Abstract
A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes. Numerical experiments in 2D underline the applicability of the theoretical results in adaptive computations.
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References
Afif, M., Amaziane, B.: Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in flow in porous media. Comput. Methods Appl. Mech. Eng. 191, 5265–5286 (2002)
Afif, M., Bergam, A., Mghazli, Z., Verfürth, R.: A posteriori estimators for the finite volume discretization of an elliptic equation. Numer. Algorithms 34, 127–136 (2003)
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000)
Angermann, L.: Balanced a-posteriori error estimates for finite volume type discretizations of convection-dominated elliptic problems. Computing 55, 305–323 (1995)
Apel, Th.: Anisotropic Finite Elements: Local Estimates and Applications, Advances in Numerical Mathematics. Teubner, Leipzig (1999)
Bank, R.E., Rose, D.: Some error estimates for the box method. SIAM J. Numer. Anal. 24, 777–787 (1987)
Bergam, A., Mghazli, Z.: Estimateurs a posteriori d’un schéma de volumes finis pour un problème non linéaire. C. R. Acad. Sci. 331, 475–478 (2000)
Bergam, A., Mghazli, Z., Verfürth, R.: Estimations a posteriori d’un schéma de volumes finis pour un problème non linéaire. Numer. Math. 95, 599–624 (2003)
Bey, J.: Finite-Volumen- und Mehrgitterverfahren für Elliptische Randwertprobleme. Advances in Numerical Mathematics. Teubner, Stuttgart (1998)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Formaggia, L., Perotto, S., Zunino, P.: An anisotropic a-posteriori error estimate for a convection-diffusion problem. Comput. Vis. Sci. 4, 99–104 (2001)
Jonathan, R.S.: Delaunay refinement algorithms for triangular mesh generation. Comput. Geom., Theory Appl. 22(1–3), 21–74 (2002)
Kröner, D., Ohlberger, M.: A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math. Comput. 69, 25–39 (2000)
Kunert, G.: An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86, 471–490 (2000)
Kunert, G.: A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39, 668–689 (2001)
Kunert, G.: Robust a posteriori error estimation for a singularly perturbed reaction–diffusion equation on anisotropic tetrahedral meshes. Adv. Comput. Math. 15, 237–259 (2001)
Kunert, G.: Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes. Math. Model. Numer. Anal. 35, 1079–1109 (2001)
Kunert, G.: A note on the energy norm for a singularly perturbed model problem. Computing 69, 265–272 (2002)
Kunert, G.: A posteriori error estimation for convection dominated problems on anisotropic meshes. Math. Methods Appl. Sci. 26, 589–617 (2003)
Lazarov, R., Tomov, S.: A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations. Comput. Geosci. 6, 483–503 (2002)
Mackenzie, J.A., Mayers, D.F., Mayfield, A.J.: Error estimates and mesh adaption for a cell vertex finite volume scheme. Notes Numer. Fluid Mech. 44, 290–310 (1993)
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: 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)
Morton, K.W.: Numerical Solution of Convection-Diffusion Problems. Chapman & Hall, London (1996)
Ohlberger, M.: A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math. 87, 737–761 (2001)
Ohlberger, M.: A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. Math. Model. Numer. Anal. 35, 355–387 (2001)
Papastavrou, A., Verfürth, R.: A posteriori error estimators for stationary convection–diffusion problems: a computational comparison. Comput. Methods Appl. Mech. Eng. 189, 449–462 (2000)
Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer, Berlin (1996)
Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner, Chichester/Stuttgart (1996),
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Afif, M., Amaziane, B., Kunert, G. et al. A Posteriori Error Estimation for a Finite Volume Discretization on Anisotropic Meshes. J Sci Comput 43, 183–200 (2010). https://doi.org/10.1007/s10915-010-9352-7
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DOI: https://doi.org/10.1007/s10915-010-9352-7