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Numerical precision for differential inclusions with uniqueness. (English) Zbl 1036.34012

Here, the authors extend the existence and uniqueness results of Brezis to the following differential inclusion \[ \dot{u}(t)+B(u)+A(u) \ni f(t,u) \text{ a.e. on } ]0,T[, \quad u(0)=u_0, \] where \(A\) is maximal monotone, \(B\) is continuous and coercive, and \(f\) is Lipschitz continuous with respect to its second argument, and its derivative operator maps bounded sets to bounded sets. Using the following numerical scheme, under the Gelfand triple, \[ \frac {U^{p+1}-U^p}{h}+B(U^{p+1})+A(U^{p+1}\ni f(t_p,U^p), \] where \(t_p=ph\) and \(h\) is the time-step, they prove that the above scheme is convergent, hence, a by-product of this result is the existence of a solution to the differential inclusion, which had not been previously treated. The order of the scheme is 1/2 in general and 1 in a special case, i.e., \(A\) is the subdifferential of the indicatrix of a closed convex set.

MSC:

34A60 Ordinary differential inclusions
34G25 Evolution inclusions
47J35 Nonlinear evolution equations
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
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