Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities

1990 ◽  
Vol 3 (2) ◽  
pp. 173-188 ◽  
Author(s):  
Ken'ichi Nagasaki ◽  
Takashi Suzuki
Keyword(s):  
Asymptotic Analysis ◽  
Eigenvalue Problems ◽  
Two Dimensional ◽  
Elliptic Eigenvalue Problems

scholarly journals A Fourier Continuation Method for the Solution of Elliptic Eigenvalue Problems in General Domains

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Oscar P. Bruno ◽  
Timothy Elling ◽  
Ayon Sen
Keyword(s):  
Eigenvalue Problems ◽  
Continuation Method ◽  
Gibbs Phenomenon ◽  
Variable Coefficients ◽  
Computational Method ◽  
The Novel ◽  
Two Dimensional ◽  
Fourier Approximation ◽  
Fourier Continuation ◽  
Elliptic Eigenvalue Problems

We present a new computational method for the solution of elliptic eigenvalue problems with variable coefficients in general two-dimensional domains. The proposed approach is based on use of the novel Fourier continuation method (which enables fast and highly accurate Fourier approximation of nonperiodic functions in equispaced grids without the limitations arising from the Gibbs phenomenon) in conjunction with an overlapping patch domain decomposition strategy and Arnoldi iteration. A variety of examples demonstrate the versatility, accuracy, and generality of the proposed methodology.

scholarly journals Perturbation of eigenvalues due to gaps in two-dimensional boundaries

2006 ◽  
Vol 463 (2079) ◽  
pp. 759-786 ◽  
Author(s):  
Anthony M.J Davis ◽  
Stefan G Llewellyn Smith
Keyword(s):  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Neumann Boundary ◽  
Diffusion Problem ◽  
Length Scale ◽  
Two Dimensional ◽  
Constructive Approach ◽  
Term Outcome ◽  
Long Term Outcome

Motivated by problems involving diffusion through small gaps, we revisit two-dimensional eigenvalue problems with localized perturbations to Neumann boundary conditions. We recover the known result that the gravest eigenvalue is O (|ln  ϵ | −1 ), where ϵ is the ratio of the size of the hole to the length-scale of the domain, and provide a simple and constructive approach for summing the inverse logarithm terms and obtaining further corrections. Comparisons with numerical solutions obtained for special geometries, both for the Dirichlet ‘patch problem’ where the perturbation to the boundary consists of a different boundary condition and for the gap problem, confirm that this approach is a simple way of obtaining an accurate value for the gravest eigenvalue and hence the long-term outcome of the underlying diffusion problem.

An obstacle problem for elliptic membrane shells

2018 ◽  
Vol 24 (5) ◽  
pp. 1503-1529 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare ◽  
Paolo Piersanti
Keyword(s):  
Vector Field ◽  
Asymptotic Analysis ◽  
Obstacle Problem ◽  
Displacement Vector ◽  
Middle Surface ◽  
Two Dimensional ◽  
Displacement Vector Field ◽  
Membrane Shells ◽  
Signorini Condition ◽  
Confinement Condition

Our objective is to identify two-dimensional equations that model an obstacle problem for a linearly elastic elliptic membrane shell subjected to a confinement condition expressing that all the points of the admissible deformed configurations remain in a given half-space. To this end, we embed the shell into a family of linearly elastic elliptic membrane shells, all sharing the same middle surface [Formula: see text], where [Formula: see text] is a domain in [Formula: see text] and [Formula: see text] is a smooth enough immersion, all subjected to this confinement condition, and whose thickness [Formula: see text] is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as [Formula: see text] approaches zero, the corresponding “limit” two-dimensional variational problem. This problem takes the form of a set of variational inequalities posed over a convex subset of the space [Formula: see text]. The confinement condition considered here considerably departs from the Signorini condition usually considered in the existing literature, where only the “lower face” of the shell is required to remain above the “horizontal” plane. Such a confinement condition renders the asymptotic analysis substantially more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.

scholarly journals Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows

2003 ◽  
Vol 10 (1-2) ◽  
pp. 137-163
Author(s):  
Cristian A. Coclici ◽  
Jörg Heiermann ◽  
Gh. Moroşanu ◽  
W. Wendland
Keyword(s):  
Asymptotic Analysis ◽  
Viscous Flows ◽  
Two Dimensional ◽  
Coupled Problem ◽  
Compressible Viscous Flows
Sign in / Sign up

Export Citation Format

Share Document