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Convergent semidiscretization of a nonlinear fourth order parabolic system. (English) Zbl 1026.35045

Fourth order nonlinear parabolic equations arise in different field of applied mathematics, for example lubrication problems for spreading droplets or in modern quantum semiconductor design. Due to their high order nature and the lack of a suitable maximum principle, they pose challenging analytical and numerical problems. In this article a classical quantum hydrodynamical model,which is of hyperbolic type is significantly improved by introducing an additional nonlinear potential. The resulting model, is a fourth order nonlinear parabolic equation for the electron density, which is self-consistently coupled to Poisson’s equation for the potential. This article extends authors previous results for this problem obtained for one space dimension to multidimensional case. An implicit time discretization by a backward Euler scheme for this system is suggested. The resulting sequence of elliptic systems proves to be uniquely solvable at each time step and moreover the semidiscrete solution is strictly positive as long as the lattice temperature is sufficiently large. Convergence of this numerical solution is presented, and under some assumption on the regularity of the solution error estimates which exhibit the optimal order of convergence for the implicit Euler scheme is proved.

MSC:

35K35 Initial-boundary value problems for higher-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
35K55 Nonlinear parabolic equations
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