A numerical verification method for solutions of singularly perturbed problems with nonlinearity
This paper is concerned with the computer assisted proof to verify solutions of two-point singularly perturbed boundary value problems of type: \( L u \equiv - \epsilon u'' -b(x) u' + c(x) u = f(u),\) \( x \in (0,1)\), \( u(0)=u(1)=0\), where \(\epsilon\) is a small positive parameter, \(f\) is a bounded continuous non linear map, \( b(x), c(x) \in W_{\infty}^1(0,1)\) with \( c(x) \geq \gamma >0\). The proposed technique is based in previous researches of one of the authors [\textit{M. T. Nakao}, Numer. Funct. Anal. Optimization 22, No. 3--4, 321--356 (2001; Zbl 1106.65315)] and requires some a priori estimates of related linear problems. However for the above nonlinear singularly perturbed problems, standard estimates contain negative powers of the small parameter and therefore cannot be applied. In the paper under consideration, the authors derive a priori estimates for the L-spline method based on an exponential fitting with the Green's function, that are not badly affected by the small parameter. Such estimates allow accurate verification of solutions of the above problems. Finally, the results of two numerical examples are presented to show the effectiveness of the proposed verification results.
nonlinear two point boundary value problems
verification of finite element solutions
scientific article; zbMATH DE number 2162990
scientific article