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Continuous finite element methods which preserve energy properties for nonlinear problems. (English) Zbl 0716.65084

This paper considers the general problem of developing numerical methods for ordinary and partial differential equations which preserve the time dependence of certain functionals (for example energy conservation).
For example with \(\dot u+F'(u)=0,\quad u(0)\quad given,\) then \((*)\quad d/dt[\dot u^ 2+F(u)]=0\) and a method of solution can be obtained by a discretization which gives a discrete version of (*).
More sophisticated examples considered include the Cahn-Hilliard equations, Klein-Gordon equations, Korteweg-de Vries equations and Schrödinger equations. Comparisons are made with existing schemes.
Reviewer: B.Burrows

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
34A34 Nonlinear ordinary differential equations and systems
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
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