Some of the most challenging eigenvalue problems arise in the stability analysis of solutions to parameter-dependent nonlinear partial diierential equations. Surface tension gradients along the free boundary of the oat-zone in crystal growth give rise to a stationary thermocapillary convection. Loss of stability leads to undesirable material imperfections. Stability analyses employing both ener...

We analyze the use of edge finite element methods to approximate Maxwell’s equations in a bounded cavity. Using the theory of collectively compact operators, we prove h-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are...

Martin COSTABEL & Monique DAUGE Abstract. The 2 2 system of integral equations corresponding to the biharmonic single layer potential in R2 is known to be strongly elliptic. It is also known to be positive definite on a space of functions orthogonal to polynomials of degree one. We study the question of its unique solvability without this orthogonality condition. To each curve , we associate a ...

The goal of our report is to compare a number of algorithms for computing a large number of eigenvectors of the generalized eigenvalue problem arising from a modal analysis of elastic structures using preconditioned iterative methods. The shift-invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however a sparse direct method is require...

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting (LTL) method with the configuration equation to develop a new effective and convergent algori...

Using transfer matrix method to solve 3D Ising model is generalized straightforwardly from 2D case. We follow the B.Kaufman's approach. No approximation is made except the largest eigenvalue cannot be identified. This problem comes from the fact that we follow the choice of directions of 2-dimensional rotations in direct product space of 2D Ising model such that all eigenvalue equations reduce ...

We analyse an optimal control with the following features: the dynamical system is linear, and the dependence upon the control parameter is affine. More precisely we consider ẋα(t) = (G+α(t)F )xα(t), where G and F are 3×3 matrices with some prescribed structure. In the case of constant control α(t) ≡ α, we show the existence of an optimal Perron eigenvalue with respect to varying α under some a...

Three iteration methods are proposed for the computation of eigenvalues and eigenfunctions in the linear stability of solitary waves. These methods are based on iterating certain time evolution equations associated with the linear stability eigenvalue problem. The first method uses the fourth-order Runge–Kutta method to iterate the pre-conditioned linear stability operator, and it usually conve...