On the asymptotic convergence of collocation methods. (English) Zbl 0541.65075

Arnold, Douglas N.; Wendland, Wolfgang L.

Math. Comput. 41, 349-381 (1983).

In this paper an interesting and well documented proof of convergence for a class of boundary element methods is presented. The analysis is restricted to one independent variable problems, including nodal collocation by odd degree polynomial splines, and is applied to systems of strongly elliptic pseudodifferential equations (including two-point boundary value problems for ordinary differential equations).
The error analysis is based on an equivalence between collocation and certain non-standard Galerkin methods and provides means of comparing collocation with the standard Galerkin approach from the point of view of numerical efficiency.

Reviewer: J.C.F.Telles

MSC:

65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65R20	Numerical methods for integral equations
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
35S15	Boundary value problems for PDEs with pseudodifferential operators
34B05	Linear boundary value problems for ordinary differential equations
45B05	Fredholm integral equations
45E05	Integral equations with kernels of Cauchy type
45J05	Integro-ordinary differential equations
65N15	Error bounds for boundary value problems involving PDEs
65L10	Numerical solution of boundary value problems involving ordinary differential equations

Keywords:

quasioptimal order estimates; linear strongly elliptic pseudodifferential equations; method of nodal collocation; odd degree polynomial splines; boundary element methods; Galerkin method; rates of convergence