Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes

Kunert, Gerd

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 1079-1109.

Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

[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. | Zbl

[2] M. Ainsworth and J. Oden, A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000). | MR | Zbl

[3] L. Angermann, Balanced a-posteriori error estimates for finite volume type discretizations of convection-dominated elliptic problems. Computing 55 (1995) 305-323. | Zbl

[4] T. Apel and G. Lube, Anisotropic mesh refinement in stabilized Galerkin methods. Numer. Math. 74 (1996) 261-282. | Zbl

[5] T. 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. | Zbl

[6] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. | Zbl

[7] N.S. Bakhvalov, Optimization of methods for the solution of boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. i Mat. Fiz. 9 (1969) 841-859. In Russian. | Zbl

[8] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283-301. | Zbl

[9] M. Beckers, Numerical Integration in High Dimensions. Ph.D. Thesis, Katholieke Universiteit Leuven / Louvain, Belgium (1992).

[10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR | Zbl

[11] M. Dobrowolski, S. Gräf and C. Pflaum, On a posteriori error estimators in the finite element method on anisotropic meshes. ETNA, Electron. Trans. Numer. Anal. 8 (1999) 36-45. | Zbl

[12] P. Keast, Moderate-degree tetrahedral quadrature formulas. Comput. Methods Appl. Mech. Engrg. 55 (1986) 339-348. | Zbl

[13] G. Kunert, A Posteriori Error Estimation for Anisotropic Tetrahedral and Triangular Finite Element Meshes. Logos Verlag, Berlin (1999). Also Ph.D. Thesis, TU Chemnitz, http:/$$/archiv.tu-chemnitz.de/pub/1999/0012/index.html | Zbl

[14] G. Kunert, An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86 (2000) 471-490. DOI 10.1007/s002110000170. | Zbl

[15] G. Kunert, Towards anisotropic mesh construction and error estimation in the finite element method | MR | Zbl

[16] G. Kunert, A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39 (2001) 668-689. | Zbl

[17] G. Kunert, A note on the energy norm for a singularly perturbed model problem. Preprint SFB393/01-02, TU Chemnitz (2001). Also http:/$$/archiv.tu-chemnitz.de/pub/2001/0006/index.html | MR

[18] G. Kunert, Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. To appear in Adv. Comp. Math. | MR | Zbl

[19] 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. DOI 10.1007/s002110000152. | Zbl

[20] J. Peraire, M. Vahdati, K. Morgan and O.C. Zienkiewicz, Adaptive remeshing for compressible flow computation. J. Comput. Phys. 72 (1987) 449-466. | Zbl

[21] W. Rick, H. Greza and W. Koschel, FCT-solution on adapted unstructured meshes for compressible high speed flow computations. in Flow Simulation with High-Performance Computers I, in Notes Numer. Fluid Mech. 38, E.H. Hirschel, Ed., Vieweg (1993) 334-438 .

[22] H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer, Berlin (1996). | MR | Zbl

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

[24] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67-83. | Zbl

[25] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, Stuttgart (1996). | Zbl

[26] R. Verfürth, Robust a posteriori error estimators for singularly perturbed reaction-diffusion equations. Numer. Math. 78 (1998) 479-493. | Zbl

[27] R. Vilsmeier and D. Hänel, Computational aspects of flow simulation in three dimensional, unstructured, adaptive grids, in Flow Simulation with High-Performance Computers II, in Notes Numer. Fluid Mech. 52, E.H. Hirschel, Ed., Vieweg (1996) 431-44. | Zbl

[28] O.C. Zienkiewicz and J. Wu, Automatic directional refinement in adaptive analysis of compressible flows. Internat. J. Numer. Methods Engrg. 37 (1994) 2189-2210 . | Zbl