Stability of a streamline diffusion finite element method for turning point problems
In this paper, the numerical solution of one-dimensional singularly perturbed problems of the type \[ - \varepsilon u'' - b(x) u' = f(x),\;x \in (0,1),\;u(0)= u(1)=0 \] by means of finite difference methods is considered. Here, \( 0 << \varepsilon <<1\) and \( b(0)=0\), \( b(x)>0\), \( x \in [0,1]\). By introducing a uniform grid in \([0,1]\), denoted by \( x_j = j h \), \( j=0,\ldots ,N\), \( N h = 1 \) and the finite element space \( V^h \) of piecewise linear functions \( \varphi_i, i=1, \ldots N\) at the vertex \(x_i\), the authors prove that the numerical approximation \( u_h\) satisfies a bound of the type \[ \| u - u_h \|_ {\infty} \leq C | \log \varepsilon | \inf \{ \| u - v_h \|_ {\infty}, v_h \in V^h \} \] where \(C\) is independent of \( \varepsilon\). This stability result is applied to derive the almost \(\varepsilon\)-uniform second order accuracy of the layer adapted grids of Shishkin and Bakhvalov type for \( b(x)= x^p\), \( f(x)= x^p g(x)\), \( g \in C^1 \left( [0,1] \right)\) [\textit{G. I. Shishkin}, Grid approximation of singularly perturbed elliptic and parabolic equations. Ekaterinburg: Russian Academy of Sciences, Ural Branch; Moscow: Keldysh Institute of Applied Mathematics, RAS (Habilitation 1991). 232~p. (1992; Zbl 1397.65005); \textit{N. S. Bakhalov}, Towards optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl. Mater. Mater. Fiz 9, 841--859 (1969)].
Singularly perturbed second order problems
Uniform stability of turning point problems
scientific article; zbMATH DE number 5315430
scientific article