×

zbMATH — the first resource for mathematics

A superconsistent Chebyshev collocation method for second-order differential operators. (English) Zbl 0991.65071
Summary: A standard way to approximate the model problem \(-u''= f\), with \(u(\pm 1)= 0\), is to collocate the differential equation at the zeros of \(T_n':x_i\), \(i= 1,\dots, n-1\), having denoted by \(T_n\) the \(n\)th Chebyshev polynomial. We introduce an alternative set of collocation nodes \(z_i\), \(i= 1,\dots, n-1\), which will provide better numerical performances. The approximated solution is still computed at the nodes \(\{x_i\}\), but the equation is required to be satisfied at the new nodes \(\{z_i\}\) which are determined by asking an extra degree of consistency in the discretization of the differential operator.

MSC:
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI