Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.Comment: 22 page

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods

Generalized Duffy transformation for integrating vertex singularities

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u β , y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation

Second Order Darboux Displacements

The potentials for a one dimensional Schroedinger equation that are displaced along the x axis under second order Darboux transformations, called 2-SUSY invariant, are characterized in terms of a differential-difference equation. The solutions of the Schroedinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential. It is proven that a particular case of the periodic Lame-Ince potential is 2-SUSY invariant. Both Bloch solutions of the corresponding Schroedinger equation equation are found for any value of the energy. A simple analytic expression for a family of two-gap potentials is derived

Connection between the Green functions of the supersymmetric pair of Dirac Hamiltonians

The Sukumar theorem about the connection between the Green functions of the supersymmetric pair of the Schr\"odinger Hamiltonians is generalized to the case of the supersymmetric pair of the Dirac Hamiltonians.Comment: 12 pages,Latex, no figure

Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity

In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adopt the Hellinger--Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. On applying the divergence theorem to the weak strain-displacement relations, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement-based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions. However, for flexibility in choosing basis functions, we also present a formulation that uses a penalty term to enforce the element equilibrium conditions. This method is referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the $L^2$ norm of the displacement, energy seminorm, and the $L^2$ norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter.Comment: 49 pages, 49 figure

Multichannel coupling with supersymmetric quantum mechanics and exactly-solvable model for Feshbach resonance

A new type of supersymmetric transformations of the coupled-channel radial Schroedinger equation is introduced, which do not conserve the vanishing behavior of solutions at the origin. Contrary to usual transformations, these ``non-conservative'' transformations allow, in the presence of thresholds, the construction of potentials with coupled scattering matrices from uncoupled potentials. As an example, an exactly-solvable potential matrix is obtained which provides a very simple model of Feshbach-resonance phenomenon.Comment: 10 pages, 2 figure