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An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes. (English) Zbl 1120.65118
The author considers the singularly perturbed reaction-diffusion problems which yield solutions with local anisotropic behaviour, e.g. boundary and (or) interior layers. In these cases a special mesh adaptivity is required. He investigates the so-called anisotropic finite element meshes which utilize triangles streched along the boundary or in the direction of the interior layer.
The main purpose of the work is to consider a posteriori error estimates and to construct upper and lower error bounds. The author suggests a modification of the equilibrated residual method based on the stretching ratios of the mesh elements and proves that the resulting estimator is robust. A suitable approximate solution of the local problem is proposed and its equivalence to the exact solution is shown. A numerical example is given.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
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References:
[1] M. Ainsworth and I. Babuška , Reliable and robust a posteriori error estimation for singularly perturbed reaction-diffusion problems . SIAM J. Numer. Anal. 36 ( 1999 ) 331 - 353 (electronic). See also Corrigendum at http://www.maths.strath.ac.uk/ aas98107/papers.html. Zbl 0948.65114 · Zbl 0948.65114 · doi:10.1137/S003614299732187X
[2] M. Ainsworth and J.T. Oden , A unified approach to a posteriori error estimation using element residual methods . Numer. Math. 65 ( 1993 ) 23 - 50 . Article | Zbl 0797.65080 · Zbl 0797.65080 · doi:10.1007/BF01385738 · eudml:133721
[3] M. Ainsworth and J.T. Oden , A Posteriori Error Estimation in Finite Element Analysis . Wiley ( 2000 ). MR 1885308 | Zbl 1008.65076 · Zbl 1008.65076
[4] T. Apel , Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements . Computing 60 ( 1998 ) 157 - 174 . Zbl 0897.65003 · Zbl 0897.65003 · doi:10.1007/BF02684363
[5] T. Apel , Treatment of boundary layers with anisotropic finite elements . Z. Angew. Math. Mech. ( 1998 ). Zbl 0925.65188 · Zbl 0925.65188 · doi:10.1002/(SICI)1521-4001(199812)78:12<855::AID-ZAMM855>3.0.CO;2-0
[6] T. Apel , Anisotropic finite elements: local estimates and applications . B.G. Teubner, Stuttgart ( 1999 ). MR 1716824 | Zbl 0934.65121 · Zbl 0934.65121
[7] T. Apel , S. Grosman , P.K. Jimack and A. Meyer , A new methodology for anisotropic mesh refinement based upon error gradients . Appl. Numer. Math. 50 ( 2004 ) 329 - 341 . Zbl 1050.65122 · Zbl 1050.65122 · doi:10.1016/j.apnum.2004.01.006
[8] T. Apel and G. Lube , Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem . Appl. Numer. Math. 26 ( 1998 ) 415 - 433 . Zbl 0933.65136 · Zbl 0933.65136 · doi:10.1016/S0168-9274(97)00106-2
[9] I. Babuška and W. Rheinboldt , A posteriori error estimates for the finite element method . Int. J. Numer. Meth. Eng. 12 ( 1978 ) 1597 - 1615 . Zbl 0396.65068 · Zbl 0396.65068 · doi:10.1002/nme.1620121010
[10] R. Bank and A. Weiser , Some a posteriori error estimators for elliptic partial differential equations . Math. Comp. 44 ( 1985 ) 283 - 301 . Zbl 0569.65079 · Zbl 0569.65079 · doi:10.2307/2007953
[11] H. Bufler and E. Stein , Zur Plattenberechnung mittels finiter Elemente . Ingenier Archiv 39 ( 1970 ) 248 - 260 . Zbl 0197.21901 · Zbl 0197.21901 · doi:10.1007/BF00532457
[12] P.G. Ciarlet , The finite element method for elliptic problems . North-Holland Publishing Co., Amsterdam. Studies in Mathematics and its Applications, Vol. 4 , ( 1978 ). MR 520174 | Zbl 0383.65058 · Zbl 0383.65058
[13] M. Dobrowolski , S. Gräf and C. Pflaum , On a posteriori error estimators in the infinte element method on anisotropic meshes . Electron. Trans. Numer. Anal. 8 ( 1999 ) 36 - 45 . Zbl 0934.65122 · Zbl 0934.65122 · emis:journals/ETNA/vol.8.1999/pp36-45.dir/pp36-45.html · eudml:120197
[14] S. Grosman , The robustness of the hierarchical a posteriori error estimator for reaction-diffusion equation on anisotropic meshes . SFB393-Preprint 2, Technische Universität Chemnitz, SFB 393 (Germany), ( 2004 ).
[15] R. Hagen , S. Roch , and B. Silbermann , C *-algebras and numerical analysis. Marcel Dekker Inc., New York (2001). MR 1792428 | Zbl 0964.65055 · Zbl 0964.65055
[16] H. Han and R.B. Kellogg , Differentiability properties of solutions of the equation \(-\epsilon ^ 2\delta u+ru=f(x,y)\) in a square . SIAM J. Math. Anal. 21 ( 1990 ) 394 - 408 . Zbl 0732.35020 · Zbl 0732.35020 · doi:10.1137/0521022
[17] G. Kunert , 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. Zbl 0919.65066 · Zbl 0919.65066
[18] G. Kunert , An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes . Numer. Math. 86 ( 2000 ) 471 - 490 . Zbl 0965.65125 · Zbl 0965.65125 · doi:10.1007/s002110000170
[19] G. Kunert , A local problem error estimator for anisotropic tetrahedral finite element meshes . SIAM J. Numer. Anal. 39 ( 2001 ) 668 - 689 . Zbl 1004.65112 · Zbl 1004.65112 · doi:10.1137/S003614299935615X
[20] G. Kunert , Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes . Adv. Comput. Math. 15 ( 2001 ) 237 - 259 . Zbl 1049.65121 · Zbl 1049.65121 · doi:10.1023/A:1014248711347
[21] G. Kunert , Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes . ESAIM: M2AN 35 ( 2001 ) 1079 - 1109 . Numdam | Zbl 1041.65072 · Zbl 1041.65072 · doi:10.1051/m2an:2001149 · numdam:M2AN_2001__35_6_1079_0 · eudml:197504
[22] G. Kunert and R. Verfürth , Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes . Numer. Math. 86 ( 2000 ) 283 - 303 . Zbl 0964.65120 · Zbl 0964.65120 · doi:10.1007/s002110000152
[23] P. Ladevèze and D. Leguillon , Error estimate procedure in the finite element method and applications . SIAM J. Numer. Anal. 20 ( 1983 ) 485 - 509 . Zbl 0582.65078 · Zbl 0582.65078 · doi:10.1137/0720033
[24] K.G. Siebert , An a posteriori error estimator for anisotropic refinement . Numer. Math. 73 ( 1996 ) 373 - 398 . Zbl 0873.65098 · Zbl 0873.65098 · doi:10.1007/s002110050197
[25] R. Verfürth , A review of a posteriori error estimation and adaptive mesh-refinement techniques . Wiley-Teubner Series Advances in Numerical Mathematics. Chichester: John Wiley & Sons. Stuttgart: B.G. Teubner ( 1996 ). Zbl 0853.65108 · Zbl 0853.65108
[26] M. Vogelius and I. Babuška , On a dimensional reduction method . I. The optimal selection of basis functions. Math. Comp. 37 ( 1981 ) 31 - 46 . Zbl 0495.65049 · Zbl 0495.65049 · doi:10.2307/2007498
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