On singular BVPs with nonsmooth data: convergence of the collocation schemes
type: Item
instance of: scholarly article
title: On singular BVPs with nonsmooth data: convergence of the collocation schemes
zbMATH Open document ID: 1421.65021
author: Jana Burkotová
author: E. B. Weinmüller
author: Irena Rachůnková
published in: BIT
publication date: 2018-01-12T00:00:00Z
review text: Linear systems of singular linear ordinary differential equations with variable coefficient matrices and non-smooth inhomogeneities arise in several applications, either as models by themselves or as discretizations of certain partial differential equations. Based on previous results on the existence and uniqueness of smooth solutions for this class of systems, based in particular on the properties of the spectrum of the coefficient matrix, in the present paper a convergence analysis of the piecewise polynomial collocation method as a numerical integrator is carried out. The class of collocation methods considered here to approximate the solution of the problem is based on the construction on a grid of size \(h\) of a piecewise continuous polynomial function that reduces in each subinterval \([t_j, t_{j+1} = t_j + h]\) of the grid to a polynomial of degree less or equal to \(k\), the number of equidistantly spaced collocation nodes in each subinterval. The treatment proceeds by stages: first a class of boundary value problems that can be expressed as a well-posed initial value problem (with all boundary conditions posed at \(t=0\)) is considered; then the collocation scheme is shown to converge for terminal value problems, and finally a general value problem is analyzed. The main result is contained in Theorem 7.2, which states that, under quite general assumptions, the unique solution \(p\) of the collocation scheme verifies \(\|p - y\| \leq \mathrm{const}\cdot h^k\), where \(y\) is the unique solution of the boundary value problem. In other words, the collocation method with \(k\) collocation points retains its classical stage order \(k\) uniformly in \(t\). The numerical examples provided clearly illustrate this behavior. In particular, if Gaussian points are taken as collocation points, then only order \(k+1\) is exhibited (instead of the typical order \(2k\)).
reviewed by: Fernando Casas
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L10
zbMATH DE Number: 6825308
zbMATH Keywords: linear systems of ordinary differential equations
zbMATH Keywords: singular boundary value problems
zbMATH Keywords: time singularity of the first kind
zbMATH Keywords: nonsmooth inhomogeneity
zbMATH Keywords: collocation method
zbMATH Keywords: convergence
describes a project that uses: bvp4c
describes a project that uses: COLSYS
describes a project that uses: COLNEW
describes a project that uses: bvpsuite
describes a project that uses: avalanche.f
MaRDI profile type: Publication
full work available at URL: s10543-017-0686-5
OpenAlex ID: W2766375184
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DOI: 10.1007/S10543-017-0686-5
description: scientific article; zbMATH DE number 6825308