Global convergence of \u3ci\u3ea posteriori\u3c/i\u3e error estimates for a discontinuous Galerkin method for one-dimensional linear hyperbolic problems

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2 -norm under mesh refinement. The order of convergence is proved to be k + 2, when k-degree piecewise polynomials with k ≥ 1 are used. As a consequence, we prove that the DG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hk+2) superconvergent solutions. We perform numerical experiments to demonstrate that the rate of convergence is optimal. We further prove that the global effectivity indices in the L2 -norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using Pk polynomials with k ≥ 1 and for both the periodic boundary condition and the initial-boundary value problem. Several numerical simulations are performed to validate the theory

Superconvergence and \u3ci\u3ea posteriori\u3c/i\u3e error estimates of a local discontinuous Galerkin method for the fourth-order initial-boundary value problems arising in beam theory

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 that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for Pk polynomials with k ≥ 1, and for various types of boundary conditions

The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations

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 the DG error is proportional to the $(p + 1)$-degree right Radau polynomial. These results allow us to develop a residual-based a posteriori error estimator which is computationally simple, efficient, and asymptotically exact. The proposed a posteriori error estimator is proved to converge to the actual error in the $L^2$-norm with order $p+2$. Computational results indicate that the theoretical orders of convergence are optimal. Finally, a local adaptive mesh refinement procedure that makes use of our local a posteriori error estimate is also presented. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement

Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension

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 method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used

Asymptotically exact local discontinuous Galerkin error estimates for the linearized Korteweg-de Vries equation in one space dimension

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 is proved to be k + 3/2, when polynomials of total degree not exceeding k are used. These results allow us to prove that the significant parts of the spatial discretization errors for the LDG solution and its spatial derivatives (up to second order) are proportional to (k + 1)-degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates and prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives, at a fixed time t, converge to the true errors at O(hk+3/2) rate in the L2 -norm. Finally, we prove that the global effectivity indices, for the solution and its spatial derivatives, converge to unity at O (h1/2) rate. Numerical results are presented to validate the theory

Analysis of \u3ci\u3ea posteriori\u3c/i\u3e error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations

We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal O(hp+1) convergence rate in the L2-norm when p-degree piece-wise polynomials with p ≥1 are used. We further prove that the DG solution is O(h2p+1) superconvergent at the downwind points. We use these results to prove that the p-degree DG solution is O(hp+2) super close to a particular projection of the exact solution. This superconvergence result allows us to show that the true error can be divided into a significant part and a less significant part. The significant part of the discretization error for the DG solution is proportional to the (p +1)-degree right Radau polynomial and the less significant part converges at O(hp+2) rate in the L2-norm. Numerical experiments demonstrate that the theoretical rates are optimal. Based on the global superconvergent approximations, we construct asymptotically exact a posteriori error estimates and prove that they converge to the true errors in the L2-norm under mesh refinement. The order of convergence is proved to be p +2. Finally, we prove that the global effectivity index in the L2-norm converges to unity at O(h)rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement. A local adaptive procedure that makes use of our local a posteriori error estimate is also presented

AN OPTIMAL A POSTERIORI ERROR ESTIMATES OF THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE SECOND-ORDER WAVE EQUATION IN ONE SPACE DIMENSION

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 + 2, when piecewise polynomials of degree at most p are used. We use these results to show that the leading error terms on each element for the solution and its derivative are proportional to (p + 1)-degree right and left Radau polynomials. These new results enable us to construct residual-based a posteriori error estimates of the spatial errors. We further prove that, for smooth solutions, these a posteriori LDG error estimates converge, at a fixed time, to the true spatial errors in the L 2 -norm at O(h p+2) rate. Finally, we show that the global effectivity indices in the L 2 -norm converge to unity at O(h) rate. The current results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be p+ 3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using P p polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results

A posteriori error control for discontinuous Galerkin methods for parabolic problems

We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which requires the derivation of novel a posteriori error bounds. For the analysis of the time-dependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure