## Robust a-posteriori estimator for advection-diffusion-reaction problems.(English)Zbl 1130.65083

The paper deals with an a posteriori estimator for a Galerkin discretization of the Dirichlet boundary value problem ${\mathcal L}u:=-\varepsilon u^{\prime \prime}+\beta u^{\prime}+\rho u=f(x), \; x \in (0,L), \quad u(0)=u(L)=0$ with $$\varepsilon >0$$ and $$\beta, \rho \geq 0$$. The numerical error is measured with respect to a norm possessing the structure like $\| \cdot \| _V \approx (\varepsilon | \cdot | _{H_0^1}^2+| \beta| | \cdot | ^2_{1/2}+\rho \| \cdot\| _{L_2}^2)^{1/2},$ where $$| \cdot | _{1/2}$$ is a seminorm of order 1/2. The a posteriori estimator includes the residual and the first derivative of the approximate solution and is robust up to a factor $$\sqrt{\log{(Pe)}},$$ where $$Pe$$ is the global Péctlet number $$Pe=| \beta| L/\varepsilon$$. Numerical tests in one dimension confirm the theoretical results and the performance of the estimator. A possible two-dimensional extension of the results under consideration is discussed.

### MSC:

 65L70 Error bounds for numerical methods for ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L10 Numerical solution of boundary value problems involving ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations
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### References:

 [1] Mark Ainsworth and Ivo Babuška, Reliable and robust a posteriori error estimating for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal. 36 (1999), no. 2, 331 – 353. · Zbl 0948.65114 [2] Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. · Zbl 1008.65076 [3] Rodolfo Araya, Edwin Behrens, and Rodolfo Rodríguez, An adaptive stabilized finite element scheme for the advection-reaction-diffusion equation, Appl. Numer. Math. 54 (2005), no. 3-4, 491 – 503. · Zbl 1078.65101 [4] Rodolfo Araya, Abner H. Poza, and Ernst P. Stephan, A hierarchical a posteriori error estimate for an advection-diffusion-reaction problem, Math. Models Methods Appl. Sci. 15 (2005), no. 7, 1119 – 1139. · Zbl 1087.65101 [5] I. Babuška, A. Miller, and M. Vogelius, Adaptive methods and error estimation for elliptic problems of structural mechanics, Adaptive computational methods for partial differential equations (College Park, Md., 1983) SIAM, Philadelphia, PA, 1983, pp. 57 – 73. · Zbl 0581.73080 [6] Stefano Berrone, Robustness in a posteriori error analysis for FEM flow models, Numer. Math. 91 (2002), no. 3, 389 – 422. · Zbl 1009.76049 [7] Stefano Berrone and Claudio Canuto, Multilevel a posteriori error analysis for reaction-convection-diffusion problems, Appl. Numer. Math. 50 (2004), no. 3-4, 371 – 394. · Zbl 1065.65124 [8] Franco Brezzi and Alessandro Russo, Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci. 4 (1994), no. 4, 571 – 587. · Zbl 0819.65128 [9] Alexander N. Brooks and Thomas J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), no. 1-3, 199 – 259. FENOMECH ’81, Part I (Stuttgart, 1981). · Zbl 0497.76041 [10] Guillermo Hauke, Mohamed H. Doweidar, and Mario Miana, The multiscale approach to error estimation and adaptivity, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 13-16, 1573 – 1593. · Zbl 1122.76057 [11] Paul Houston, Rolf Rannacher, and Endre Süli, A posteriori error analysis for stabilised finite element approximations of transport problems, Comput. Methods Appl. Mech. Engrg. 190 (2000), no. 11-12, 1483 – 1508. · Zbl 0970.65115 [12] Gerd Kunert, A posteriori error estimation for convection dominated problems on anisotropic meshes, Math. Methods Appl. Sci. 26 (2003), no. 7, 589 – 617. · Zbl 1020.65078 [13] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. · Zbl 0223.35039 [14] J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5 – 68 (French). · Zbl 0148.11403 [15] Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631 – 658 (2003). Revised reprint of ”Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466 – 488 (electronic); MR1770058 (2001g:65157)]. · Zbl 1016.65074 [16] Gerd Rapin and Gert Lube, A stabilized scheme for the Lagrange multiplier method for advection-diffusion equations, Math. Models Methods Appl. Sci. 14 (2004), no. 7, 1035 – 1060. · Zbl 1185.76816 [17] Alessandro Russo, A posteriori error estimators via bubble functions, Math. Models Methods Appl. Sci. 6 (1996), no. 1, 33 – 41. · Zbl 0853.65109 [18] Giancarlo Sangalli, A robust a posteriori estimator for the residual-free bubbles method applied to advection-diffusion problems, Numer. Math. 89 (2001), no. 2, 379 – 399. · Zbl 1026.65089 [19] Giancarlo Sangalli, Construction of a natural norm for the convection-diffusion-reaction operator, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7 (2004), no. 2, 336 – 355 (English, with English and Italian summaries). · Zbl 1150.35455 [20] -, On robust a posteriori estimators for the advection-diffusion-reaction problem, Tech. Report 04-55, ICES Report, 2004. [21] Giancarlo Sangalli, A uniform analysis of nonsymmetric and coercive linear operators, SIAM J. Math. Anal. 36 (2005), no. 6, 2033 – 2048. · Zbl 1114.35060 [22] Hans Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. · Zbl 0830.46028 [23] R. Verfürth, A posteriori error estimators for convection-diffusion equations, Numer. Math. 80 (1998), no. 4, 641 – 663. · Zbl 0913.65095 [24] R. Verfürth, Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation, Numer. Math. 78 (1998), no. 3, 479 – 493. · Zbl 0887.65108 [25] R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1766 – 1782. · Zbl 1099.65100 [26] M. Vohralík, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, to appear in SIAM J. Numer. Anal. · Zbl 1151.65084
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