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

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Abstract

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 

Notes

Acknowledgements

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.

References

  1. 1.
    Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1999)MATHGoogle Scholar
  2. 2.
    Apel, T., Heinrich, B.: Mesh refinement and windowing near edges for some elliptic problem. SIAM J. Numer. Anal. 31(3), 695–708 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Apel, T., Nicaise, S.: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21, 519–549 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Apel, T., Schöberl, J.: Multigrid methods for anisotropic edge refinement. SIAM J. Numer. Anal 40(5), 1993–2006 (2002). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Apel, T., Sändig, A.-M., Whiteman, J.: Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19(1), 63–85 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Apel, T., Lombardi, A.L., Winkler, M.: Anisotropic mesh refinement in polyhedral domains: error estimates with data in \(L^2(\Omega )\). ESAIM Math. Model. Numer. Anal. 48(4), 1117–1145 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Babuška, I., Aziz, A.K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13(2), 214–226 (1976)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bacuta, C., Li, H., Nistor, V.: Anisotropic graded meshes and quasi-optimal rates of convergence for the FEM on polyhedral domains in 3D. In: CCOMAS 2012—European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers, pp 9003–9014 (2012)Google Scholar
  9. 9.
    Bacuta, C., Nistor, V., Zikatanov, L.: Improving the rate of convergence of high-order finite elements on polyhedra. II. Mesh refinements and interpolation. Numer. Funct. Anal. Optim. 28(7–8), 775–824 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 2nd edn. Springer, New York (2002)CrossRefMATHGoogle Scholar
  11. 11.
    Buffa, A., Costabel, M., Dauge, M.: Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron. C. R. Math. Acad. Sci. Paris 336(7), 565–570 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4. North-Holland, Amsterdam (1978)CrossRefMATHGoogle Scholar
  13. 13.
    Costabel, M., Dauge, M., Nicaise, S.: Weighted analytic regularity in polyhedra. Comput. Math. Appl. 67(4), 807–817 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dauge, M.: Elliptic boundary value problems on corner domains. In: Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)Google Scholar
  15. 15.
    De Coster, C., Nicaise, S.: Singular behavior of the solution of the Helmholtz equation in weighted \(L^p\)-Sobolev spaces. Adv. Differ. Equ. 16(1–2), 165–198 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Fritzsch, R.: Optimale finite-elemente-approximationen für Funktionen mit Singularitäten. Thesis (Ph.D.)–TU Dresden (1990)Google Scholar
  17. 17.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)CrossRefMATHGoogle Scholar
  18. 18.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)MATHGoogle Scholar
  19. 19.
    Grisvard, P.: Edge behavior of the solution of an elliptic problem. Math. Nachr. 132, 281–299 (1987)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kondrat’ev, V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16, 209–292 (1967)MathSciNetGoogle Scholar
  21. 21.
    Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. Mathematical Surveys and Monographs, vol. 52. American Mathematical Society, Providence (1997)MATHGoogle Scholar
  22. 22.
    Křížek, M.: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29(2), 513–520 (1992)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Li, H.: An anisotropic finite element method on polyhedral domains: interpolation error analysis. Math. Comput. (2017)Google Scholar
  24. 24.
    Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181 (1972)Google Scholar
  25. 25.
    Schötzau, D., Schwab, C., Wihler, T.P.: \(hp\)-dGFEM for second-order mixed elliptic problems in polyhedra. Math. Comp. 85(299), 1051–1083 (2016)MathSciNetCrossRefMATHGoogle Scholar

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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