The finite element approximation of parabolic quasi-variational inequalities related to impulse control problem (Q2892382)

The authors propose a discrete iterative algorithm to prove the existence and uniqueness and devote the asymptotic behavior using the semi-implicit time scheme combined with a finite element spatial approximation for the following parabolic quasi-variational inequalities (PQVIs):NEWLINENEWLINEFind \(w \in K(u)\) solution of NEWLINE\[NEWLINE\frac{\partial u}{\partial t} + Au \leq f(u)\tag{1}NEWLINE\]NEWLINE where \(\Omega\) is convex domain in \(\mathbb R^N\), with smooth boundary \(\Gamma\) and \(\Sigma\) set in \(\mathbb R \times \mathbb R^N\), \(\Sigma = \Omega \times [0, T]\) with \(T < +\infty, (.,.)\) denotes the inner product in \(L_2 (\Omega)\), \(A\) is an operator defined over \(H^1(\Omega)\).NEWLINENEWLINEThe implicit convex set has the form \(K(u)=\{u \in L_2 (0,T, H^1_\diamond (\Omega)), ~u \leq Mu, ~u(0,x)= u_\diamond\) in \(\Omega\}\), where \(M(u) = \Psi + S(u)\) (\(\Psi\) is a smooth function and \(S\) is a nonlinear continuous operator from \(L^\infty (\Omega)\) into itself satisfying the assumptions: \(S(u) \leq S(\widetilde{u})\) whenever \(u \leq \widetilde{u}\) a.e. in \(\Omega\), and \(S(u + \delta) \leq S(u)+ \delta\) for a positive constant \(\delta, f, \frac{\partial f}{\partial t} \in L^2 (0,T, L^\infty (\Omega)) \cap C^1(0,T,H^1 (\Omega)), f \geq 0\)). Here, the class of PQVIs includes at least two important problems: the variational inequality of obstacle type problems (when \(S\) is identically equal to zero) and the quasi-variational inequality of impulse control problems (when \(\Psi\) is identically equal to zero and \(S(u) = K + \inf (u+\xi),~ x \in \Omega,~ \xi \geq 0,~ x + \xi \in \Omega\)). The authors show that the asymptotic behavior can be properly approximated by a semi-implicit time scheme combined with a finite element spatial using the discrete iterative algorithm. With the discrete PQVIs problem a fixed point mapping is associated. Main result: A new approach for the finite element approximation of PQVIs is presented. Convergence and asymptotic behavior (using the semi-implicit time scheme combined with a finite element spatial approximation of PQVIs in the \(L^\infty\)-norm) are established. The type of estimation which is obtained here is important for the calculus of the quasi-stationary state for the simulation of petroleum or gaseous deposit.