A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator

Bao, Weizhu; Chen, Lizhen; Jiang, Xiaoyun; Ying, Ma

doi:10.1016/j.jcp.2020.109733

Cited by 7 publications

(6 citation statements)

References 60 publications

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“…Since it contains at most thirteen non-zero entries in each column of B m and C m , and at most two non-zero entries in each column of D m , only 53M 3 6 + O(M 2 ) operations are required to solve (3.14). In return, the Clenshaw algorithm shares the same order of complexity.…”

Section: Clenshaw Algorithm For Koornwinder Expansionsmentioning

confidence: 99%

“…To assemble the stiffness matrix, we need to evaluate the integral (∇ϕ ℓ , ∇ϕ k ) T . Indeed, the linear mapping Ψ defined in (4.1) has the following explicit form: (1) x1 + x (2) x2 + x (3) x3 .…”

Section: Stiffness Matrix Assemblingmentioning

confidence: 99%

“…Next, we turn to explore different gaps of these reliable numerical eigenvalues. We introduce the following definitions [16,3]:…”

Section: Numerical Examples For Eigenvalue Problemsmentioning

confidence: 99%

“…Spectral element methods with unstructured mesh have been widely used in the study of computational fluid dynamics, elastodynamics, resistivity modeling and many other fields due to their "spectral accuracy" [30,17,35,22]. Their virtue of high accuracy also makes spectral (element) methods powerful tools for solving eigenvalue problems as they are able to provide more reliable eigen-solutions than the low order methods such as finite element methods and finite difference methods [34,3]. Meanwhile, as simplices are one kind of the basic geometric elements, their use gives flexibility in the discretization of complex domains.…”

Section: Introductionmentioning

confidence: 99%

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Sparse Spectral-Galerkin Method on An Arbitrary Tetrahedron Using Generalized Koornwinder Polynomials

Jia¹,

Li²,

Zhang³

2021

Preprint

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In this paper, we propose a sparse spectral-Galerkin approximation scheme for solving the second-order partial differential equations on an arbitrary tetrahedron. Generalized Koornwinder polynomials are introduced on the reference tetrahedron as basis functions with their various recurrence relations and differentiation properties being explored. The method leads to well-conditioned and sparse linear systems whose entries can either be calculated directly by the orthogonality of the generalized Koornwinder polynomials for differential equations with constant coefficients or be evaluated efficiently via our recurrence algorithm for problems with variable coefficients. Clenshaw algorithms for the evaluation of any polynomial in an expansion of the generalized Koornwinder basis are also designed to boost the efficiency of the method. Finally, numerical experiments are carried out to illustrate the effectiveness of the proposed Koornwinder spectral method.

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Section: Clenshaw Algorithm For Koornwinder Expansionsmentioning

confidence: 99%

Section: Stiffness Matrix Assemblingmentioning

confidence: 99%

“…Next, we turn to explore different gaps of these reliable numerical eigenvalues. We introduce the following definitions [16,3]:…”

Section: Numerical Examples For Eigenvalue Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sparse Spectral-Galerkin Method on An Arbitrary Tetrahedron Using Generalized Koornwinder Polynomials

Jia¹,

Li²,

Zhang³

2021

Preprint

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“…(2.20) and (2.25)). Indeed, the A-GHFs can provide a viable tool for the solutions of fractional Schrödinger problems (see, e.g., [22,47,5,6]).…”

Section: Introductionmentioning

confidence: 99%

Nontensorial generalised hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators

Sheng

et al. 2021

ESAIM: M2AN

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In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schr\"{o}dinger operators in R^d. As a generalisation of the G. Szego's family in 1D (1939), the first family of multivariate GHPs (resp. GHFs) are orthogonal with respect to the weight function |x|^{2\mu} e^{-|x|^2} (resp. |x|^{2\mu}) in R^d. We further construct the adjoint generalised Hermite functions (A-GHFs), which have an interwoven connection with the corresponding GHFs through the Fourier transform, and are orthogonal with respect to the inner product [u,v]_{H^s(R^d)}=((-\Delta)^{s/2}u, (-\Delta)^{s/2} v)_{R^d} associated with the IFL of order s>0. As an immediate consequence, 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 remarkably efficient in solving PDEs with the fractional Schrodinger operator: (-\Delta)^s +|x|^{2\mu} with s\in (0,1] and \mu>-1/2 in R^d. We construct the second family of multivariate nontensorial Muntz-type GHFs, which are orthogonal with respect to an inner product associated with the underlying Schrodinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Muntz-type GHF spectral method leads to sparse matrices and spectrally accurate solution to some Schrodinger eigenvalue problems.

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Sparse Spectral-Galerkin Method on An Arbitrary Tetrahedron Using Generalized Koornwinder Polynomials

Jia

Zhang

2022

J Sci Comput

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A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator

Cited by 7 publications

References 60 publications

Sparse Spectral-Galerkin Method on An Arbitrary Tetrahedron Using Generalized Koornwinder Polynomials

Sparse Spectral-Galerkin Method on An Arbitrary Tetrahedron Using Generalized Koornwinder Polynomials

Nontensorial generalised hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators

Sparse Spectral-Galerkin Method on An Arbitrary Tetrahedron Using Generalized Koornwinder Polynomials

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