Abstract
This paper presents selected topics from the author's study on the linear water wave problem formulated by the theory of gravitational wave in the book of Stoker (1957). The outline of the theoretical results of the study is the following. The finite-element method applied to the linear initial-boundary value problem for the potential function of the velocity of water in a bounded region is analyzed. After the problem is formulated operator-theoretically, the approximation problems are investigated. Convergence properties and error estimates together with a stability criterion for the central difference full-discrete scheme are obtained. The analysis of stationary problem in the study is closely connected with the classical paper of Bramble and Osborn (1972). The paper emphasizes the points typically different from the finite-element treatment of the usual linear elliptic problem and corresponding linear evolution problem. The convergence property and the error estimate for the semi-discrete approximation problem, and the stability criterion of the full-discrete evolution problem are mainly discussed in the present paper.
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