Abstract
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.
Highlights
Over the past few decades, singularly perturbed problems have attracted considerable attention in the scientific community
For reaction–diffusion problems, difficulties arise owing to the presence of boundary layers in the solution
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
Summary
Over the past few decades, singularly perturbed problems have attracted considerable attention in the scientific community. The regularity of the solution is complex This adds many difficulties to the theoretical analysis, such as in the construction of layer-adapted meshes and the estimates of various approximation errors. Better than the convergence order obtained in [9] for the convection–diffusion problem, we can establish an optimal convergence of order k + 1 for the LDG method in the energy norm through a more elaborate analysis for the two-dimensional Gauss–Radau projections on anisotropic meshes. To date, balanced-norm error estimates are only available for the Galerkin finite-element method (FEM) [23,24], mixed FEM [22], and hp-FEM [25], but not for the LDG method.
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