Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

Guzmán, Johnny

doi:10.1090/s0025-5718-08-02067-x

Cited by 10 publications

(4 citation statements)

References 30 publications

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“…They used dyadic decomposition of the domain and require local energy estimates together with sharp pointwise estimates for the corresponding components of the Green's matrix. For smooth domains this technique was successfully used in [7] for mixed methods, in [17,20] for discontinuous Galerkin (DG) methods and in [6] for local DG methods. This technique was also applied for the Stokes equations, see Guzmán and Leykekhman [18].…”

Section: Introductionmentioning

confidence: 99%

Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D

Chen,

Monk,

Zhang

2021

Preprint

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In [SIAM J. Numer. Anal., 59 (2), 720-745], we proved quasi-optimal L ∞ estimates (up to logarithmic factors) for the solution of Poisson's equation by a hybridizable discontinuous Galerkin (HDG) method. However, the estimates only work in 2D. In this paper, we use the approach in [Numer. Math., 131 (2015), pp. 771-822] and obtain sharp (without logarithmic factors) L ∞ estimates for the HDG method in both 2D and 3D. Numerical experiments are presented to confirm our theoretical result.

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Section: Introductionmentioning

confidence: 99%

Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D

Chen,

Monk,

Zhang

2021

Preprint

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“…Because it has many advantages, such as high order accuracy, flexibility for mesh refinement, localizability, stability, parallelizability and less numerical diffusion/dipersion, the DG method has been widely extended and used to solve many partial differential equations. For solving the nonlinear systems of the general conservation laws, the total variation bounded (TVB) RungeKutta local projection discontinuous Galerkin (RKDG) method with high order of accuracy is developed [4−9] Another popular DG method is the local discontinuous Galerkin method, see [10][11][12][13][14][15][16][17] and references therein. Additionally, many other DG methods, such as generalized DG (GDG) method [18−19] , coupled DG method [20] , DG finite volume element method [21] , EulerianLagrangian DG method [22] , characteristic DG method [23] and compact DG method [24] , have been developed.…”

Section: Introductionmentioning

confidence: 99%

Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient

Lin

2010

J Syst Sci Complex

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This paper applies bilinear immersed finite elements (IFEs) in the interior penalty discontinuous Galerkin (DG) methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bilinear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.

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“…Recently, Burman et al [3] used weighted norms for a continuous interior penalty method and obtained a local error estimate for singularly perturbed problems. Also, Chen [4] and Guzman [7] provided pointwise error estimates of a conforming mixed method and discontinuous Galerkin method for the Stokes equation using weighted norm respectively. For other weighted norm error estimates, we refer to [6,11,[13][14][15] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Pointwise error estimates for a stabilized Galerkin method: non-selfadjoint problems

2010

Bit Numer Math

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Highly localized pointwise error estimates for a stabilized Galerkin method are provided for second-order non-selfadjoint elliptic partial differential equations. The estimates show a local dependence of the error on the derivative of the solution u and weak dependence on the global norm. The results in this paper are an extension of the previous pointwise error estimates for the self-adjoint problems. In order to provide pointwise error estimates in the presence of the first-order term in the differential equations, we prove that the stabilized Galerkin solution is higher order perturbation to the Ritz projection of the true solutions. Then, we proceed to obtain pointwise estimates using the so-called discrete Green's function. Application to error expansion inequalities and a posteriori error estimators are briefly discussed.

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Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

Cited by 10 publications

References 30 publications

Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D

Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D

Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient

Pointwise error estimates for a stabilized Galerkin method: non-selfadjoint problems

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