Anisotropic mesh refinement in polyhedral domains: error estimates with data in L 2 (Ω)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, p. 1117-1145
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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.

DOI : https://doi.org/10.1051/m2an/2013134
Classification:  65N30
Keywords: elliptic boundary value problem, edge and vertex singularities, finite element method, anisotropic mesh grading, optimal control problem, discrete compactness property
@article{M2AN_2014__48_4_1117_0,
     author = {Apel, Thomas and Lombardi, Ariel L. and Winkler, Max},
     title = {Anisotropic mesh refinement in polyhedral domains: error estimates with data in $L^2(\Omega )$},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {4},
     year = {2014},
     pages = {1117-1145},
     doi = {10.1051/m2an/2013134},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_4_1117_0}
}
Apel, Thomas; Lombardi, Ariel L.; Winkler, Max. Anisotropic mesh refinement in polyhedral domains: error estimates with data in $L^2(\Omega )$. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, pp. 1117-1145. doi : 10.1051/m2an/2013134. http://www.numdam.org/item/M2AN_2014__48_4_1117_0/

[1] Th. Apel, Interpolation of non-smooth functions on anisotropic finite element meshes. ESAIM: M2AN 33 (1999) 1149-1185. | Numdam | MR 1736894 | Zbl 0984.65113

[2] Th. Apel and M. Dobrowolski, Anisotropic interpolation with applications to the finite element method. Computing 47 (1992) 277-293. | MR 1155498 | Zbl 0746.65077

[3] Th. Apel and B. Heinrich, Mesh refinement and windowing near edges for some elliptic problem. SIAM J. Numer. Anal. 31 (1994) 695-708. | MR 1275108 | Zbl 0807.65122

[4] Th. Apel and S. Nicaise, 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. | MR 1615426 | Zbl 0911.65107

[5] Th. Apel, A.-M. Sändig, and J.R. Whiteman, 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. | MR 1365264 | Zbl 0838.65109

[6] Th. Apel and D. Sirch, L2-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes. Appl. Math. 56 (2011) 177-206. | MR 2810243 | Zbl 1224.65252

[7] 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. | MR 3060484

[8] F. Assous, P. Ciarlet, 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. | MR 1762079 | Zbl 1007.78014

[9] I. Babuška, Finite element method for domains with corners. Computing 6 (1970) 264-273. | MR 293858 | Zbl 0224.65031

[10] A.E. Beagles and J.R. Whiteman, Finite element treatment of boundary singularities by augmentation with non-exact singular functions. Numer. Methods Partial Differ. Eqs. 2 (1986) 113-121. | MR 867853 | Zbl 0626.65112

[11] H. Blum and M. Dobrowolski, On finite element methods for elliptic equations on domains with corners. Computing 28 (1982) 53-63. | MR 645969 | Zbl 0465.65059

[12] C. Băcuţă, V. Nistor and L.T. Zikatanov, 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. | MR 2347683 | Zbl 1122.65109

[13] A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 29-65. | MR 2194717 | Zbl 1116.78020

[14] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 2 (1975) 77-84. | Numdam | Zbl 0368.65008

[15] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441-463. | MR 559195 | Zbl 0423.65009

[16] P. Grisvard, Singularities in boundary value problems, vol. 22. Research Notes Appl. Math. Springer, New York (1992). | MR 1173209 | Zbl 0766.35001

[17] M. Hinze. A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45-61. | MR 2122182 | Zbl 1074.65069

[18] P. Jamet, Estimations d'erreur pour des éléments finis droits presque dégénérés. R.A.I.R.O. Anal. Numer. 10 (1976) 43-61. | Numdam | MR 455282 | Zbl 0346.65052

[19] V. John and G. Matthies, MooNMD-a program package based on mapped finite element methods. Comput. Visual. Sci. 6 (2004) 163-169. | MR 2061275 | Zbl 1061.65124

[20] F. Kikuchi, On a discrete compactness property for the nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989) 479-490. | MR 1039483 | Zbl 0698.65067

[21] A.L. Lombardi, The discrete compactness property for anisotropic edge elements on polyhedral domains. ESAIM: M2AN 47 (2013) 169-181. | Numdam | MR 2979513 | Zbl 1281.78014

[22] J. M.-S. Lubuma and S. Nicaise, Dirichlet problems in polyhedral domains II: approximation by FEM and BEM. J. Comput. Appl. Math. 61 (1995) 13-27,. | MR 1358044 | Zbl 0840.65110

[23] P. Monk, Finite Element Methods for Maxwell's Equations. Oxford University Press, New York (2003). | MR 2059447 | Zbl 1024.78009

[24] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Éditeurs, Paris, Academia, Éditeurs, Paris, Prague (1967). | MR 227584 | Zbl 1225.35003

[25] S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784-816. | MR 1860445 | Zbl 1001.65122

[26] L.A. Oganesyan and L.A. Rukhovets, 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. | Zbl 0267.65070

[27] T. Von Petersdorff and E.P. Stephan. Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes. Math. Methods Appl. Sci. 12 (1990) 229-249. | MR 1043756 | Zbl 0722.35017

[28] G. Raugel, Résolution numérique de problèmes elliptiques dans des domaines avec coins. Ph.D. thesis. Université de Rennes (1978).

[29] A.H. Schatz and L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 2: Refinements. Math. Comput. 33 (1979) 465-492. | MR 502067 | Zbl 0417.65053

[30] H. Schmitz, K. Volk and W.L. Wendland, On three-dimensional singularities of elastic fields near vertices. Numer. Methods Partial Differ. Eqs. 9 (1993) 323-337. | MR 1216118 | Zbl 0771.73014

[31] L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483-493. | MR 1011446 | Zbl 0696.65007

[32] K. Siebert, An a posteriori error estimator for anisotropic refinement. Numer. Math. 73 (1996) 373-398. | MR 1389492 | Zbl 0873.65098

[33] G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Englewood Cliffs, NJ (1973). | MR 443377 | Zbl 0356.65096