Numerische Mathematik

, Volume 139, Issue 1, pp 47–92 | Cite as

Regularity and a priori error analysis on anisotropic meshes of a Dirichlet problem in polyhedral domains

  • Hengguang Li
  • Serge Nicaise


Consider the Poisson equation on a polyhedral domain with the given data in a weighted \(L^2\) space. We establish new regularity results for the solution with possible vertex and edge singularities and propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with less geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires smoother given data. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations by allowing almost-\(L^2\) data. Numerical tests validate the theoretical analysis.

Mathematics Subject Classification

65N30 65N50 65N15 35J15 35J75 



The first author was supported in part by the NSF Grant DMS-1418853, by the Natural Science Foundation of China (NSFC) Grant 11628104, and by the Wayne State University Grants Plus Program.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.Institut des Sciences et Techniques of ValenciennesUniversité de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956Valenciennes Cedex 9France

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