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The localized finite element method and its application to the two- dimensional sea-keeping problem. (English) Zbl 0656.76008
The ‘sea-keeping problem’ (used here as a model problem) is the problem of determining the motion of a rigid body which undergoes the action of a given incidental swell of small amplitude in a fluid of uniform depth H. Thus the (two dimensional) calculation domain $$\Omega$$ is a horizontally unbounded fluid strip bounded by the bottom B from below and by the free surface FS and the contour $$\Gamma$$ of the body from above, and an oscillatory motion can be described by its velocity potential $$\phi (x,y,t)=Re(u(x,y)\exp (-i\omega t))$$, where u is a solution of the following problem (P): Find u in $$H^ 1_{loc}(\Omega)$$ such that $$\Delta u=0$$ in $$\Omega$$ ; $$\partial u/\partial n=\nu u$$ on FS, where $$\nu =\omega^ 2/g$$; $$\partial u/\partial n=f$$ on $$\Gamma$$, where f is given in $$H^{-1/2}(\Gamma)$$; $$\partial u/\partial n=0$$ on B; $$\lim_{| x| \to \infty}\int^{0}_{-H}| \partial u/\partial n-i\nu_ 0u|^ 2dy=0$$ with $$\nu_ 0H=\nu /\nu_ 0$$, $$\nu_ 0>0$$. Problem (P) is treated by a localized finite element method, i.e. a finite element appoximation in the finite subdomain $$\Omega \cap \{| x| <r\}$$ and analytical representations of the solution in the infinite subdomains $$\Omega \cap \{| x| >r\}$$ are matched along the lines $$x=\pm r$$. The convergence properties of the method are extensively studied.
Reviewer: W.Müller

##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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