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Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems. (English) Zbl 1019.65091
This paper deals with the standard finite element method combined with a Shishkin mesh strategy for a convection-diffusion problem \[ -\varepsilon\Delta u+ \vec\beta\nabla u+ cu= f\text{ in }\Omega= (0,1)\times (0, 1),\;u= 0\text{ on }\partial\Omega, \] where \(\varepsilon\) is a small positive number. Superconvergence in a discrete \(\varepsilon\)-weighted energy norm in the presence of exponential boundary layers is analyzed. As a consequence of the superconvergence result, the author obtains convergence of the same order in the \(L^2\)-norm and pointwise convergence of order \(N^{-3/2} \ln^{5/2} N+\varepsilon N^{-1}\ln^{1/2}N\) at some mesh points inside the boundary layer under the same regularity assumption. Some numerical results are presented.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
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