A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

Baccouch, Mahboub

doi:10.1016/j.camwa.2014.08.023

Cited by 30 publications

(9 citation statements)

References 62 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The above LDG method is called the md-LDG method. It is studied in References [23,28,[37][38][39][40][41]. We remark that in the md-LDG method, the stabilization parameters associated with the numerical trace of the flux are identically equal to zero for all interior edges; this is why its dissipation is said to be minimal.…”

Section: Definition Of the Ldg Methodsmentioning

confidence: 99%

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

Baccouch

2020

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

In this paper, we study the local discontinuous Galerkin (LDG) methods for two-dimensional nonlinear second-order elliptic problems of the type u xx + u yy = f (x, y, u, u x , u y), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L 2 norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss-Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1, and for both mixed Dirichlet-Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal. KEYWORDS a priori error estimates, local discontinuous Galerkin method, nonlinear second-order elliptic boundary-value problems, supercloseness Dedicated to Professor Slimane Adjerid on the occasion of his 65th birthday.

show abstract

Section: Definition Of the Ldg Methodsmentioning

confidence: 99%

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

Baccouch

2020

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

“…Since then, LDG schemes have been successfully applied to hyperbolic, elliptic, and parabolic partial differential equations [2-4, 6, 7, 14, 16-18, 25-28, 33, 34, 37, 38], to mention a few. A review of the LDG methods is given in [10,15,16,[22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension

Baccouch¹

2016

JCM

Self Cite

View full text Add to dashboard Cite

In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. 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. These results are used 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. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L 2-norm at O(h p+2) rate. Finally, we prove that the global effectivity indices in the L 2-norm converge to unity at O(h) rate. 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 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.

show abstract

“…More recently, the author constructed and analyzed new superconvergence‐based a posteriori error estimates for the LDG methods in for convection‐diffusion equations, for the KdV equation, and for the fourth‐order Euler–Bernoulli equations. The first superconvergence‐based a posteriori LDG error estimates for two‐dimensional convection‐diffusion and wave problems on Cartesian grids were developed in for convection‐diffusions problems and for the wave equation. These a posteriori error estimates are computationally simple and are obtained by solving local steady problems with no boundary conditions on each element.…”

Section: Introductionmentioning

confidence: 99%

“…The theoretical analysis of superconvergence in higher dimensions and for the general case is a subject of ongoing research. Superconvergence properties of the LDG method applied to wave equation on rectangular meshes are reported in . The generalization to nonlinear equations and to two space dimensions on triangular meshes involve several technical difficulties and will be investigated in the future.…”

Section: Introductionmentioning

confidence: 99%

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

Baccouch

2014

Numer. Methods Partial Differential Eq.

Self Cite

View full text Add to dashboard Cite

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 L 2 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 L 2 -norm under mesh refinement. The order of convergence is proved to be p + 3/2, when p-degree piecewise polynomials with p ≥ 1 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 O(h p+3/2 ) superconvergent solutions. Our computational results show higher O(h p+2 ) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L 2 -norm converge to unity at O(h 1/2 ) rate while numerically they exhibit O(h 2 ) and O(h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

show abstract

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

Cited by 30 publications

References 62 publications

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension

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

Contact Info

Product

Resources

About