The local discontinuous Galerkin method for a singularly perturbed convection–diffusion problem with characteristic and exponential layers

A singularly perturbed convection–diffusion problem, posed on the unit square in $${\mathbb {R}}^2$$ , is studied; its solution has both exponential and characteristic boundary layers. The problem solved numerically using local discontinuous Galerkin (LDG) method Shishkin meshes. Using tensor-product piecewise polynomials of degree at most $$k>0$$ each variable, error between LDG true proved to converge, uniformly singular perturbation parameter, a rate $$O\left( \left( N^{-1}\ln N\right) ^{k+1/2}\right) $$ an associated energy norm, where N number mesh intervals coordinate direction. (This first uniform convergence result for applied with layers.) Furthermore, we prove that this order increases ^{k+1}\right) when one measures energy-norm difference Gauss-Radau projection into finite element space. This supercloseness property implies optimal $$L^2$$ estimate $$\left( ^{k+1}$$ our method. Numerical experiments show sharpness theoretical results.