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IMA Journal of Numerical Analysis Advance Access originally published online on March 20, 2008
IMA Journal of Numerical Analysis 2009 29(2):257-283; doi:10.1093/imanum/drm052
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

Olaf Hansen

Department of Mathematics, California State University San Marcos

Kendall Atkinson{dagger}

Departments of Mathematics and Computer Science, The University of Iowa

David Chien

Department of Mathematics, California State University San Marcos

{dagger} Email: atkinson{at}math.uiowa.edu

Received on 31 August 2006. Revised on 29 November 2007.


  Abstract

In this article, we study the properties of the hyperinterpolation operator on the unit disc D in R2. We show how hyperinterpolation can be used in connection with the Kumar–Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disc (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove the convergence of the discrete Galerkin method in the maximum norm if the solution of the Poisson equation is in the class C1, {delta}(D), {delta} > 0. Finally, we present numerical examples which show that the discrete Galerkin method converges faster than O(n–k), for every k isin N if the solution of the nonlinear Poisson equation is in C{infty}(D).

Key Words: hyperinterpolation operator; discrete Galerkin method; projector norm; nonlinear Poisson equation


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