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IMA Journal of Numerical Analysis Advance Access originally published online on May 2, 2008
IMA Journal of Numerical Analysis 2009 29(2):376-403; doi:10.1093/imanum/drn021
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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.

Analysis of a mixed finite-volume discretization of fourth-order equations on general surfaces

Qiang Du{dagger}

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Lili Ju{ddagger} and Li Tian§

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

{dagger} Corresponding author. Email: qdu{at}math.psu.edu

{ddagger} Email: ju{at}math.sc.edu

§ Email: tianl{at}math.sc.edu

Received on 5 July 2007. Revised on 3 March 2008.


  Abstract

In this paper, we study a finite-volume method for the numerical solution of a model fourth-order partial differential equation defined on a smooth surface. The discretization is done via a surface mesh consisting of piecewise planar triangles and its dual surface polygonal tessellation. We provide an error estimate for the approximate solution under the H1-norm on general regular meshes. Numerical experiments are carried out on various sample surfaces to verify the theoretical results. In addition, when the underlying mesh is constructed by the so-called constrained centroidal Voronoi meshes, we propose a numerically demonstrated superconvergent scheme to compute gradients more accurately.

Key Words: mixed finite-volume discretization; PDEs on surfaces; fourth-order equations; error estimates


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