Uniform norm error estimate for rectangular finite element approximation of a 2D turning point problem

This work presents error analysis for a finite element method applied to a two-dimensional singularly perturbed convection-diffusion turning point problem. Utilizing a layer-adapted Shishkin mesh, we prove uniform convergence in the maximum norm in the x-layer regions and $\varepsilon$-independent bounds for the coarse region. The analysis, critically based on the properties of a discrete Green’s function, guarantees the method’s robustness and accuracy in capturing sharp solution layers.

💡 Research Summary

The paper addresses the numerical solution of a two‑dimensional singularly perturbed convection‑diffusion problem that possesses an interior turning point along the line x = 0. The governing equation is
    −ε Δu + x a(x,y) uₓ + c(x,y) u = f(x,y) in Ω = (−1,1)×(−1,1),
with homogeneous Dirichlet boundary conditions. The convection coefficient vanishes at x = 0, creating an interior layer, while exponential boundary layers appear near y = ±1. The small diffusion parameter ε ≪ 1 makes the problem stiff and demands a mesh that resolves these layers.

Main contributions

1. Shishkin layer‑adapted mesh – The authors construct a tensor‑product Shishkin mesh with separate transition parameters in the x‑ and y‑directions:
    λₓ = min{2 ε^{α} log(1/ε), ½}, λᵧ = min{2 r ε^{β} log(1/ε)^{3/2}, ¼}.
   These parameters depend on the layer strengths α and β and on ε. The mesh is refined in the sub‑intervals