Global Effectivity Indices Research Articles

A posteriori error analysis of the local discontinuous Galerkin method for the sine–Gordon equation in one space dimension

In this paper, we present asymptotically exact a posteriori error estimators of the local discontinuous Galerkin (LDG) method for the nonlinear sine–Gordon equation in one space dimension. We apply our recent optimal L2 error estimates and superconvergence results (Baccouch, 2018) to show that the LDG errors on each element can be split into two parts. The first part is proportional to the (p+1)-degree Radau polynomial and the second part converges with order p+3∕2 in the L2-norm, when piecewise polynomials of degree at most p are used. This new result is used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. We further prove that, for smooth solutions, these a posteriori error estimates for the solution and its derivative, at a fixed time, converge to the exact spatial errors in the L2-norm under mesh refinement. The theoretical order of convergence is proved to be p+3∕2 while our computational results show higher O(hp+2) convergence rate. Finally, we prove that the global effectivity index converges to unity at O(h1∕2) rate. Several numerical results are presented to validate the theoretical results.

Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems

In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the $$L^2$$ -norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve $$p+1$$ order of convergence for the solution and its spatial derivative in the $$L^2$$ -norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order $$p+1$$ towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order $$p+3/2$$ towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the $$(p+1)$$ -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the $$L^2$$ -norm at $$\mathcal {O}(h^{p+3/2})$$ rate. Finally, we prove that the global effectivity index in the $$L^2$$ -norm converge to unity at $$\mathcal {O}(h^{1/2})$$ rate. Our proofs are valid for arbitrary regular meshes using $$P^p$$ polynomials with $$p\ge 1$$ . Finally, several numerical examples are given to validate the theoretical results.

A posteriori local discontinuous Galerkin error estimates for the one-dimensional sine-Gordon equation

ABSTRACTIn this paper, we present a posteriori error analysis of the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equations with smooth solutions. We show that the dominant components of the local LDG errors on each element are proportional to right and left Radau polynomials of degree p+1. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are superconvergent at the roots of -degree right and left Radau polynomials, respectively. Numerical experiments indicate that our superconvergence results hold globally. We use the superconvergence results to construct simple, efficient, and asymptotically exact a posteriori LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving local steady problems with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative, at any fixed time, converge to the true errors at rate, respectively. We also demonstrate that the global effectivity indices for the solution and its derivative in the -norm converge to unity. We present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimates.

Analysis of a posteriori 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 piecewise 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.

A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids

In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651---1671, 2015). We first prove that the DG solution converges in the $$L^2$$L2-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be $$p+2$$p+2, when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two $$(p+1)$$(p+1)-degree right Radau polynomials in the x and y directions. The less significant part converges to zero at $${\mathcal {O}}\left( h^{p+2}\right) $$Ohp+2. These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the $$L^2$$L2-norm at $${\mathcal {O}}\left( h^{p+2}\right) $$Ohp+2 rate. Finally, we prove that the global effectivity indices in the $$L^2$$L2-norm converge to unity at $${\mathcal {O}}(h)$$O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.

Superconvergence and a posteriori error estimates of the DG method for scalar hyperbolic problems on Cartesian grids

In this paper, we analyze the discontinuous Galerkin (DG) finite element method for the steady two-dimensional transport-reaction equation on Cartesian grids. We prove the L2 stability and optimal L2 error estimates for the DG scheme. We identify a special numerical flux for which the L2-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. We further prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. We also provide a very simple derivative recovery formula which is O(hp+1) superconvergent approximation to the directional derivative. Moreover, we establish an O(h2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element. Finally, we prove that the proposed a posteriori DG error estimates converge to the true errors in the L2-norm at O(hp+1) rate and that the global effectivity index converges to unity at O(h) rate. Our results are valid without the flow condition restrictions. We perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal.

Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation

In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L2 error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862–901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, 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 , when p-degree piecewise polynomials with are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and superconvergent solutions. Our computational results show higher convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L2-norm converge to unity at rate while numerically they exhibit and rates, respectively. Numerical experiments are shown to validate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461–1491, 2015

A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws

In this paper, new a posteriori error estimates for a discontinuous Galerkin (DG) formulation applied to nonlinear scalar conservation laws in one space dimension are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local problem with no boundary condition on each element of the mesh. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree Radau polynomial, when p-degree piecewise polynomials with p≥1 are used. This result allows us 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 p+5/4. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h1/2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

Superconvergence and a posteriori error estimates for the LDG method for convection–diffusion problems in one space dimension

In this paper we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to transient convection–diffusion problems in one space dimension. We show that the leading terms of the local discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are O(hp+2) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. The superconvergence results are used to construct asymptotically correct a posteriori error estimates. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at O(hp+3) and O(hp+2) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the L2-norm converge to unity at O(h2) and O(h) rates, respectively. Finally, we show that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hp+2) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1, and for periodic, Dirichlet, and mixed Dirichlet–Neumann boundary conditions. Several numerical simulations are performed to validate the theory.

Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection–diffusion problems

In this paper, new a posteriori error estimates for the local discontinuous Galerkin (LDG) formulation applied to transient convection–diffusion problems in one space dimension are presented and analyzed. These error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree right Radau polynomial while the leading error term for the solution’s derivative is proportional to a (p+1)-degree left Radau polynomial, when polynomials of degree at most p are used. These results are used 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. More precisely, we prove that our LDG error estimates converge to the true spatial errors at O(hp+5/4) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h1/2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part II: A Posteriori Error Estimation

In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler---Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to $$(k+1)$$ ( k + 1 ) -degree Radau polynomials, when polynomials of total degree not exceeding $$k$$ k are used. These results allow us to prove that the $$k$$ k -degree LDG solution and its derivatives are $$\mathcal O (h^{k+3/2})$$ O ( h k + 3 / 2 ) superconvergent at the roots of $$(k+1)$$ ( k + 1 ) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time $$t$$ t converge to the true errors at $$\mathcal O (h^{k+5/4})$$ O ( h k + 5 / 4 ) rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the $$L^2$$ L 2 -norm converge to unity at $$\mathcal O (h^{1/2})$$ O ( h 1 / 2 ) rate. Our proofs are valid for arbitrary regular meshes and for $$P^k$$ P k polynomials with $$k\ge 1$$ k ? 1 , and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.

A posteriori error estimation and adaptive finite element computation of the Helmholtz equation in exterior domains

This paper presents adaptive finite element methods for the Helmholtz equation in exterior domains, with application to time-harmonic acoustics problems. Particular focus is given on the a posteriori error estimator, which is an explicit function of residuals and provides an upper bound on global L 2-norm of the error. An h-adaptive strategy, which does not require knowledge of the scaling constant appearing in the error estimator, is described. Numerical results for a two-dimensional model problem show that significant problem size reduction can be obtained through this adaptive approach. An algorithm is presented to compute the scaling constant, which leads to computable error estimates. The accuracy of the error estimates, indicated by the global effectivity index, varies with the wave number.

Explicit residual-based a posteriori error estimation for finite element discretizations of the Helmholtz equation: Computation of the constant and new measures of error estimator quality

This paper continues the study of an explicit residual-based a posteriori error estimator developed for finite element discretizations of the Helmholtz equation, in the context of time-harmonic exterior acoustics problems. In previous papers [1–3] the error estimator was derived and subsequently applied to h-adaptive computations of model problems in two dimensions. In the present paper, a methodology is established for computing error estimates; these error estimates are then analyzed in detail. The error estimates are made possible by computing the (scaling) constant, which exists in error estimators of this type. In our case, this constant is difficult to obtain. An algorithm for computing the constant is described. (It is noted that the error indicators used for adaptive computations do not require knowledge of this constant.) Several measures are used to analyze the quality of the error estimates, providing a complete description of the error estimator in the context of these problems. We compute global effectivity indices, and utilize the local (element-wise) effectivity index in computing three additional measures. The first of these is the robustness index. The others are the modified robustness index and the distribution index, which are introduced for the first time. It is found that the global effectivity index varies with problem parameters, in particular the wave number. Robustness indices also vary due to the local error definition and wave number. Values of the distribution index are much better, however, indicating the suitability of the error estimator in guiding efficient adaptive mesh refinement. The analysis of the error estimator is carried out on relatively coarse meshes, typical of those used in practical engineering computations.