An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion,

Institut Neel, CNRS-UJF, Nanosciences Department, 25 Av. des Martyrs, BP 166, 38042 Grenoble cedex 9, France Institut National Polytechnique de Grenoble, 46 Av. Félix Viallet, 38031 Grenoble cedex 1, France Laboratoire SPINTEC, CEA-CNRS-UJF-INPG, Bât. 1005, 17 Rue des Martyrs, 38054 Grenoble cedex 9, France Département de Mathématiques, Faculté des Sciences d'Orsay, Université Paris-Sud 11, Bât...

In this paper we deal with a constrained variational equation associated with the usual weak formulation of an elliptic boundary value problem in the context of Banach spaces, which generalizes the classical results of existence and uniqueness. Furthermore, we give a precise estimation of the norm of the solution.

Element-wise techniques based on SVD or QR Decomposition completely get rid of the usual ill-conditioning inherent to plane-wave discretizations Ultra-Weak Variational Formulation (UWVF) for Helmholtz equation. Associated preconditioning strategies lead very low condition numbers corresponding linear system matrices, without any limitation number plane waves per element. In addition, some these...

Given a càdlàg process X on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let Psem be the set of all probability measures P under which X is a semimartingale. We construct processes (B , C, ν ) which are jointly measurable in time, space, and the probability law P , an...

In this paper we consider the problem of scattering of time-harmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions, studying dependence on the wave number in two classical formulations of this problem. The first is the standard variational/weak formulation in the part of the exterior domain contained in a large sphere, with an exact Dirichletto-Neumann map applied ...

We will present a set of equations for General Relativity adapted to weak simplicial spacetimes. The equations will be obtained via a weak formulation of the field equations similar to the methods commonly used in finite element schemes. The equations will be shown to be a weighted sum of the Regge equations.

The system of shallow water equations admits infinitely many conservation laws. This paper investigates weak local residuals as smoothness indicators of numerical solutions to the shallowwater equations. To get a weak formulation, a test function and integration are introduced into the shallow water equations. We use a finite volume method to solve the shallow water equations numerically. Based...

There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed in terms of an appropriate bilinear form to prove existence of a unique solution. The selection of the solution space and the bilinear form controls the PDE and boundary conditions that the solution will satisfy. We will give...