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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.

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
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