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Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces

Author: Laurent Véron
Journal: Math. Comp. 39 (1982), 325-337
MSC: Primary 47H15; Secondary 34A45
MathSciNet review: 669633
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Abstract: Let $ \partial \Phi $ be the subdifferential of some lower semicontinuous convex function $ \Phi $ of a real Hilbert space H, $ f \in {L^2}(0,T;H)$ and $ {u_n}$ a continouous piecewise linear approximate solution of $ du/dt + \partial \Phi (u) \ni f$, obtained by an implicit scheme. If $ {u_0} \in \operatorname{Dom} (\Phi )$, then $ d{u_n}/dt$ converges to $ du/dt$ in $ {L^2}(0,T;H)$. Moreover, if $ {u_0} \in \overline {\operatorname{Dom} (\partial \Phi )} $, we construct a step function $ {\eta _n}(t)$ approximating t such that $ {\lim _{n \to + \infty }}\smallint _0^T{\eta _n}\vert d{u_n}/dt - du/dt{\vert^2}\;dt = 0$. When $ \Phi $ is inf-compact and when the sequence of approximation of f is weakly convergent to f, then $ {u_n}$ converges to u in $ C([0,T];H)$ and $ {\eta _n}d{u_n}/dt$ is weakly convergent to $ tdu/dt$.

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Article copyright: © Copyright 1982 American Mathematical Society

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