An analysis of a conforming exponentially fitted finite element method for a convection-diffusion problem
This very interesting and important paper is devoted to theory of projective-grid methods for the stationary linear convection-diffusion equation in a polygonal region \(\Omega\) under the standard Dirichlet conditions. The equation is written as \(-\nabla\cdot [\varepsilon\nabla u-u\vec au]+gu=f\) and it is assumed that: \(\vec a\in [C^1(\overline\Omega)]^2\), \(\nabla\vec a +2g\geq 0 \), \(g\in L^{\infty}(\Omega)\), \(f\in L_2(\Omega)\), and that the solution \(u\in H^2(\Omega)\), Then \(\|u\|_1=O(\varepsilon^{-1/2})\) and \(\|u\|_2=O(\varepsilon^{-3/2})\). To obtain effective approximations (in the presence of the boundary layer) special projective-grid methods (the Galerkin finite element methods) are used on the base of triangulations \(T_h(\overline \Omega)\). The approximating subspaces instead of the standard piecewise linear splines make use of special more involved functions on the triangular cells via exponential fitting in dependence of the given vector \(\vec a\). In the case of regular triangulations \(T_h(\overline \Omega)\) estimates of errors in the energy norm are given of the form \(\|u-u_h\|_E=O(h\varepsilon^{-1/2})\). In case of the unit square and \(\vec a\equiv[a_1,a_2]\) with \(a_i\asymp 1\) and a layer-fitted grids with refinement, the authors obtained remarkable estimates of the type \(\|u-u_h\|_E=O(h^{1/2}[1+h^{1/2}|\ln \varepsilon])\) almost independent of the singular parameter. The number of the triangles is \(O(h^{-2})\).
scientific article; zbMATH DE number 1787642