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Homogeneous Dirichlet Boundary Condition (homogeneous + dirichlet_boundary_condition)

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Mathematics and Statistics	100%

Selected Abstracts

On an initial-boundary value problem for a wide-angle parabolic equation in a waveguide with a variable bottom

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2009
V. A. Dougalis
Abstract We consider the third-order Claerbout-type wide-angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range-dependent topography. There is strong indication that the initial-boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well-posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second-order boundary condition, derived by assuming that the standard (narrow-angle) PE holds on B, and establish a priori H2 estimates for the solution of the resulting initial-boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second-order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge-type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson,Kreiss boundary condition in the wide-angle case. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Time asymptotics for the polyharmonic wave equation in waveguides

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2003
P. H. Lesky
Abstract Let , denote an unbounded domain in ,n having the form ,=,l×D with bounded cross-section D,,n,l, and let m,, be fixed. This article considers solutions u to the scalar wave equation ,u(t,x) +(,,)mu(t,x) = f(x)e,i,t satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t,, is investigated. Depending on the choice of f ,, and ,, two cases occur: Either u shows resonance, which means that ,u(t,x),,, as t,, for almost every x , ,, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd. [source]

On parallel solution of linear elasticity problems.

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2002
Part II: Methods, some computer experiments
Abstract This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block- diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg-method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M -matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block-diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)-factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher-order finite elements. Copyright © 2002 John Wiley & Sons, Ltd. [source]

The boundary element method with Lagrangian multipliers,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2009
Gabriel N. Gatica
Abstract On open surfaces, the energy space of hypersingular operators is a fractional order Sobolev space of order 1/2 with homogeneous Dirichlet boundary condition (along the boundary curve of the surface) in a weak sense. We introduce a boundary element Galerkin method where this boundary condition is incorporated via the use of a Lagrangian multiplier. We prove the quasi-optimal convergence of this method (it is slightly inferior to the standard conforming method) and underline the theory by a numerical experiment. The approach presented in this article is not meant to be a competitive alternative to the conforming method but rather the basis for nonconforming techniques like the mortar method, to be developed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

Lagrange interpolation and finite element superconvergence,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004
Bo Li
Abstract We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d -dimensional Qk -type elements with d , 1 and k , 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H1 norm. For d -dimensional Pk -type elements, we consider the standard Lagrange interpolation,the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d , 2 and k , d + 1 that such interpolation and the finite element solution are not superclose in both H1 and L2 norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33,59, 2004. [source]

Infinitely many solutions for polyharmonic elliptic problems with broken symmetries

MATHEMATISCHE NACHRICHTEN, Issue 1 2003
Sergio Lancelotti
Abstract By means of a perturbation argument devised by P. Bolle, we prove the existence of infinitely many solutions for perturbed symmetric polyharmonic problems with non,homogeneous Dirichlet boundary conditions. An extension to the higher order case of the estimate from below for the critical values due to K. Tanaka is obtained. [source]