The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system
The numerical solution for singularly perturbed initial boundary value problem of the system of nonlinear differential equations of the form \[ \begin{cases} \varepsilon u_1'+f_1(x,u_1,u_2)=0,&0<x\leq\ell\\ -u_2''+f_2(x,u_1,u_2)=0,&0<x<\ell\\ u_1(0)=u_*,\\ u_2(0)=A,\;u_2(\ell)=B\end{cases} \tag{\(*\)} \] is obtained, where \(0<\varepsilon<1\) is a small parameter, \(u_*\), \(A, B\) are given constants, \(f_1\) and \(f_2\) are given smooth functions satisfying certain regularity conditions. A method based on finite elements with piecewise constant and piecewise linear basis functions and appropriate quadrature formulae with remainder term in integral form is derived. It is shown that the solution of the finite difference scheme on a special non-uniform mesh converges point-wise independently of the singular perturbation parameter. Numerical results supporting the theory developed are presented.
scientific article; zbMATH DE number 2137733