\(L^\infty\)-error estimate for a system of elliptic quasivariational inequalities (Q1811929)

This paper is devoted to the \(L^\infty\)-convergence of the standard finite element approximation for the following system of quasi-variational inequalities, that is: find \(U=(u_2,\dots,u_m)\in (H_0^1(\Omega))^J\) satisfying \[ a^i(u^i,v-u^i)\geq (f^i,v-u^i) \quad\forall v\in H_0^1(\Omega), \qquad u^i\geq 0, \quad u^i\leq Mu^i, \quad v\leq Mu^i, \] where \(\Omega\) is a bounded smooth domain of \(\mathbb R^N\), \(N\geq 1\), with boundary \(\partial\Omega\), \(a^i(u,v)\) are \(J\)-elliptic bilinear forms continuous on \(H^1(\Omega)\times H^1(\Omega)\), \((\cdot,\cdot)\) is the inner product in \(L^1(\Omega)\), and \(f^i\) are \(J\)-regular functions. In the case studied here, \(Mu^i\) represents a ``cost function'' and prototype encountered is \[ Mu^i(x)= k+\inf_{\mu\neq i} u^\mu(x), \] where \(k\) represents the switching cost. Note also that the operator \(M\) provides the coupling between the unknowns \(u^1,\dots,u^J\). Under \(W^{2,p}\) regularity of the continuous solution, a quasi-optimal \(L^\infty\)-convergence of piecewise linear finite element is established.