×

Toward anisotropic mesh construction and error estimation in the finite element method. (English) Zbl 1041.65097

This article uses the Hessian strategy for mesh construction in the solution of partial differential equations. The finite element method uses a family \({\mathcal F}\) at tetraedras in the 3-dimensional case and triangles in the 2-dimensional case. A posteriori error estimation is obtained under the condition that the matching function is bounded: \(m_1(u- u_h,{\mathcal T})\leq 1\). Numerical experiments are carried out for two examples and confirm the results: 1) \(-\Delta u= f\) and \(u|_\Gamma= u_0\) when \(\Omega\) consist on three quarters of a cylinder; 2) \(-\varepsilon\Delta u+ u=0\) and \(u|_\Gamma= u_0\) when \(\Omega= [0,1]^3\).

MSC:

65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dörfler, SIAM J Num Anal 33 pp 1106– (1996)
[2] and Data oscillation and convergence of adaptive fem. Report 17/1999, University of Freiburg, July 1999.
[3] Anisotropic finite elements: Local estimates and applications, Advances in Numerical Mathematics. Teubner, Stuttgart, 1999, Habilitationsschrift. · Zbl 0917.65090
[4] and Finite volume methods with local mesh alignment in 2-D. Adaptive Methods?Algorithms, Theory and Applications, Volume 46 of Notes on Num. Fluid Mechanics, Braunschweig, Vieweg, 1994, pp. 38-53. · Zbl 0808.65097 · doi:10.1007/978-3-663-14246-1_3
[5] Dolej?í, Comput Vis Sci 1 pp 165– (1998)
[6] and Selfadaptive finite element computations with smooth time controller and anisotropic refinement, Report 96-16, ZIB, 1996.
[7] Kornhuber, IMPACT of Comput Sci Engrg 2 pp 40– (1990)
[8] Nochetto, Math Comp 64 pp 1– (1995)
[9] Peraire, J Comp Phys 72 pp 449– (1987)
[10] Rachowicz, Comput Methods Appl Mech Engrg 109 pp 169– (1993)
[11] and FCT-solution on adapted unstructured meshes for compressible high speed flow computations, editor, Flow simulation with high-performance computers I, Volume 38 of Notes on Num Fluid Mechanics, pp. 334-438, Vieweg, 1993.
[12] Siebert, Numer Math 73 pp 373– (1996)
[13] Simpson, Appl Numer Math 14 pp 183– (1994)
[14] and Computational aspects of flow simulation in three dimensional, unstructured, adaptive grids, editor, Flow simulation with high-performance computers II, Volume 52 of Notes on Num Fluid Mechanics, pp. 431-446. Vieweg, 1996. · doi:10.1007/978-3-322-89849-4_31
[15] Zienkiewicz, Int J Numer Methods Engrg 37 pp 2189– (1994)
[16] A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes. Logos Verlag, Berlin, 1999. Also PhD thesis, TU Chemnitz. http://archiv.tu-chemnitz.de/pub/1999/0012/index.html.
[17] Kunert, Numer Math 86 pp 471– (2000)
[18] Kunert, SIAM J Numer Anal 39 pp 668– (2001)
[19] Kunert, Adv Comp Math 15 pp 237– (2001)
[20] Kunert, Math Model Numer Anal 35 pp 1079– (2001)
[21] Kunert, Numer Math 86 pp 283– (2000)
[22] Dobrowolski, Electronic Transactions Num Anal 8 pp 36– (1999)
[23] Local error estimation for singularly perturbed reaction-diffusion boundary value problem, Master’s thesis, TU Chemnitz, September 2001.
[24] and New progress in anisotropic grid adaption for inviscid and viscous flow simulations. Proceedings of the 4th Annual International Meshing Roundtable, pp 73-85, Albuquerque, NM, 1995. Sandia National Laboratories. Also Report 2671 at INRIA.
[25] and Anisotropic error estimates for elliptic problems, Report 18, Ecole Polytechnique Federale de Lausanne, 2000.
[26] and A new methodology for anisotropic mesh refinement based upon error gradients. Preprint SFB393/01-11, TU Chemnitz, 2001.
[27] and A pliant method for anisotropic mesh generation. Proceedings of the 5th Annual International Meshing Roundtable, Pittsburgh, PA, 1996. Sandia National Laboratories.
[28] Apel, Numer Math 89 pp 193– (2001)
[29] Apel, Computing 47 pp 277– (1992)
[30] Anisotropic mesh construction and error estimation in the finite element method, Preprint SFB393/00_01, TU Chemnitz, 2000. Also http://archiv.tu-chemnitz.de/pub/2000/0066/index.html.
[31] Apel, Math Modeling Numer Anal 33 pp 1149– (1999)
[32] Bakhvalov, Zh. Vychisl. Mat. Mat. Fiz. 9 pp 841– (1969)
[33] BAMG: Bidimensional Anisotropic Mesh Generator. Version 0.58. INRIA, Roquencourt, 1998. http://www-rocq.inria.fr/gamma/cdrom/ftp/bamg/bamg.ps.gz.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.