Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-12T08:17:25.295Z Has data issue: false hasContentIssue false
  • English
  • Français

Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω)

Published online by Cambridge University Press:  04 July 2014

Thomas Apel,
Ariel L. Lombardi and
Max Winkler
Thomas Apel
Affiliation:
Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Germany. thomas.apel@unibw.de; max.winkler@unibw.de
Ariel L. Lombardi
Affiliation:
Departamento de Matemática, Universidad de Buenos Aires, and Instituto de Ciencias, Universidad Nacional de General Sarmiento. Member of CONICET, Argentina; aldoc7@dm.uba.ar
Max Winkler
Affiliation:
Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Germany. thomas.apel@unibw.de; max.winkler@unibw.de

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equations and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.

Keywords

Elliptic boundary value problemedge and vertex singularitiesfinite element methodanisotropic mesh gradingoptimal control problemdiscrete compactness property

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014, pp. 1117 - 1145
Copyright
© EDP Sciences, SMAI 2014

References

Apel, Th., Interpolation of non-smooth functions on anisotropic finite element meshes. ESAIM: M2AN 33 (1999) 1149–1185. Google Scholar
Apel, Th. and Dobrowolski, M., Anisotropic interpolation with applications to the finite element method. Computing 47 (1992) 277–293. Google Scholar
Apel, Th. and Heinrich, B., Mesh refinement and windowing near edges for some elliptic problem. SIAM J. Numer. Anal. 31 (1994) 695–708. Google Scholar
Apel, Th. and Nicaise, S., The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21 (1998) 519–549. Google Scholar
Apel, Th., Sändig, A.-M., and Whiteman, J.R., Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19 (1996) 63–85. Google Scholar
Apel, Th. and Sirch, D., L 2-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes. Appl. Math. 56 (2011) 177–206. Google Scholar
Th. Apel and D. Sirch, A priori mesh grading for distributed optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, vol. 160. Edited by G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich. Int. Ser. Numer. Math.. Springer, Basel (2011) 377–389.
Assous, F., Ciarlet, P., Jr. and J. Segré, Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the Singular Complement Method. J. Comput. Phys. 161 (2000) 218–249. Google Scholar
Babuška, I., Finite element method for domains with corners. Computing 6 (1970) 264–273. Google Scholar
Beagles, A.E. and Whiteman, J.R., Finite element treatment of boundary singularities by augmentation with non-exact singular functions. Numer. Methods Partial Differ. Eqs. 2 (1986) 113–121. Google Scholar
Blum, H. and Dobrowolski, M., On finite element methods for elliptic equations on domains with corners. Computing 28 (1982) 53–63. Google Scholar
Băcuţă, C., Nistor, V. and Zikatanov, L.T., Improving the rate of convergence of high-order finite elements in polyhedra II: mesh refinements and interpolation. Numer. Funct. Anal. Optim. 28 (2007) 775–824. Google Scholar
Buffa, A., Costabel, M. and Dauge, M., Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 29–65. Google Scholar
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 2 (1975) 77–84. Google Scholar
Dupont, T. and Scott, R., Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441–463. Google Scholar
P. Grisvard, Singularities in boundary value problems, vol. 22. Research Notes Appl. Math. Springer, New York (1992).
Hinze, M.. A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–61. Google Scholar
Jamet, P., Estimations d’erreur pour des éléments finis droits presque dégénérés. R.A.I.R.O. Anal. Numer. 10 (1976) 43–61. Google Scholar
John, V. and Matthies, G., MooNMD-a program package based on mapped finite element methods. Comput. Visual. Sci. 6 (2004) 163–169. Google Scholar
Kikuchi, F., On a discrete compactness property for the nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989) 479–490. Google Scholar
Lombardi, A.L., The discrete compactness property for anisotropic edge elements on polyhedral domains. ESAIM: M2AN 47 (2013) 169–181. Google Scholar
Lubuma, J. M.-S. and Nicaise, S., Dirichlet problems in polyhedral domains II: approximation by FEM and BEM. J. Comput. Appl. Math. 61 (1995) 13–27,. Google Scholar
P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003).
J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Éditeurs, Paris, Academia, Éditeurs, Paris, Prague (1967).
Nicaise, S., Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784–816. Google Scholar
Oganesyan, L.A. and Rukhovets, L.A., Variational-difference schemes for linear second-order elliptic equations in a two-dimensional region with piecewise smooth boundary. Zh. Vychisl. Mat. Mat. Fiz. 8 (1968) 97–114. In Russian. English translation in USSR Comput. Math. and Math. Phys. 8 (1968) 129–152. Google Scholar
von Petersdorff, T. and Stephan, E.P.. Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes. Math. Methods Appl. Sci. 12 (1990) 229–249. Google Scholar
G. Raugel, Résolution numérique de problèmes elliptiques dans des domaines avec coins. Ph.D. thesis. Université de Rennes (1978).
Schatz, A.H. and Wahlbin, L.B., Maximum norm estimates in the finite element method on plane polygonal domains. Part 2: Refinements. Math. Comput. 33 (1979) 465–492. Google Scholar
Schmitz, H., Volk, K. and Wendland, W.L., On three-dimensional singularities of elastic fields near vertices. Numer. Methods Partial Differ. Eqs. 9 (1993) 323–337. Google Scholar
Scott, L.R. and Zhang, S., Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. Google Scholar
Siebert, K., An a posteriori error estimator for anisotropic refinement. Numer. Math. 73 (1996) 373–398. Google Scholar
G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Englewood Cliffs, NJ (1973).