Corrected collocation methods for periodic pseudodifferential equations
The paper presents refinements of approximate solutions to pseudodifferential equations on a smooth curve. Hereby the authors' aim is the construction of methods whose numerical realization is nearly as simple as that of collocation methods and whose convergence in negative Sobolev norms is nearly as good as that of Galerkin methods. A sequence of approximate solutions \(u_ n\), converging to the solution \(u\) of the pseudodifferential equation to be solved, is constructed using a (spline or trigonometric) collocation solution \(z_ n\) and a correction by a solution \(y_ m\) of a ``small'' (spline or trigonometric) Galerkin system, \(m \ll n\). Hence, the computational amount for the Galerkin system is defensible. Stability and convergence investigations show an improvement of the convergence, in most of the cases the convergence result just presented even improves that of the qualocation method, which is also a compromise between collocation and Galerkin methods, see e.g. \textit{I. H. Sloan} [Numer. Math. 54, No. 1, 41-56 (1988; Zbl 0668.65111)]. Representing the error terms -- as usually -- as series of squared Fourier coefficients we see, that actually the aim of the correction of the collocation method is directed to the improvement of the ``lower'' Fourier coefficients, which is also the aim of the qualocation method. Thus, the convergence rate of any qualocation cannot be better than the rate of recent corrected collocation method for a suitably chosen size of the Galerkin system. Numerical examples with Cauchy singular integral equations, both for trigonometric and for spline approximation, show the improvement of convergence rates and of the absolute values of the errors, too. One of the attributes of the present paper is the embedding of the numerical method proposed here in an excellently handled profound analysis. Reading the paper gives a thorough insight into backgrounds and genesis of ideas and methods to be applied here.
scientific article; zbMATH DE number 770166