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A Weak Galerkin Finite Element Method for a Type of Fourth Order Problem Arising from Fluorescence Tomography

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Abstract

In this paper, an innovative and effective numerical algorithm by the use of weak Galerkin (WG) finite element methods is proposed for a type of fourth order problem arising from fluorescence tomography. Fluorescence tomography is emerging as an in vivo non-invasive 3D imaging technique reconstructing images that characterize the distribution of molecules tagged by fluorophores. Weak second order elliptic operator and its discrete version are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. An error estimate of optimal order is derived in an \(H^2_{\kappa }\)-equivalent norm for the WG finite element solutions. Error estimates of optimal order except the lowest order finite element in the usual \(L^2\) norm are established for the WG finite element approximations. Numerical tests are presented to demonstrate the accuracy and efficiency of the theory established for the WG numerical algorithm.

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Acknowledgements

Funding was provided by National Science Foundation (Grant Nos. DMS-1522586, DMS-1620345, DMS-1042998, DMS-1419027), Office of Naval Research (Grant No. N000141310408), National Natural Science Foundation of China (Grant No. 11526113), Jiangsu Key Lab for NSLSCS (Grant No. 201602), and by Jiangsu Provincial Foundation Award (No. BK20050538).

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Authors and Affiliations

  1. Department of Mathematics, Texas State University, San Marcos, TX, 78666, USA

    Chunmei Wang

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Haomin Zhou

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  1. Chunmei Wang
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  2. Haomin Zhou
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Correspondence to Chunmei Wang.

Additional information

The research of Chunmei Wang was partially supported by National Science Foundation Award DMS-1522586, National Natural Science Foundation of China Award #11526113, Jiangsu Key Lab for NSLSCS Grant #201602, and by Jiangsu Provincial Foundation Award #BK20050538.

The research of Haomin Zhou was supported by NSF Faculty Early Career Development(CAREER) Award DMS-1620345, DMS-1042998, DMS-1419027, and ONR Award N000141310408.

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Wang, C., Zhou, H. A Weak Galerkin Finite Element Method for a Type of Fourth Order Problem Arising from Fluorescence Tomography. J Sci Comput 71, 897–918 (2017). https://doi.org/10.1007/s10915-016-0325-3

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  • DOI: https://doi.org/10.1007/s10915-016-0325-3

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