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Global a posteriori error estimates for convection-reaction-diffusion problems. (English) Zbl 1176.65126

Summary: We propose a nonstandard technique for constructing global a posteriori error estimates for the stationary convection-reaction-diffusion equation. In order to estimate the approximation error in appropriate weighted energy norms, which measures the overall quality of the approximations, the underlying bilinear form is decomposed into several terms which can be directly computed or easily estimated from above using elementary tools of functional analysis.
Several auxiliary parameters are introduced to construct such a splitting and tune the resulting upper error bound. It is demonstrated how these parameters can be chosen in some natural and convenient way for computations so that the weighted energy norm of the error is almost recovered, which shows that the estimates proposed are, in fact, quasi-sharp.
The presented methodology is completely independent of numerical techniques used to compute approximate solutions. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g., due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors etc. Moreover, the only constant that appears in the proposed error estimates is of global nature and comes from the Friedrichs-Poincaré inequality.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] Axelsson, O.; Karátson, J., Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators, Numer. Math., 99, 197-223 (2004) · Zbl 1061.65043
[2] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 45, 285-312 (1984) · Zbl 0526.76087
[3] Křížek, M.; Neittaanmäki, P., Finite Element Approximation of Variational Problems and Applications (1990), Longman Scientific & Technical: Longman Scientific & Technical Harlow · Zbl 0708.65106
[4] Morton, K. W., Numerical Solution of Convection-Diffusion Problems (1995), Kluwer Academic Publishers.: Kluwer Academic Publishers. Dordrecht · Zbl 0861.65070
[5] Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis (2000), John Wiley & Sons · Zbl 1008.65076
[6] Babuška, I.; Strouboulis, T., The Finite Element Method and Its Reliability (2001), Oxford University Press Inc.: Oxford University Press Inc. New York · Zbl 0997.74069
[7] Bangerth, W.; Rannacher, R., Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics ETH Zürich (2003), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1020.65058
[8] Becker, R.; Rannacher, R., A feed-back approach to error control in finite element methods: Basic approach and examples, East-West J. Numer. Math., 4, 237-264 (1996) · Zbl 0868.65076
[9] Korotov, S., A posteriori error estimation of goal-oriented quantities for elliptic type BVPs, J. Comput. Appl. Math., 191, 216-227 (2006) · Zbl 1089.65120
[10] Neittaanmäki, P.; Repin, S., Reliable Methods for Computer Simulation: Error Control and A Posteriori Estimates (2004), Elsevier · Zbl 1076.65093
[11] Rüter, M.; Korotov, S.; Steenbock, Ch., Goal-oriented error estimates based on different FE-solution spaces for the primal and the dual problem with applications to fracture mechanics, Comput. Mech., 39, 787-797 (2007) · Zbl 1178.74172
[12] Vejchodský, T., Guaranteed and locally computable a posteriori error estimate, IMA J. Numer. Anal., 26, 525-540 (2006) · Zbl 1096.65112
[13] Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996), Wiley-Teubner · Zbl 0853.65108
[14] Eriksson, K.; Johnson, C., Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems, Math. Comp., 60, 167-188 (1993) · Zbl 0795.65074
[15] John, V., A numerical study of a posteriori error estimators for convection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 190, 757-781 (2000) · Zbl 0973.76049
[16] Verfürth, R., A posteriori error estimators for convection-diffusion equations, Numer. Math., 80, 641-663 (1998) · Zbl 0913.65095
[17] Verfürth, R., Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal., 43, 1766-1782 (2005) · Zbl 1099.65100
[18] Carstensen, C.; Funken, S. A., Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods, East-West J. Numer. Math., 8, 153-175 (2000) · Zbl 0973.65091
[19] Angermann, L., A posteriori error estimates for FEM with violated Galerkin orthogonality, Numer. Methods Partial Differential Equations, 18, 241-259 (2002) · Zbl 1003.65058
[20] (Kuzmin, D.; Löhner, R.; Turek, S., Flux-Corrected Transport: Principles, Algorithms, and Applications (2005), Springer) · Zbl 1062.76003
[21] Repin, S.; Sauter, S.; Smolianski, A., A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions, Computing, 70, 205-233 (2003) · Zbl 1128.35319
[22] Repin, S.; Sauter, S.; Smolianski, A., A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions, J. Comput. Appl. Math., 164/165, 601-612 (2004) · Zbl 1038.65114
[23] A. Hannukainen, S. Korotov, Computational technologies for reliable control of global and local errors for linear elliptic type boundary value problems. Technical Report A494, Helsinki University of Technology, J. Numer. Anal. Industr. Appl. Math., in press.; A. Hannukainen, S. Korotov, Computational technologies for reliable control of global and local errors for linear elliptic type boundary value problems. Technical Report A494, Helsinki University of Technology, J. Numer. Anal. Industr. Appl. Math., in press. · Zbl 1161.65081
[24] Korotov, S., Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions, Appl. Math., 52, 235-249 (2007) · Zbl 1164.65485
[25] D. Kuzmin, A. Hannukainen, S. Korotov, A new a posteriori error estimate for convection-reaction-diffusion problems, J. Comput. Appl. Math., in press.; D. Kuzmin, A. Hannukainen, S. Korotov, A new a posteriori error estimate for convection-reaction-diffusion problems, J. Comput. Appl. Math., in press. · Zbl 1143.65086
[26] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043
[27] Mikhlin, S. G., Constants in Some Inequalities of Analysis (1986), John Wiley & Sons Ltd.: John Wiley & Sons Ltd. Chichester · Zbl 0603.65068
[28] Brandts, J.; Křížek, M., Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J. Numer. Anal., 23, 489-505 (2003) · Zbl 1042.65081
[29] Möller, M.; Kuzmin, D., Adaptive mesh refinement for high-resolution finite element schemes, Int. J. Numer. Meth. Fluids, 52, 545-569 (2006) · Zbl 1108.65115
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