A Spectral Element/Laguerre Coupled Method to the Elliptic Helmholtz Problem on the Half Line

A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.     

A coupled finite and boundary spectral element method for linear water-wave propagation problems

A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p-convergence of spectral methods.     

Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation

A modified Laguerre pseudospectral method is proposed for differential equations on the half-line. The numerical solutions are refined by multidomain Legendre pseudospectral approximation. Numerical results show the spectral accuracy of this approach. Some approximation results on the modified Laguerre and Legendre interpolations are established. The convergence of proposed method is proved.     

Spectral methods using generalized Laguerre functions for second and fourth order problems

Spectral methods using generalized Laguerre functions are proposed for second-order equations under polar (resp. spherical) coordinates in ?² (resp. ?³) and fourth-order equations on the half line. Some Fourier-like Sobolev orthogonal basis functions are constructed for our Laguerre spectral methods for elliptic problems. Optimal error estimates of the Laguerre spectral methods are obtained for both second-order and fourth-order elliptic equations. Numerical experiments demonstrate the effectiveness and the spectral accuracy.     

Legendre spectral collocation method for volterra-hammerstein integral equation of the second kind

This paper is concerned with obtaining the approximate solution for VolterraHammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function ω(x) = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L~2 norm and L~∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.     

On semi-infinite spectral elements for Poisson problems with re-entrant boundary singularities

A spectral element method is described which enables Poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi-infinite rectangular regions. Two choices of trial functions are considered, namely the eigenfunctions of the differential operator and Chebyshev polynomials. The coefficients in the series expansions are obtained by imposing weak C¹ matching conditions across element interfaces. Singularities at re-entrant corners are treated by a post-processing technique which makes use of the known asymptotic behaviour of the solution at the singular point. Accurate approximations are obtained with few degrees of freedom.     

The spectrum of Jacobi matrices in terms of its associated weight function

Using the so-called Lanczos procedure of orthogonalization a method is developed to calculate the elements of a N-dimensional Jacobi matrix and/or the coefficients of the three-term recurrence relation of a system of orthogonal polynomials {P_m(x), m = 0, 1, 2, ?, N} in terms of the moments μ_r^′(1) of its associated weight function. The eigenvalue density ?^(N)(x) and its asymptotical limit, i.e. when N tends to infinite, are also calculated in terms of μ_r^′(1). The method is used to determine the functions ?^(N)(x) and ?(x) for some known weight functions, like the normal distribution, the uniform distribution, the semicircular distribution and the gamma or Pearson type III distribution. As a byproduct the asymptotical density of zeros of Chebyshev, Legendre and generalized Laguerre polynomials are found.     

Petrov-Galerkin Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems on Polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.     

Stokes' first problem for a Newtonian fluid in a non‐Darcian porous half‐space using a Laguerre–Galerkin method

A Laguerre–Galerkin method is proposed and analysed for the Stokes' first problem of a Newtonian fluid in a non‐Darcian porous half‐space on a semi‐infinite interval. It is well known that Stokes' first problem has a jump discontinuity on boundary which is the main obstacle in numerical methods. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi‐infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations of the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Global superconvergence of simplified hybrid combinations of the Ritz–Galerkin and FEMs for elliptic equations with singularities II. Lagrange elements and Adini's elements

To solve the elliptic problems with singularities, the simplified hybrid combinations of the Ritz–Galerkin method and the finite element method (RGM–FEM) are explored, which lead to the high global superconvergence rates on the entire solution domain. Let the solution domain be split into a singular subdomain involving a singular point, and a regular subdomain where the true solution is smooth enough. In the singular subdomain, the singular particular functions are chosen to be admissible functions. In the regular subdomain either the k-order Lagrange rectangles or Adini's elements are adapted. Along their common boundary, the simplified hybrid techniques are employed to couple two different numerical methods. It is proven in this paper that the global superconvergence rates, O(h^k+3/2), on the entire domain can be achieved for k(⩾2)-order Lagrange rectangles, and that the global superconvergence rates O(h^3.5) for the Adini's elements. Numerical experiments are reported for the combinations of the Ritz–Galerkin and Adini's methods. This paper presents a development of [Z.C. Li, Computing 65 (2000) 27–44] in high accurate solutions for the general case of the Poisson problems on a polygonal domain S estimates for the Sobolev norm ∥·∥₁, given in a much more general sense than known before, cf. [P.G. Ciarlet, J.L. Lions (Eds.) Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991, pp. 17–351, 501–522; SIAM J. Sci. Statist. Comput. 11 (1990) 343; J. Comput. Appl. Math. 20 (1987) 341; Numer. Math. 63 (1992) 483; L. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Springer, Berlin, 1995; Numer. Methods Partial Differential Equations 3 (1987) 65, 357].     

Uniform convergence of the legendre spectral method for the Zakharov equations

The Legendre spectral and pseudospectral approximations are proposed for the standard Zakharov equations with initial boundary conditions. Optimal H¹ error estimate of the method is given for both semidiscrete and fully discrete schemes. The uniform convergence for the parameter ε relative to the acoustic speed is proved. Moreover, the multidomain Legendre spectral scheme is also constructed, which can be implemented in parallel. Finally, numerical results in single domain and multidomain verify the high accuracy of the Legendre spectral method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Laguerre Isopararmetric Hypersurfaces in R^4

Let x : M → Rn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C , and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R4 .     

A new robust C-type nonconforming triangular element for singular perturbation problems

In this paper, a new robust C⁰ triangular element is proposed for the fourth order elliptic singular perturbation problem with double set parameter method and bubble function technique, and a general convergence theorem for C⁰ nonconforming elements is presented. The convergence of the new element is proved in the energy norm uniformly with respect to the perturbation parameter. Numerical experiments are also carried out to demonstrate the efficiency of the new element.     

A Legendre spectral element method on a large spatial domain to solve the predator–prey system modeling interacting populations

A Legendre spectral element method is developed for solving a one-dimensional predator–prey system on a large spatial domain. The predator–prey system is numerically solved where the prey population growth is described by a cubic polynomial and the predator’s functional response is Holling type I. The discretization error generated from this method is compared with the error obtained from the Legendre pseudospectral and finite element methods. The Legendre spectral element method is also presented where the predator response is Holling type II and the initial data are discontinuous.     

C 0 elements for generalized indefinite Maxwell equations

In this paper we develop the C ⁰ finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H ^r regularity for some r?<?1. The ingredients of our method are that two ??mass-lumping?? L ² projectors are applied to curl and div operators in the problem and that C ⁰ linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C ⁰ Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H ^r regularity where r may vary in the interval [0, 1), we obtain the error bound ${{\mathcal O}(h^r)}$ in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.     

Chebyshev‐Legendre spectral method for solving the two‐dimensional vorticity equations with homogeneous Dirichlet conditions

The Chebyshev‐Legendre spectral method for the two‐dimensional vorticity equations is considered. The Legendre Galerkin Chebyshev collocation method is used with the Chebyshev‐Gauss collocation points. The numerical analysis results under the L²‐norm for the Chebyshev‐Legendre method of one‐dimensional case are generalized into that of the two‐dimensional case. The stability and optimal order convergence of the method are proved. Numerical results are given. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Laguerre polynomial expansions in indefinite inner product spaces

When ?j ? 1 < α < ?j, where j is a positive integer, the Laguerre polynomials {L_n^(α)}_{n = 0}^∞ form a complete orthogonal set in a nondegenerate inner product space H which is defined by employing an appropriate regularized linear functional on H^(j)[[0, ∞); x^{α + j}e^?x]. Expansions in terms of these Laguerre polynomials are exhibited. The Laguerre differential operator is shown to be self-adjoint with real, discrete, integer eigenvalues. Its spectral resolution and resolvent are exhibited and discussed.     

Approximation of analytic functions by Laguerre functions

We will solve the inhomogeneous Laguerre differential equation and apply this result to prove that if a function can be represented by a power series whose radius of convergence is larger than 1, then the function can be approximated, on the interval [0, 1), by a Laguerre function with an error bound C x for some constant C > 0.     

The method of infinite ascent applied on A 4 ± nB 3 = C 2

Each of the Diophantine equations A ⁴ ± nB ³ = C ² has an infinite number of integral solutions (A,B,C) for any positive integer n. In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when A, B and C are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions (A,B,C) of the Diophantine equation aA ³ +cB ³ = C ² for any co-prime integer pair (a, c).     

Sobolev orthogonal Legendre rational spectral methods for problems on the half line

Modified Legendre rational spectral methods for solving second-order differential equations on the half line are proposed. Some Sobolev orthogonal Legendre rational basis functions are constructed, which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. Numerical results demonstrate the effectiveness and the spectral accuracy of this approach.