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A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential

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Abstract

We develop an a posteriori error estimate for mixed boundary value problems of the form \((-\Delta +\fancyscript{V})u=f\), where the potential \(\fancyscript{V}\) may possess inverse-square singularities at finitely many points in the domain. We prove that our error estimate can be efficiently computed and is asymptotically identical to the actual error in the energy norm, on a family of geometrically graded meshes appropriate for singular solutions of such problems. Therefore, our estimate can be used for a practical stopping criterion. A variety of numerical experiments support our theoretical results. We also offer a direct convergence and effectivity comparison between the geometrically-graded meshes, which are based on a priori knowledge of possible singularities in the solution, and adaptively refined meshes driven by local error indicators associated with our a posteriori error estimate.

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Acknowledgments

The authors thank Dr. Alan Demlow for helpful remarks concerning Lemma 4.3.

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Hengguang Li

  2. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, 97201, USA

    Jeffrey S. Ovall

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  1. Hengguang Li
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Correspondence to Jeffrey S. Ovall.

Additional information

H. Li was partially supported by the NSF Grant DMS-1158839. J.S. Ovall was partially supported by the NSF Grant DMS-1216672.

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Li, H., Ovall, J.S. A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential. Numer. Math. 128, 707–740 (2014). https://doi.org/10.1007/s00211-014-0628-y

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  • DOI: https://doi.org/10.1007/s00211-014-0628-y

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