Legendre Galerkin Methods Research Articles

An exponential spectral deferred correction method for multidimensional parabolic problems

We present some efficient algorithms based on an exponential time differencing spectral deferred correction (ETDSDC) method for multidimensional second and fourth-order parabolic problems with non-periodic boundary conditions including Dirichlet, Neumann, Robin boundary conditions. Similar to the Fourier spectral method for periodic problems, the key to the efficiency of our algorithms is to construct diagonal discrete linear operators via Legendre–Galerkin methods with Fourier-like basis functions. In combination with the ETDSDC scheme, the proposed methods are spectrally accurate in space and up to 10th-order accurate in time (as shown in this work). We demonstrate the high-order of convergence and efficiency of our algorithms in solving parabolic equations through a series of two-dimensional and three-dimensional examples including Ginzburg–Landau and Allen–Cahn equations.

Discrete Legendre spectral projection-based methods for Tikhonov regularization of first kind Fredholm integral equations

In this paper, we apply the discrete Legendre Galerkin and multi-Galerkin methods to find the approximate solution of the Tikhonov regularized equation of the Fredholm integral equations of the first kind. We evaluate the error bounds for the approximate solutions with the exact solution in the infinity norm. We provide an a priori parameter choice strategy to find the convergence rates under the infinity norm. Since smoothness of the solution is not known in applied problems, we discuss an adaptive parameter choice rule to choose the regularization parameter, and then using this regularization parameter, we obtain the order of convergence in infinity norm. We give test examples to justify the theoretical estimates.

Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels

Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels

Legendre–Galerkin Methods for Third Kind VIEs and CVIEs

The main purpose of this paper is to present a spectral Legendre–Galerkin method for solving Volterra integral equations of the third kind. When the operator associated with the equivalent Volterra integral equations of second kind is compact, the resulting system produced by this spectral method is uniquely solvable and the approximate solution attains the optimal convergence order. While the related operator is noncompact, that brings a serious challenge in numerical analysis. In order to overcome this difficulty, we first decompose the original operator into three operators, one is the identity operator, the other is the contraction operator and the third one is compact. Under this decomposition, we show that the proposed method guarantees the unique solvability of the approximate equation. Moreover, we establish that the approximate solution arrives at the quasi-optimal order of global convergence. In addition, we extend this spectral method to solve the associated cordial Volterra integral equations. Finally, to confirm the theoretical results, two numerical examples are presented.

Legendre Spectral Projection Methods for Hammerstein Integral Equations with Weakly Singular Kernel

In this paper, we consider the Legendre Galerkin and Legendre collocation methods for solving the Fredholm–Hammerstein integral equation with weakly singular kernels. We evaluate the convergence rates for both the methods in both $$L^2$$ and infinity-norm. To improve the convergence rates, iterated Legendre Galerkin and iterated Legendre collocation methods have been considered. We prove that iterated Legendre Galerkin methods converge faster than Legendre Galerkin methods in both $$L^2$$ and infinity-norm. Numerical examples are presented to validate the theoretical estimate.

LEAST-SQUARES METHOD FOR THE BUBBLE STABILIZATION BY THE GAUSS-NEWTON METHOD

Abstract. In the discrete formulation of the bubble stabilized Le-gendre Galerkin methods, the system of equations includes the arti-flcial viscosity term as the parameter. We investigate the estimationof this parameter to get the least-squares solution which minimizesthe sum of the squares of errors at each node points. Some numer-ical results are reported. 1. IntroductionLet us consider the one-dimensional advection-difiusion model prob-lem deflned on the interval ⁄ = [ i 1 ; 1] as following: Lu := i”u 00 + flu 0 = f (1.1) in ⁄ u ( i 1) = u (1) = 0 ; where ” is a positive constant, fl 2 W 1 ;1 (⁄) and f 2 L 2 (⁄).In [6], Canuto and Puppo suggested the bubble stabilized LegendreGalerkin method for problem (1.1). The Legendre Galerkin method([4],[5]) using the Legendre-Gauss-Lobatto(=:LGL) points on ⁄ has the dis-crete formulation as following: Find u N 2 V N such that ” ( u 0N ;v 0N )+( flu 0 (1.2) N ;v N ) = ( f;v N ) ; for all v N 2 V NHere V N is the polynomial function space of the degree

Error estimates of spectral Legendre–Galerkin methods for the fourth-order equation in one dimension

We employ spectral Legendre–Galerkin and mixed Legendre–Galerkin approximations to solve the first bi-harmonic equation in one dimension, respectively. By orthogonal properties of Legendre polynomials, we obtain an explicit a posteriori error indicator for spectral Legendre–Galerkin methods. Furthermore, in virtue of an auxiliary variable, we present spectral mixed Legendre–Galerkin methods and study the a priori estimate and a posteriori error indicator. Especially, these indicators only depend on the expansions of the right-hand item. Numerical examples are presented to verify our theoretical analysis.

OPTIMIZATION FOR THE BUBBLE STABILIZED LEGENDRE GALERKIN METHODS BY STEEPEST DESCENT METHOD

In the discrete formulation of the bubble stabilized Legendre Galerkin methods, the system of equations includes the artificial viscosity term as the parameter. We investigate the estimation of this parameter to get the optimal solution which minimizes the maximum error. Some numerical results are reported.

Discrete Legendre spectral projection methods for Fredholm–Hammerstein integral equations

In this paper we discuss the discrete Legendre Galerkin and discrete Legendre collocation methods for Fredholm–Hammerstein integral equations with smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain optimal convergence rates for both discrete Legendre Galerkin and discrete Legendre collocation solutions in both infinity and L2-norm. Numerical examples are given to illustrate the theoretical results.

Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case

We analyze the theoretical properties of an adaptive Legendre–Galerkin method in the multidimensional case. After the recent investigations for Fourier–Galerkin methods in a periodic box and for Legendre–Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/ $$hp$$ discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in $$H^1$$ , based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type.

Legendre spectral projection methods for Urysohn integral equations

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n12−r) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n−2r) in both L2-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n−r) in both L2-norm and infinity norm, n being the highest degree of the Legendre polynomial employed in the approximation and r being the smoothness of the kernel. We are able to obtain similar superconvergence rates for the iterated Galerkin solution for Urysohn integral equations with smooth kernel as in the case of piecewise polynomial basis functions.

Contraction and optimality properties of adaptive Legendre–Galerkin methods: The one-dimensional case

As a first step towards a mathematically rigorous understanding of adaptive spectral/hp discretizations of elliptic boundary-value problems, we study the performance of adaptive Legendre–Galerkin methods in one space dimension. These methods offer unlimited approximation power only restricted by solution and data regularity. Our investigation is inspired by a similar study that we recently carried out for Fourier–Galerkin methods in a periodic box. We first consider an “ideal” algorithm, which we prove to be convergent at a fixed rate. Next we enhance its performance, consistently with the expected fast error decay of high-order methods, by activating a larger set of degrees of freedom at each iteration. We guarantee optimality (in the non-linear approximation sense) by incorporating a coarsening step. Optimality is measured in terms of certain sparsity classes of the Gevrey type, which describe a (sub-)exponential decay of the best approximation error.

Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials

It is well known that for the discretization of the biharmonic operator with spectral methods (Galerkin, tau, or collocation) we have a condition number of O ( N 8 ) , where N is the number of retained modes of approximations. This paper presents some efficient spectral algorithms, for reducing this condition number to O ( N 4 ) , based on the Jacobi–Galerkin methods for fourth-order equations in one variable. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. Jacobi–Galerkin methods for fourth-order equations in two dimension are considered. Numerical results indicate that the direct solvers presented in this paper are significantly more accurate at large N values than that based on the Chebyshev– and Legendre–Galerkin methods.

The spectral method for the flow between two concentric rotating spheres, part I: The full discrete legendre-galerkin methods

In this paper, we consider the flow between two concentric rotating spheres, and use Legendre and adjoint Legendre functions to approximate r and 0 directions, respectively, thus construct the full discrete Legendre-Galerkin spectral scheme, and derive the stability conditions and its error estimate.