \mu,\nu \leq 2\). It is proved also that the hypotheses of the main theorem are satisfied by linear and quasilinear operators, by the pseudo-Laplacian and other nonlinear operators. A numerical example is provided.
Reviewer: S.Mirica

MSC:
65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
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