Physical Sciences and Mathematics Commons^™

Analysis Of Optimal Superconvergence Of A Local Discontinuous Galerkin Method For Nonlinear Second-Order Two-Point Boundary-Value Problems, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1toward the derivatives of Gauss-Radau projections of …

An Optimal A Posteriori Error Estimates Of The Local Discontinuous Galerkin Method For The Second-Order Wave Equation In One Space Dimension, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L 2 -norm to (p + 1)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p + …

Analysis Of Optimal Error Estimates And Superconvergence Of The Discontinuous Galerkin Method For Convection-Diffusion Problems In One Space Dimension, Mahboub Baccouch, Helmi Temimi

Mathematics Faculty Publications

In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(h^p+1) and O(h^p) convergence rates in the L 2 -norm, respectively, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the p-degree DG solution and its derivative are O(h^2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG …

The Discontinuous Galerkin Finite Element Method For Ordinary Differential Equations, Mahboub Baccouch

Mathematics Faculty Publications

We present an analysis of the discontinuous Galerkin (DG) finite element method for nonlinear ordinary differential equations (ODEs). We prove that the DG solution is $(p + 1) $th order convergent in the $L^2$-norm, when the space of piecewise polynomials of degree $p$ is used. A $ (2p+1) $th order superconvergence rate of the DG approximation at the downwind point of each element is obtained under quasi-uniform meshes. Moreover, we prove that the DG solution is superconvergent with order $p+2$ to a particular projection of the exact solution. The superconvergence results are used to show that the leading term of …

Order-Preserving Derivative Approximation With Periodic Radial Basis Functions, Edward Fuselier, Grady B. Wright

Mathematics Faculty Publications and Presentations

In this exploratory paper we study the convergence rates of an iterated method for approximating derivatives of periodic functions using radial basis function (RBF) interpolation. Given a target function sampled on some node set, an approximation of the m ^th derivative is obtained by m successive applications of the operator “interpolate, then differentiate”- this process is known in the spline community as successive splines or iterated splines. For uniformly spaced nodes on the circle, we give a sufficient condition on the RBF kernel to guarantee that, when the error is measured only at the nodes, this iterated method approximates …

Recovery Techniques For Finite Element Methods And Their Applications, Hailong Guo

Wayne State University Dissertations

Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery.

Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based …

Asymptotically Exact Local Discontinuous Galerkin Error Estimates For The Linearized Korteweg-De Vries Equation In One Space Dimension, Mahboub Baccouch

Mathematics Faculty Publications

We present and analyze a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the linearized Korteweg-de Vries (KdV) equation in one space dimension. These estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We extend the work of Hufford and Xing [J. Comput. Appl. Math., 255 (2014), pp. 441-455] to prove new superconvergence results towards particular projections of the exact solutions for the two auxiliary variables in the LDG method that approximate the first and second derivatives of the solution. The order of convergence …

Superconvergence And A Posteriori Error Estimates Of A Local Discontinuous Galerkin Method For The Fourth-Order Initial-Boundary Value Problems Arising In Beam Theory, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k + 1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h^k+2) superconvergent at the roots of (k + 1)-degree Radau polynomials. Computational results indicate …

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang

Mathematics Research Reports

We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two dimensional Poisson equation. Results of this paper are two folds. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered.gradient by the Polynomial Preserving Recovery (PPR) is superconvergent. Secondly, we demonstrate by numerical examples that an adaptive procedure with a posteriori error estimator based on PPR does produce adaptive meshes satisfy our mesh density assumption, and the recovered gradient by PPR is indeed supercoveregent in the adaptive process.

Superconvergence Of Iterated Solutions For Linear And Nonlinear Integral Equations: Wavelet Applications, Boriboon Novaprateep

Mathematics & Statistics Theses & Dissertations

In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern …

Validation Of The A Posteriori Error Estimator Based On Polynomial Preserving Recovery For Linear Elements, Zhimin Zhang, Ahmed Naga

Mathematics Research Reports

In this paper the quality of the error estimator based on the Polynomial Preserving Recovery (PPR) is investigated using the computer-based approach proposed by Babiiska et al. A comparison is made between the error estimator based on the PPR and the one based on the Superconvergence Patch Recovery (SPR). It was found that the PPR is at least as good as the SPR.

Gradient Recovery And A Posteriori Estimate For Bilinear Element On Irregular Quadrilateral Meshes, Zhimin Zhang

Mathematics Research Reports

A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under general quadrilateral meshes. It has been proven that the recovered gradient converges at a rate O(h^{1+rho) for rho = min(alpha, 1) when the mesh is distorted O(h1+alpha) (alpha > 0) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.}

Analysis Of Recovery Type A Posteriori Error Estimators For Mildly Structured Grids, Jinchao Xu, Zhimin Zhang

Mathematics Research Reports

Some recovery type error estimators for linear finite element method are analyzed under O(h^{1+alpha) (alpha greater than 0) regular grids. Superconvergence is established for recovered gradients by three different methods when solving general non-self-adjoint second-order elliptic equations. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.}

A Meshless Gradient Recovery Method Part I: Superconvergence Property, Zhiming Zhang, Ahmed Naga

Mathematics Research Reports

A new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz-Zhu patch recovery method. In addition, under uniform triangular meshes, the method is superconvergent for the Chevron pattern, and ultraconvergence at element edge centers for the regular pattern.

Sharp L2-Error Estimates And Superconvergence Of Mixed Finite Element Methods For Non-Fickian Flows In Porous Media, Richard E. Ewing, Yanping Lin, Tong Sun, Junping Wang, Shuhua Zhang

Mathematics and Statistics Faculty Publications

No abstract provided.

Ultraconvergence Of Zz Patch Recovery At Mesh Symmetry Points, Zhimin Zhang, Runchang Lin

Mathematics Research Reports

Ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed. Superconvergence recovery for the Q8 element is proved and ultraconvergence numerical examples are demonstrated.

Superconvergence In Iterated Solutions Of Integral Equations, Peter A. Padilla

Mathematics & Statistics Theses & Dissertations

In this thesis, we investigate the superconvergence phenomenon of the iterated numerical solutions for the Fredholm integral equations of the second kind as well as a class of nonliner Hammerstein equations. The term superconvergence was first described in the early 70s in connection with the solution of two-point boundary value problems and other related partial differential equations. Superconvergence in this context was understood to mean that the order of convergence of the numerical solutions arising from the Galerkin as well as the collocation method is higher at the knots than we might expect from the numerical solutions that are obtained …

Superconvergence Of The Iterated Collocation Methods For Hammerstein Equations, Hideaki Kaneko, Richard D. Noren, Peter A. Padilla

Mathematics & Statistics Faculty Publications

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18].

Superconvergence Of The Iterated Galerkin Methods For Hammerstein Equations, Hideaki Kaneko, Yuesheng Xu

Mathematics & Statistics Faculty Publications

In this paper, the well-known iterated Galerkin method and iterated Galerkin-Kantorovich regularization method for approximating the solution of Fredholm integral equations of the second kind are generalized to Hammerstein equations with smooth and weakly singular kernels. The order of convergence of the Galerkin method and those of superconvergence of the iterated methods are analyzed. Numerical examples are presented to illustrate the superconvergence of the iterated Galerkin approximation for Hammerstein equations with weakly singular kernels. © 1996, Society for Industrial and Applied Mathematics

Numerical Solutions For Weakly Singular Hammerstein Equations And Their Superconvergence, Hideaki Kaneko, Richard D. Noren, Yuesheng Xu

Mathematics & Statistics Faculty Publications

In the recent paper [7], it was shown that the solutions of weakly singular Hammerstein equations satisfy certain regularity properties. Using this result, the optimal convergence rate of a standard piecewise polynomial collocation method and that of the recently proposed collocationtype method of Kumar and Sloan [10] are obtained. Superconvergence of both of these methods are also presented. In the final section, we discuss briefly a standard productintegration method for weakly singular Hammerstein equations and indicate its superconvergence property. © 1992 Rocky Mountain Mathematics Consortium.