The paper considers a singularly perturbed reaction-diffusion problem posed on the unit square in R^2. Typical solutions of this class of problems exhibit sharp boundary layers along the sides of the domain, which can cause difficulties for numerical methods.
The authors propose an LDG method that handles the boundary layers by using a Shishkin mesh and introducing a new "layer-upwind flux" in the discretization. This flux is chosen on the fine mesh (inside the boundary layers) in the direction where the layer weakens, while a standard central flux is used on the coarse mesh.
The main contributions are:
Derivation of an optimal-order error analysis in a balanced norm, which is a stronger norm than the usual energy norm and is a more appropriate measure for errors in computed solutions of singularly perturbed problems.
Proof of O((N^-1 ln N)^(k+1)) convergence in the balanced norm for even values of the polynomial degree k, and the same rate for odd k under a mild condition on the mesh size and perturbation parameter.
Introduction of a new layer-upwind flux in the LDG method, which avoids the need for penalty terms and leads to the optimal-order balanced-norm error bounds.
Construction of a sophisticated combination of local Gauss-Radau and L2 projectors, which play a crucial role in the error analysis and enable the extension of the 1D results to the 2D case.
The error analysis is highly technical but the authors demonstrate the sharpness of the theoretical results through numerical experiments.
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