Numerische Mathematik

, Volume 112, Issue 2, pp 221-243

First online:

Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

  • J. GuzmánAffiliated withDivision of Applied Mathematics, Brown University
  • , D. LeykekhmanAffiliated withDepartment of Mathematics, University of Connecticut Email author 
  • , J. RossmannAffiliated withInstitut für Mathematik, Universität Rostock
  • , A. H. SchatzAffiliated withDepartment of Mathematics, Cornell University

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A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder estimates for the corresponding Green’s function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in \({W^1_{\infty}}\) . In contrast to previously known results, \({W_p^{2}}\) regularity for p > 3, which does not hold for general convex polyhedral domains, is not required. Furthermore, the new Green’s function estimates allow us to obtain localized error estimates at a point.

Mathematics Subject Classification (2000)

65N30 65N15