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Analysis Of Optimal Superconvergence Of A Local Discontinuous Galerkin Method For Nonlinear Second-Order Two-Point Boundary-Value Problems, Mahboub Baccouch
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
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
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(hp+1) and O(hp) 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(h2p) 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
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 …
Asymptotically Exact Local Discontinuous Galerkin Error Estimates For The Linearized Korteweg-De Vries Equation In One Space Dimension, Mahboub Baccouch
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(hk+2) superconvergent at the roots of (k + 1)-degree Radau polynomials. Computational results indicate …