Local Analysis of Local Discontinuous Galerkin Method for the Time-Dependent Singularly Perturbed Problem

Yao, Cheng; Zhang, Feng; Zhang, Qiang

doi:10.1007/s10915-014-9901-6

Cited by 14 publications

(6 citation statements)

References 17 publications

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“…Various parameter-uniform convergence results have been established in this way; notably, the order of convergence and error constant are independent of the singular perturbation parameters. The second approach is to use a stabilised numerical method, such as the streamline diffusion finite-element method, interior-penalty discontinuous Galerkin method, or local discontinuous Galerkin (LDG) method [3,4,10]; well-behaved local error estimates have been investigated using uniform or quasi-uniform meshes. The third approach is to combine the aforementioned stabilised numerical methods with layer-adapted meshes.…”

Section: Introductionmentioning

confidence: 99%

“…Because the LDG method shares many advantages of the DG methods and can effectively simulate the acute change of a singular solution, it is particularly suited to solving singularly perturbed problems. For example, Cheng et al performed double-optimal local error estimates for two explicit, fully discrete LDG methods on quasi-uniform meshes [3,4]. Xie et al established uniform convergence and super-convergence analyses of the LDG method on a standard Shishkin mesh [19,21,22].…”

Section: Introductionmentioning

confidence: 99%

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Local discontinuous Galerkin method on layer-adapted meshes for singularly perturbed reaction-diffusion problems in two dimensions

Mei¹,

Yao²,

Wang³

et al. 2021

Preprint

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We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction-diffusion problems. A class of layer-adapted meshes, including Shishkin-and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and "balanced"norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the "balanced" norm is artifically introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L 2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss-Radau projections. Numerical experiments are presented.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local discontinuous Galerkin method on layer-adapted meshes for singularly perturbed reaction-diffusion problems in two dimensions

Mei¹,

Yao²,

Wang³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Various parameter-uniform convergence results have been established in this way; notably, the order of convergence and error constant are independent of the singular perturbation parameters. The second approach is to use a stabilised numerical method, such as the streamline diffusion finite-element method, interior-penalty discontinuous Galerkin method, or local discontinuous Galerkin (LDG) method [6][7][8]; well-behaved local error estimates have been investigated using uniform or quasi-uniform meshes. The third approach [2,[9][10][11] is to combine the aforementioned stabilised numerical methods with layer-adapted meshes.…”

Section: Introductionmentioning

confidence: 99%

“…Because the LDG method shares many advantages of the DG methods and can effectively simulate the acute change of a singular solution, it is particularly suited to solving singularly perturbed problems. For example, Cheng et al performed double-optimal local error estimates for two explicit, fully discrete LDG methods on quasi-uniform meshes [6,7]. Xie et al established uniform convergence and super-convergence analyses of the LDG method on a standard Shishkin mesh [10,11,20].…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Mei

Wang

et al. 2021

Symmetry

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We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

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“…In [16], Xie et al demonstrated that the LDG solution does not produce any oscillations on a uniform mesh. In [2,3], Cheng et al realized a local stability analysis and proved doubleoptimal local error estimates for two explicit fully-discrete LDG methods on quasiuniform meshes. In [18], Zhu et al performed a uniform convergence analysis of the LDG method on a standard Shishkin mesh for a stationary problem.…”

Section: Introductionmentioning

confidence: 99%

Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems

Yao

Song

Mei

2020

Computational Methods in Applied Mathematics

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Local discontinuous Galerkin method is considered for time-dependent singularly perturbed semilinear problems with boundary layer. The method is equipped with a general numerical flux including two kinds of independent parameters. By virtue of the weighted estimates and suitably designed global projections, we establish optimal {(k+1)}-th error estimate in a local region at a distance of {\mathcal{O}(h\log(\frac{1}{h}))} from domain boundary. Here k is the degree of piecewise polynomials in the discontinuous finite element space and h is the maximum mesh size. Both semi-discrete LDG method and fully discrete LDG method with a third-order explicit Runge–Kutta time-marching are considered. Numerical experiments support our theoretical results.

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Local Analysis of Local Discontinuous Galerkin Method for the Time-Dependent Singularly Perturbed Problem

Cited by 14 publications

References 17 publications

Local discontinuous Galerkin method on layer-adapted meshes for singularly perturbed reaction-diffusion problems in two dimensions

Local discontinuous Galerkin method on layer-adapted meshes for singularly perturbed reaction-diffusion problems in two dimensions

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems

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