Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods

In this study, the bounds for eigenvalues of the Laplacian operator on an L-shaped domain are determined. By adopting some special functions in Goerisch method for lower bounds and in traditional Rayleigh–Ritz method for upper bounds, very accurate bounds to eigenvalues for the problem are obtained. Numerical results show that these functions can also be successfully used to solve the problem on the region with other reentrant angle.     

Numerical solution of the Orr–Sommerfeld equation using the viscous Green function and split-Gaussian quadrature

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green functions, building on the successful use of split-Gauss-type quadrature schemes. Here we adapt the method for eigenvalue problems, in particular the Orr–Sommerfeld equation of hydrodynamic stability theory. Use of the Green function for the viscous part of the problem reduces the fourth-order ordinary differential equation to an integro-differential equation which we then discretize using the split-Gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators. As the latter are only second-order the resulting discrete equations are much more stable than those obtained from the original differential equation. This permits us to obtain results for the standard test problem (plane Poiseuille flow at unit streamwise wavenumber and Reynolds number 10 000) that we believe are the most accurate to date.     

DIRECT MINIMIZATION FOR CALCULATING INVARIANT SUBSPACES IN DENSITY FUNCTIONAL COMPUTATIONS OF THE ELECTRONIC STRUCTURE

In this article, we analyse three related preconditioned steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the admissible manifold, i.e., the invariance with respect to unitary transformations, to reformulate the problem on the Grassmann manifold as the admissible set. We then prove asymptotical linear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the Grassmann manifold at the minimizer.     

球面域上高阶拉普拉斯的特征值估计

本文研究了球面域上高阶拉普拉斯的特征值问题. 利用Rayleigh-Ritz不等式, 获得了球面域上高阶拉普拉斯的第(k+1)个特征值的上界估计, 这个估计式由前k个特征值给出.     

Some remarks on the Dirac–Witten operator on pseudo-Riemannian manifolds

In this paper, we get optimal lower bounds for the eigenvalues of the Dirac–Witten operator. These estimates are given in terms of the intrinsic and extrinsic curvature expressions. In the limiting case, we prove that the spacelike submanifold is Einstein if the normal bundle is flat.