Discrete Fourier analysis with lattices on planar domains

A discrete Fourier analysis associated with translation lattices is developed recently by the authors. It permits two lattices, one determining the integral domain and the other determining the family of exponential functions. Possible choices of lattices are discussed in the case of lattices that tile \RR^2 and several new results on cubature and interpolation by trigonometric, as well as algebraic, polynomials are obtained

Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.Comment: 29 page

Discrete Fourier Analysis and Chebyshev Polynomials with G2G_2 Group

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type

An efficient and effective nonlinear solver in a parallel software for large scale petroleum reservoir simulation

Abstract. We study a parallel Newton-Krylov-Schwarz (NKS) based algorithm for solving large sparse systems resulting from a fully implicit discretization of partial differential equations arising from petroleum reservoir simulations. Our NKS algorithm is designed by combining an inexact Newton method with a rank-2 updated quasi-Newton method. In order to improve the computational efficiency, both DDM and SPMD parallelism strategies are adopted. The effectiveness of the overall algorithm depends heavily on the performance of the linear preconditioner, which is made of a combination of several preconditioning components including AMG, relaxed ILU, up scaling, additive Schwarz, CRPlike(constraint residual preconditioning), Watts correction, Shur complement

Flexible but Refractory Single-Crystalline Hyperbolic Metamaterials

The fabrication of flexible single-crystalline plasmonic or photonic components in a scalable way is fundamentally important to flexible electronic and photonic devices with high speed, high energy efficiency, and high reliability. However, it remains to be a big challenge so far. Here, we have successfully synthesized flexible single-crystalline optical hyperbolic metamaterials by directly depositing refractory nitride superlattices on flexible fluoro phlogopite-mica substrates with magnetron sputtering. Interestingly, these flexible hyperbolic metamaterials show dual-band hyperbolic dispersion of dielectric constants with low dielectric losses and high figure-of-merit in the visible to near-infrared ranges. More importantly, the optical properties of these nitride-based flexible hyperbolic metamaterials show remarkable stability under either heating or bending. Therefore, the strategy developed in this work offers an easy and scalable route to fabricate flexible, high-performance, and refractory plasmonic or photonic components, which can significantly expand the applications of current electronic and photonic devices.Comment: 15 page

Genome and pan-genome assembly of asparagus bean (Vigna unguiculata ssp. sesquipedialis) reveal the genetic basis of cold adaptation

Asparagus bean (Vigna unguiculata ssp. sesquipedialis) is an important cowpea subspecies. We assembled the genomes of Ningjiang 3 (NJ, 550.31 Mb) and Dubai bean (DB, 564.12 Mb) for comparative genomics analysis. The whole-genome duplication events of DB and NJ occurred at 64.55 and 64.81 Mya, respectively, while the divergence between soybean and Vigna occurred in the Paleogene period. NJ genes underwent positive selection and amplification in response to temperature and abiotic stress. In species-specific gene families, NJ is mainly enriched in response to abiotic stress, while DB is primarily enriched in respiration and photosynthesis. We established the pan-genomes of four accessions (NJ, DB, IT97K-499-35 and Xiabao II) and identified 20,336 (70.5%) core genes present in all the accessions, 6,507 (55.56%) variable genes in two individuals, and 2,004 (6.95%) unique genes. The final pan genome is 616.35 Mb, and the core genome is 399.78 Mb. The variable genes are manifested mainly in stress response functions, ABC transporters, seed storage, and dormancy control. In the pan-genome sequence variation analysis, genes affected by presence/absence variants were enriched in biological processes associated with defense responses, immune system processes, signal transduction, and agronomic traits. The results of the present study provide genetic data that could facilitate efficient asparagus bean genetic improvement, especially in producing cold-adapted asparagus bean

New schemes with fractal error compensation for PDE eigenvalue computations

With an error compensation term in the fractal Rayleigh quotient of PDE eigen-problems, we propose a new scheme by perturbing the mass matrix M (h) to , where K (h) is the corresponding stiff matrix of a 2m - 1 degree conforming finite element with mesh size h for a 2m-order self-adjoint PDE, and the constant C exists in the priority error estimation lambda (j) (h) - lambda (j) similar to Ch (2m) lambda (j) (2) . In particular, for Laplace eigenproblems over regular domains in uniform mesh, e.g., cube, equilateral triangle and regular hexagon, etc., we find the constant and show that in this case the computation accuracy can raise two orders, i.e., from lambda (j) (h) - lambda (j) = O(h (2)) to O(h (4)). Some numerical tests in 2-D and 3-D are given to verify the above arguments.With an error compensation term in the fractal Rayleigh quotient of PDE eigen-problems, we propose a new scheme by perturbing the mass matrix M (h) to , where K (h) is the corresponding stiff matrix of a 2m - 1 degree conforming finite element with mesh size h for a 2m-order self-adjoint PDE, and the constant C exists in the priority error estimation lambda (j) (h) - lambda (j) similar to Ch (2m) lambda (j) (2) . In particular, for Laplace eigenproblems over regular domains in uniform mesh, e.g., cube, equilateral triangle and regular hexagon, etc., we find the constant and show that in this case the computation accuracy can raise two orders, i.e., from lambda (j) (h) - lambda (j) = O(h (2)) to O(h (4)). Some numerical tests in 2-D and 3-D are given to verify the above arguments

Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes, a new preprocessing approach, called multineighboring grids (MNG), is proposed for solving PDE eigen-problems on an existing grid G(Delta). The linear or multi-linear element, based on box-splines, are taken as the first stage (K1Uh)-U-h = lambda(1M1Uh)-M-h-U-h. In this paper, the j-th stage neighboring-grid scheme is defined as (KjUh)-U-h = lambda(jMjUh)-M-h-U-h, where K-j(h) := M-j-1(h) circle times K-1(h) and (MjUh)-U-h is to be found as a better mass distribution over the j-th stage neighboring-grid G(Delta), and K-j(h) can be seen as an expansion of K-1(h) on the j-th neighboring-grid with respect to the (j - 1)-th mass distribution M-j-1(h). It is shown that for an ODE model eigen-problem, the j-th stage scheme with 2j-th order B-spline basis can reach 2j-th order accuracy and even (2j + 2)-th order accuracy by perturbing the mass matrix. The argument can be extended to high dimensions with separable variable cases. For Laplace eigen-problems with some 2-D and 3-D structured uniform grids, some 2j-th order schemes are presented for j <= 3.Instead of most existing postprocessing schemes, a new preprocessing approach, called multineighboring grids (MNG), is proposed for solving PDE eigen-problems on an existing grid G(Delta). The linear or multi-linear element, based on box-splines, are taken as the first stage (K1Uh)-U-h = lambda(1M1Uh)-M-h-U-h. In this paper, the j-th stage neighboring-grid scheme is defined as (KjUh)-U-h = lambda(jMjUh)-M-h-U-h, where K-j(h) := M-j-1(h) circle times K-1(h) and (MjUh)-U-h is to be found as a better mass distribution over the j-th stage neighboring-grid G(Delta), and K-j(h) can be seen as an expansion of K-1(h) on the j-th neighboring-grid with respect to the (j - 1)-th mass distribution M-j-1(h). It is shown that for an ODE model eigen-problem, the j-th stage scheme with 2j-th order B-spline basis can reach 2j-th order accuracy and even (2j + 2)-th order accuracy by perturbing the mass matrix. The argument can be extended to high dimensions with separable variable cases. For Laplace eigen-problems with some 2-D and 3-D structured uniform grids, some 2j-th order schemes are presented for j <= 3

new six-node and seven-node hexagonal finite elements

In this paper, we introduce a six-node and a seven-node hexagonal elements. Both elements are based on the rotated trilinear interpolation in terms of the three-directional coordinates. Optimal a priori error estimates are provided for the new elements by decomposing the consistency error into two parts that both can be estimated using the F-E-M test. Some numerical tests are carried out to verify the theoretical analysis. Comparisons to both the previously studied edge-oriented hexagonal elements and the traditional linear triangular elements are also provided to show the efficiency and superiority of the new six-node and seven-node hexagonal elements. &copy; 2012 Elsevier Inc. All rights reserved