159 research outputs found
Steklov Eigenvalue Problems on Nearly Spherical and Nearly Annular Domains
We consider Steklov eigenvalues on nearly spherical and nearly annular
domains in dimensions. By using the Green-Beltrami identity for spherical
harmonic functions, the derivatives of Steklov eigenvalues with respect to the
domain perturbation parameter can be determined by the eigenvalues of a matrix
involving the integral of the product of three spherical harmonic functions. By
using the addition theorem for spherical harmonic functions, we determine
conditions when the trace of this matrix becomes zero. These conditions can
then be used to determine when spherical and annular regions are critical
points while we optimize Steklov eigenvalues subject to a volume constraint. In
addition, we develop numerical approaches based on particular solutions and
show that numerical results in two and three dimensions are in agreement with
our analytic results
A two-dimensional flea on the elephant phenomenon and its numerical visualization
First Published in Multiscale Modeling and Simulation in 17.1 (2019): 137-166, published by the Society for Industrial and Applied Mathematics (SIAM)Localization phenomena (sometimes called flea on the elephant) for the operator Lvarepsilon = varepsilon 2Î u + p(x)u, p(x) being an asymmetric double well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-PoincarĂ© operator for Lvarepsilon, and for which error estimates are established. Such a two-dimensional discretization produces less mesh imprinting than more standard finite differences and correctly captures sharp layersEnrique Zuazuaâs research was supported by the Advanced
Grant DyCon (Dynamical Control) of the European Research Council Executive
Agency (ERC), the MTM2014-52347 and MTM2017-92996 Grants of the MINECO
(Spain) and the ICON project of the French ANR-16-ACHN-0014. L.G. thanks Profs.
François Bouchut and Roberto Natalini for some technical discussion
A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems
In this paper we revisit an approach pioneered by Auchmuty to approximate
solutions of the Laplace- Robin boundary value problem. We demonstrate the
efficacy of this approach on a large class of non-tensorial domains, in
contrast with other spectral approaches for such problems.
We establish a spectral approximation theorem showing an exponential fast
numerical evaluation with regards to the number of Steklov eigenfunctions used,
for smooth domains and smooth boundary data. A polynomial fast numerical
evaluation is observed for either non-smooth domains or non-smooth boundary
data. We additionally prove a new result on the regularity of the Steklov
eigenfunctions, depending on the regularity of the domain boundary.
We describe three numerical methods to compute Steklov eigenfunctions
Finite element schemes for elliptic boundary value problems with rough coefficients
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We consider the task of computing reliable numerical approximations of the solutions of elliptic equations and systems where the coefficients vary discontinuously, rapidly, and by large orders of magnitude. Such problems, which occur in diffusion and in linear elastic deformation of composite materials, have solutions with low regularity with the result that reliable numerical approximations can be found only in approximating spaces, invariably with high dimension, that can accurately represent the large and rapid changes occurring in the solution. The use of the Galerkin approach with such high dimensional approximating spaces often leads to very large scale discrete problems which at best can only be solved using efficient solvers. However, even then, their scale is sometimes so large that the Galerkin approach becomes impractical and alternative methods of approximation must be sought. In this thesis we adopt two approaches. We propose a new asymptotic method of approximation for problems of diffusion in materials with periodic structure. This approach uses Fourier series expansions and enables one to perform all computations on a periodic cell; this overcomes the difficulty caused by the rapid variation of the coefficients. In the one dimensional case we have constructed problems with discontinuous coefficients and computed the analytical expressions for their solutions and the proposed asymptotic approximations. The rates at which the given asymptotic approximations converge, as the period of the material decreases, are obtained through extensive computational tests which show that these rates are fundamentally dependent on the level of regularity of the right hand sides of the equations. In the two dimensional case we show how one can use the Galerkin method to approximate the solutions of the problems associated with the periodic cell. We construct problems with discontinuous coefficients and perform extensive computational tests which show that the asymptotic properties of the approximations are identical to those observed in the one dimensional case. However, the computational results show that the application of the Galerkin method of approximation introduces a discretization error which can obscure the precise asymptotic rate of convergence for low regularity right hand sides. For problems of two dimensional linear elasticity we are forced to consider an alternative approach. We use domain decomposition techniques that interface the subdomains with conjugate gradient methods and obtain algorithms which can be efficiently implemented on computers with parallel architectures. We construct the balancing preconditioner, M,, and show that it has the optimal conditioning property k(Mh(^-1)Sh) = 0 is a constant which is independent of the magnitude of the material discontinuities, H is the maximum subdomain diameter, and h is the maximum finite element diameter. These properties of the preconditioning operator Mh allow one to use the computational power of a parallel computer to overcome the difficulties caused by the changing form of the solution of the problem. We have implemented this approach for a variety of problems of planar linear elasticity and, using different domain decompositions, approximating spaces, and materials, find that the algorithm is robust and scales with the dimension of the approximating space and the number of subdomains according to the condition number bound above and is unaffected by material discontinuities. In this we have proposed and implemented new inner product expressions which we use to modify the bilinear forms associated with problems over subdomains that have pure traction boundary conditions.This work is funded by the Engineering and Physical Sciences Research Council
Min-max harmonic maps and a new characterization of conformal eigenvalues
Given a surface M and a fixed conformal class c one defines Î_k(M,c) to be the supremum of the k-th nontrivial Laplacian eigenvalue over all metrics g â c of unit volume. It has been observed by Nadirashvili that the metrics achieving Î_k(M,c) are closely related to harmonic maps to spheres. In the present paper, we identify Îâ(M,c) and Îâ(M,c) with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing Îâ(M,c), Îâ(M,c) and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition
Min-max harmonic maps and a new characterization of conformal eigenvalues
Given a surface and a fixed conformal class one defines
to be the supremum of the -th nontrivial Laplacian
eigenvalue over all metrics of unit volume. It has been observed by
Nadirashvili that the metrics achieving are closely related to
harmonic maps to spheres. In the present paper, we identify
and with min-max quantities associated to the energy
functional for sphere-valued maps. As an application, we obtain several new
eigenvalue bounds, including a sharp isoperimetric inequality for the first two
Steklov eigenvalues. This characterization also yields an alternative proof of
the existence of maximal metrics realizing ,
and, moreover, allows us to obtain a regularity theorem for maximal Radon
measures satisfying a natural compactness condition.Comment: 59 pages, minor corrections, references adde
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