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Mathematics of Computation

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On the asymptotic convergence of collocation methods
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by Douglas N. Arnold and Wolfgang L. Wendland PDF
Math. Comp. 41 (1983), 349-381 Request permission

Abstract:

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Math. Comp. 41 (1983), 349-381
  • MSC: Primary 65N35; Secondary 65L10, 65L60, 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-1983-0717691-6
  • MathSciNet review: 717691