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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.

##### 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
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