Semianalytic discretization of weakly nonlinear boundary value problems with variable coefficients (Q1188349)

The author studies two-point boundary value problems of the form \(- (a(x)y')'=f(x,y(x))\) for \(x\in\Omega=(0,1)\) and \(y(0)=y(1)=0\). Weak solution formulation is used. An iteration technique close to Newton's method is studied where the linear operator and the Fréchet-derivative of the nonlinear operator are locally approximated over the subintervals of a grid on \(\Omega\). A local convergence theorem is given. Finally, an appropriate projection of the right hand-side of the iteration method is studied. The local convergence is stated and an error estimate with respect to the step size of the underlying grid is given. A numerical example is studied.