On the convergence of adaptive feedback loops

Babuška, I., Vogelius, M.: Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44(1), 75–102 (1984). https://doi.org/10.1007/BF01389757 Bank, R.E.: PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users’ Guide 12.0. Technical report, Department of Mathematics, University of California at San Diego (2016) Bank, R.E., Deotte, C.: Adventures in adaptivity. Comput. Vis. Sci. 18(2–3), 79–91 (2017). https://doi.org/10.1007/s00791-016-0272-4 Bank, R.E., Nguyen, H.: \(hp\) adaptive finite elements based on derivative recovery and superconvergence. Comput. Vis. Sci. 14(6), 287–299 (2011). https://doi.org/10.1007/s00791-012-0179-7 Bank, R.E., Sherman, A.H., Weiser, A.: Refinement algorithms and data structures for regular local mesh refinement. In: Scientific computing (Montreal, Que., 1982), IMACS Trans. Sci. Comput., I, pp. 3–17. IMACS, New Brunswick, NJ (1983) Bank, R.E., Xu, J., Zheng, B.: Superconvergent derivative recovery for Lagrange triangular elements of degree \(p\) on unstructured grids. SIAM J. Numer. Anal. 45(5), 2032–2046 (2007). https://doi.org/10.1137/060675174 Bank, R.E., Yserentant, H.: A note on interpolation, best approximation, and the saturation property. Numer. Math. 131(1), 199–203 (2015). https://doi.org/10.1007/s00211-014-0687-0 Bürg, M., Dörfler, W.: Convergence of an adaptive \(hp\) finite element strategy in higher space-dimensions. Appl. Numer. Math. 61(11), 1132–1146 (2011). https://doi.org/10.1016/j.apnum.2011.07.008 Carstensen, C., Gallistl, D., Gedicke, J.: Justification of the saturation assumption. Numer. Math. 134(1), 1–25 (2016). https://doi.org/10.1007/s00211-015-0769-7 Demlow, A., Stevenson, R.: Convergence and quasi-optimality of an adaptive finite element method for controlling \(L_2\) errors. Numer. Math. 117(2), 185–218 (2011). https://doi.org/10.1007/s00211-010-0349-9 Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996). https://doi.org/10.1137/0733054 Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91(1), 1–12 (2002). https://doi.org/10.1007/s002110100321 Guo, B., Babuška, I.: The \(hp\) version of the finite element method. Part 1: the basic approximation results. Comput. Mech. 1(1), 21–41 (1986) Holst, M., Pollock, S.: Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems. Numer. Methods Partial Differ. Equ. 32(2), 479–509 (2016). https://doi.org/10.1002/num.22002 Holst, M., Pollock, S., Zhu, Y.: Convergence of goal-oriented adaptive finite element methods for semilinear problems. Comput. Vis. Sci. 17(1), 43–63 (2015). https://doi.org/10.1007/s00791-015-0243-1 Mitchell, W.F.: A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220, 350–364 (2013). https://doi.org/10.1016/j.amc.2013.05.068 Mitchell, W.F.: How high a degree is high enough for high order finite elements? Proc. Comput. Sci. 51, 246–255 (2015) Mitchell, W.F., McClain, M.A.: A survey of \(hp\)-adaptive strategies for elliptic partial differential equations. In: Recent Advances in Computational and Applied Mathematics. Springer, Dordrecht, pp. 227–258 (2011). https://doi.org/10.1007/978-90-481-9981-5_10 Mitchell, W.F., McClain, M.A.: A comparison of \(hp\)-adaptive strategies for elliptic partial differential equations. ACM Trans. Math. Softw. 41(1), Art. 2, 39 (2014). https://doi.org/10.1145/2629459 Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Multiscale, nonlinear and adaptive approximation. Springer, Berlin, pp. 409–542 (2009). https://doi.org/10.1007/978-3-642-03413-8_12 Nochetto, R.H., Veeser, A.: Primer of adaptive finite element methods. In: Multiscale and Adaptivity: Modeling, Numerics and Applications. Lecture Notes in Mathematics, vol. 2040. Springer, Heidelberg, pp. 125–225 (2012). https://doi.org/10.1007/978-3-642-24079-9 Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007). https://doi.org/10.1007/s10208-005-0183-0 Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). https://doi.org/10.1093/acprof:oso/9780199679423.001.0001