Abstract
We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.
Similar content being viewed by others
References
M. Ainsworth and I. Babuška, Reliable and robust a posteriori error estimation for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal. 36(2) (1999) 331–353.
M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg. 142(1-2) (1997) 1–88.
L. Angermann, Balanced a-posteriori error estimates for finite volume type discretizations of convection-dominated elliptic problems, Computing 55(4) (1995) 305–323.
Th. Apel and G. Lube, Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem, Appl. Numer. Math. 26 (1998) 415–433.
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.
N.S. Bakhvalov, Optimization of methods for the solution of boundary value problems in the presence of a boundary layer, Zh. Vychisl. Mat. Mat. Fiz. 9 (1969) 841–859 (in Russian).
R. Beinert and D. Kröner, Finite volume methods with local mesh alignment in 2-D, in: Adaptive Methods-Algorithms, Theory and Applications, Notes on Numerical Fluid Mechanics, Vol. 46 (Vieweg, Braunschweig, 1994), pp. 38–53.
P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).
P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numer. 2 (1975) 77–84.
M. Dobrowolski, S. Gräf and C. Pflaum, On a posteriori error estimators in the finite element method on anisotropic meshes, Electronic Trans. Numer. Anal. 8 (1999) 36–45.
R. Kornhuber and R. Roitzsch, On adaptive grid refinement in the presence of internal and boundary layers, IMPACT Comput. Sci. Engrg. 2 (1990) 40–72.
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.tuchemnitz. de/pub/1999/0012/index.html
G. Kunert, An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86(3) (2000) 471–490.
G. Kunert, A local problem error estimator for anisotropic tetrahedral finite element meshes, SIAM J. Numer. Anal. 39(2) (2001) 668–689.
G. Kunert, Towards anisotropic mesh construction and error estimation in the finite element method, Numer. Methods PDE (2001) (to appear).
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(2) (2000) 283–303.
J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Fitted Numerical Methods for Singularly Perturbed Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (World Scientific, Singapore, 1996).
J. Peraire, M. Vahdati, K. Morgan and O.C. Zienkiewicz, Adaptive remeshing for compressible flow computation, J. Comput. Phys. 72 (1987) 449–466.
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, ed. E.H. Hirschel, Notes on Numerical Fluid Mechanics Vol. 38 (Vieweg, 1993) pp. 334–438.
H.G. Roos and T. Linss, Gradient recovery for singularly perturbed boundary value problems II: Two-dimensional convection-diffusion, Math. Models Methods Appl. Sci. (2001) (to appear).
H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems (Springer, Berlin, 1996).
K.G. Siebert, An a posteriori error estimator for anisotropic refinement, Numer. Math. 73(3) (1996) 373–398.
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley-Teubner, Chichester/Stuttgart, 1996).
R. Verfürth, Robust a posteriori error estimators for singularly perturbed reaction-diffusion equations, Numer. Math. 78 (1998) 479–493.
R. Vilsmeier and D. Hänel, Computational aspects of flow simulation in three dimensional, unstructured, adaptive grids, in: Flow Simulation with High-Performance Computers II, ed. E.H. Hirschel, Notes on Numerical Fluid Mechanics, Vol. 52 (Vieweg, 1996) pp. 431–446.
Zh. Zhang, Superconvergent finite element method on a Shishkin mesh for convection-diffusion problems, Report 98–006, Texas Tech University (1998).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kunert, G. Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes. Advances in Computational Mathematics 15, 237–259 (2001). https://doi.org/10.1023/A:1014248711347
Issue Date:
DOI: https://doi.org/10.1023/A:1014248711347