Quasi-optimality of an Adaptive Finite Element Method for Cathodic Protection

Li, Guanglian; Xu, Yifeng

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 5, pp. 1645-1665.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2D cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the Dörfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.

[1] M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience, New York (2000). | DOI | MR | Zbl

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

[3] L. Belenki, L. Diening and C. Kreuzer, Optimality of an adaptive finite element method for the $p$-Laplacian equation. IMA J. Numer. Anal. 32 (2012) 484–510. | DOI | MR | Zbl

[4] P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. | DOI | MR | Zbl

[5] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 3rd edition. Springer, New York, NY 15 (2008). | MR | Zbl

[6] C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity. Comp. Math. Appl. 67 (2014) 1195–1253. | MR | Zbl

[7] J.M. Cascon, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. | Zbl

[8] P.G. Ciarlet, Finite element methods for elliptic problems. Siam, North-Holland, Amsterdam (1978). | Zbl

[9] L. Diening and C. Kreuzer, Linear convergence of an adaptive finite element method for the $p$-Laplacian equation. SIAM J. Numer. Anal. 46 (2008) 614–638. | DOI | MR | Zbl

[10] L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy. Found. Comput. Math. 16 (2016) 33–68. | MR | Zbl

[11] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. | MR | Zbl

[12] M. Feischl, T. Führer and D. Praetorius, Adaptive FEM with optimal convergence rates for a certain class of non-symmetric and possibly non-linear problems. SIAM J. Numer. Anal. 52 (2014) 601–625. | DOI | MR | Zbl

[13] S.A. Funken, D. Praetorius and P. Wissgott, Efficient implementation of adaptive #-FEM in Matlab. Comput. Methods Appl. Math. 11 (2011) 460–490. | MR | Zbl

[14] E.M. Garau, P. Morin and C. Zuppa, Convergence of an adaptive Kačanov FEM for quasi-linear problems. Appl. Num. Math. 61 (2011) 512–529. | DOI | MR | Zbl

[15] E.M. Garau, P. Morin and C. Zuppa, Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type. Numer. Math. Theory Methods App. 5 (2012) 131–156. | DOI | MR | Zbl

[16] L.S. Hou and W. Sun, Optimal positioning of anodes for cathodic protection. SIAM Control Optim. 34 (1996) 855–873. | DOI | MR | Zbl

[17] L.S. Hou and J.C. Turner, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls. Numer. Math. 71 (1995) 289–315. | MR | Zbl

[18] M. Karkulik, D. Pavlicek and D. Praetorius, On 2D newest vertex bisection: optimality of mesh-closure and H¹-stability of L²-projection. Constr. Approx. 38 (2013) 213–234. | DOI | MR | Zbl

[19] I. Kossaczky, A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55 (1995) 275–288. | DOI | MR | Zbl

[20] G. Li and Y. Xu, A convergent adaptive finite element method for cathodic protection. Comput. Methods Appl. Math. 17 (2017) 105–120. | DOI | MR | Zbl

[21] W.F. Mitchell, A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. 15 (1989) 326–347. | MR | Zbl

[22] R.H. Nochetto, K.G. Siebert and A. Veeser, Theory of adaptive finite element methods: an introduction. In: Multiscale, Nonlinear and Adaptive Approximation, edited by R.A. Devore and A. Kunoth. Springer, New York, NY (2009) 409–542. | DOI | MR | Zbl

[23] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

[24] R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245–269. | MR | Zbl

[25] R. Stevenson, The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77 (2008) 227–241. | DOI | MR | Zbl

[26] C. Traxler, An algorithm for adaptive mesh refinement in $n$ dimensions. Computing 59 (1997) 115–137. | DOI | MR | Zbl

[27] R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013). | DOI | MR | Zbl