Iterated discrete polynomially based Galerkin methods.
The authors consider a polynomially based iterated discrete Galerkin method for the solution of operator equations \(u-Ku=f\) and for an eigenvalue problem associated with an integral operator \((Kx)(s)=\int_{-1}^1 k(s,t)x(t)\,dm(t)\), \(s\in[-1,1]\), with a smooth kernel, \(k(\cdot,\cdot)\in C([-1,1]\times[-1,1])\). It is proved that the approximating operators in the iterated discrete Galerkin method converge to the integral operator in norm, in an appropriately chosen space. The error in the infinity norm, both for the approximation of the operator equation and of the spectral subspace, is of the order of \(n^{-r}\), where \(n\) is the degree of the polynomial approximation and \(r\) is the smoothness of the kernel. Thus the iterated discrete Galerkin solution improves upon the discrete Galerkin solution, which was shown to be of order \(n^{-r+1}\) by \textit{M. Golberg} and \textit{H. Bowman} [Appl. Math. Comput. 96, 237--271 (1998; Zbl 0944.65142)]. The authors also give a shorter proof of the result by Golberg and Bowman which states that \(\| u-v_n\|_2\) in discrete Galerkin method is of the order of \(n^{-r}\).
scientific article; zbMATH DE number 2009026
scientific article