arXiv subjects

Huiyuan Li

Publications and source records attributed to Huiyuan Li.

36 records · Page 2Linked to original sources

arXiv2021

H($\text{curl}^2$)-conforming quadrilateral spectral element method for quad-curl problems

In this paper, we propose an $H(\text{curl}^2)$-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. $H(\text{curl}^2)$-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is astonishing that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our $H(\text{curl}^2)$-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

math.NA↗

arXiv2020

Generalised Hermite spectral methods for PDEs involving integral fractional Laplacian and Schrödinger operators

In this paper, we introduce two new families of generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate generalised Hermite spectral algorithms for PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators in $\mathbb R^d.$ As a generalisation of the G. Szegö's family in 1D (1939), the first family of GHPs (resp. GHFs) are orthogonal with respect to $|\bx|^{2μ} \e^{-|\bx|^2}$ (resp. $|\bx |^{2μ}$) in $\mathbb R^d$. We further define adjoint generalised Hermite functions (A-GHFs) which have an interwoven connection with the corresponding GHFs through the Fourier transform, and which are orthogonal with respect to the inner product $[u,v]_{H^s(\mathbb R^d)}=((-Δ)^{s/ 2}u, (-Δ)^{s/2} v )_{\mathbb R^d}$ associated with the IFL of order $s>0$. Thus, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds efficient and accurate in solving PDEs with the fractional Schrödinger operator: $(-Δ)^s +|\bs x|^{2μ}$ with $s\in (0,1]$ and $μ>-1/2.$ Following the same spirit, we construct the second family of GHFs, dubbed as Müntz-type generalised Hermite functions (M-GHFs), which are orthogonal with respect to an inner product associated with the underlying Schrödinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Müntz-type GHF spectral method leads to sparse matrices and spectrally accurate to some Schrödinger eigenvalue problems.

math.NA↗

arXiv2020

Vectorial ball Prolate spheroidal wave functions with the divergence free constraint

In this paper, we introduce one family of vectorial prolate spheroidal wave functions of real order $α>-1$ on the unit ball in $R^3$, which satisfy the divergence free constraint, thus are termed as divergence free vectorial ball PSWFs. They are vectorial eigenfunctions of an integral operator related to the finite Fourier transform, and solve the divergence free constrained maximum concentration problem in three dimensions, i.e., to what extent can the total energy of a band-limited divergence free vectorial function be concentrated on the unit ball? Interestingly, any optimally concentrated divergence free vectorial functions, when represented in series in vector spherical harmonics, shall be also concentrated in one of the three vectorial spherical harmonics modes. Moreover, divergence free ball PSWFs are exactly the vectorial eigenfunctions of the second order Sturm-Liouville differential operator which defines the scalar ball PSWFs. Indeed, the divergence free vectorial ball PSWFs possess a simple and close relation with the scalar ball PSWFs such that they share the same merits. Simultaneously, it turns out that the divergence free ball PSWFs solve another second order Sturm-Liouville eigen equation defined through the curl operator $\nabla\times $ instead of the gradient operator $\nabla$.

math.NA↗

arXiv2018

Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals/derivatives, which is naturally arisen from exact representations of Chebyshev expansion coefficients, for optimal error estimates of Chebyshev approximations to functions with limited regularity. The essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterize in existing literature. We can then derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve the available results in usual Sobolev spaces of integer regularity exponentials in several senses. As a by-product, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi spectral approximations.

math.NA↗

arXiv2018

Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations

An efficient Jacobi-Galerkin spectral method for calculating eigenvalues of Riesz fractional partial differential equations with homogeneous Dirichlet boundary values is proposed in this paper. In order to retain the symmetry and positive definiteness of the discrete linear system, we introduce some properly defined Sobolev spaces and approximate the eigenvalue problem in a standard Galerkin weak formulation instead of the Petrov-Galerkin one as in literature. Poincaré and inverse inequalities are proved for the proposed Galerkin formulation which finally help us establishing a sharp estimate on the algebraic system's condition number. Rigorous error estimates of the eigenvalues and eigenvectors are then readily obtained by using Babuška and Osborn's approximation theory on self-adjoint and positive-definite eigenvalue problems. Numerical results are presented to demonstrate the accuracy and efficiency, and to validate the asymptotically exponential oder of convergence. Moreover, the Weyl-type asymptotic law $ λ_n=\mathcal{O}(n^{2α})$ for the $n$-th eigenvalue $λ_n$ of the Riesz fractional differential operator of order $2α$, and the condition number $N^{4α}$ of its algebraic system with respect to the polynomial degree $N$ are observed.

math.NA↗

arXiv2018

Ball Prolate Spheroidal Wave Functions In Arbitrary Dimensions

In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $α>-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator, and a Sturm-Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalisation of orthogonal {\em ball polynomials} in primitive variables with a tuning parameter $c>0$, through a "perturbation" of the Sturm-Liouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutions.

math.NA↗

arXiv2017

Ultrasensitive biosensor based on Nd:YAG waveguide laser: Tumor cell and Dextrose solution

This work demonstrates the Nd:YAG waveguide laser as an efficient platform for the bio-sensing. The waveguide was fabricated in the Nd:YAG crystal by the cooperation of the ultrafast laser writing and ion irradiation. As the laser oscillation in the Nd:YAG waveguide is ultra-sensitivity to the external environment of the waveguide. Even a weak disturbance would induce a large variation of the output power of the laser. According to this feature, the Nd:YAG waveguide coated with Graphene and WSe2 layers is used as substrate for the microfluidic channel. When the microflow crosses the Nd:YAG waveguide, the laser oscillation in the waveguide is disturbed, and induces the fluctuation of the output laser. Through the analysis of the fluctuation, the concentration of the dextrose solution and the size of the tumor cell are distinguished

physics.app-ph↗

arXiv2016

Spectral-Galerkin Approximation and Optimal Error Estimate for Stokes Eigenvalue Problems in Polar Geometries

In this paper we propose and analyze spectral-Galerkin methods for the Stokes eigenvalue problem based on the stream function formulation in polar geometries. We first analyze the stream function} formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction} of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one dimensional problem can be obtained. Further, we extend our method to the non-separable Stokes eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.

math.NA↗

arXiv2016

A fully diagonalized spectral method using generalized Laguerre functions on the half line

A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line. We first define the generalized Laguerre functions which are complete and mutually orthogonal with respect to an equivalent Sobolev inner product. Then the Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral method of elliptic equations. Besides, a unified orthogonal Laguerre projection is established for various elliptic equations. On the basis of this orthogonal Laguerre projection, we obtain optimal error estimates of the fully diagonalized Laguerre spectral method for both Dirichlet and Robin boundary value problems. Finally, numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized method.

math.NA↗

arXiv2016

Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential

In this article, we study numerical approximation of eigenvalue problems of the Schrödinger operator $\displaystyle -Δu + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in which case the exact solution in the radial direction can be expressed by Bessel functions of fractional degrees. This knowledge helps us to design two novel spectral methods by modifying the polynomial basis to fit the singularities of the eigenfunctions. At the second stage, we move to circular sectors in the two dimensional setting. Again the radial direction can be expressed by Bessel functions of fractional degrees. Only in the tangential direction some modifications are needed from stage one. At the final stage, we extend the idea to arbitrary polygonal domains. We propose a mortar spectral element approach: a polygonal domain is decomposed into several sub-domains with each singular corner including the origin covered by a circular sector, in which origin and corner singularities are handled similarly as in the former stages, and the remaining domains are either a standard quadrilateral/triangle or a quadrilateral/triangle with a circular edge, in which the traditional polynomial based spectral method is applied. All sub-domains are linked by mortar elements (note that we may have hanging nodes). In all three stages, exponential convergence rates are achieved. Numerical experiments indicate that our new methods are superior to standard polynomial based spectral (or spectral element) methods and $hp$-adaptive methods. Our study offers a new and effective way to handle eigenvalue problems of the Schrödinger operator including the Laplacian operator on polygonal domains with reentrant corners.

math.NA↗

arXiv2013

Spectral approximation on the unit ball

Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\ball)$, $1<p<\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces and explicit construction of approximating polynomials. One major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the Sobolev space $W_2^s(\ball)$. As an application, a direct and efficient spectral-Galerkin method based on our orthogonal polynomials is proposed for the second and the fourth order elliptic equations on the unit ball, its optimal error estimates are explicitly derived for both procedures in the Sobolev spaces and, finally, numerical examples are presented to illustrate the theoretic results.

math.CA↗

arXiv2012

Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ 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.

math.NA↗

arXiv2012

A New Triangular Spectral Element Method I: Implementation and Analysis on a Triangle

This paper serves as our first effort to develop a new triangular spectral element method (TSEM) on unstructured meshes, using the rectangle-triangle mapping proposed in the conference note [21]. Here, we provide some new insights into the originality and distinctive features of the mapping, and show that this transform only induces a logarithmic singularity, which allows us to devise a fast, stable and accurate numerical algorithm for its removal. Consequently, any triangular element can be treated as efficiently as a quadrilateral element, which affords a great flexibility in handling complex computational domains. Benefited from the fact that the image of the mapping includes the polynomial space as a subset, we are able to obtain optimal $L^2$- and $H^1$-estimates of approximation by the proposed basis functions on triangle. The implementation details and some numerical examples are provided to validate the efficiency and accuracy of the proposed method. All these will pave the way for developing an unstructured TSEM based on, e.g., the hybridizable discontinuous Galerkin formulation.

math.NA↗

arXiv2009

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.

math.NA↗

arXiv2008

Discrete Fourier analysis on fundamental domain of $A_d$ lattice and on simplex in $d$-variables

A discrete Fourier analysis on the fundamental domain $Ω_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $Ω_2$ is the regular hexagon and $Ω_3$ is the rhombic dodecahedron, and analogous results on $d$-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^d$. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.

math.CA↗

arXiv2008

Cubature formula and interpolation on the cubic domain

Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on $[-1,1]^2$, as well as new results on $[-1,1]^3$. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on $n^3/4 +\CO(n^2)$ nodes of a cubature formula on $[-1,1]^3$.

math.NA↗

arXiv2008

Discrete Fourier analysis on a dodecahedron and a tetrahedron

A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^3$.

math.NA↗

arXiv2007

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.

math.NA↗