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Error estimates of \(H^1\)-Galerkin mixed finite element method for Schrödinger equation. (English) Zbl 1199.65314

Summary: An \(H^1\)-Galerkin mixed finite element method is discussed for a class of second order Schrödinger equations. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. It is shown that the \(H^1\)-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the Ladyshenskaya-Babuška-Brezzi consistency condition.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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