The localized finite element method and its application to the two- dimensional sea-keeping problem.

*(English)*Zbl 0656.76008The ‘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 |