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Journal of Inverse and Ill-posed Problems

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A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data

1School of Mathematics and Statistics, Lanzhou University, Lanzhou, P.R. China and Department of Mathematics, Linköping University, Sweden.

2Department of Mathematics, Linköping University, Sweden.

3School of Mathematics and Statistics, Lanzhou University, Lanzhou, P.R. China.

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 18, Issue 6, Pages 617–645, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/jiip.2010.028, December 2010

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A Cauchy problem for elliptic equations with nonhomogeneous Neumann data in a cylindrical domain is investigated in this paper. For the theoretical aspect the a-priori and a-posteriori parameter choice rules are suggested and the corresponding error estimates are obtained. About the numerical aspect, for a simple case results given by two methods based on the discrete Sine transform and the finite difference method are presented; an idea of left-preconditioned GMRES (Generalized Minimum Residual) method is proposed to deal with the high dimensional case to save the time; a view of dealing with a general domain is suggested. Some ill-posed problems regularized by the quasi-boundary-value method are listed and some rules of this method are suggested.

Keywords.: Elliptic equation; a-priori; a-posteriori; discrete Sine transform; finite difference method; quasi-boundary-value method; left-preconditioned GMRES

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