Abstract
In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems.We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such as the nonlinear Poisson-Boltzmann equation and its regularizations. The algorithm we study consists of the standard SOLVE-ESTIMATE-MARK-REFINE procedure common to many adaptive finite element algorithms, but where the SOLVE step involves only a full solve on the coarsest level, and the remaining levels involve only single Newton updates to the previous approximate solution.
Supported in part by NSF Awards 0715146 and 0915220, and by CTBP and NBCR
Supported in part by NSF Award 0715146
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Bibliography
M. Arioli, E.H. Georgoulis, and D. Loghin. Convergence of inexact adaptive finite element solvers for elliptic problems. Technical Report RAL-TR-2009-021, Science and Technology Facilities Council, October 2009.
R. Bank, M. Holst, R. Szypowski, and Y. Zhu. Finite element error estimates for critical exponent semilinear problems without mesh conditions. Preprint, 2011.
R. Bank, M. Holst, R. Szypowski, and Y. Zhu. Convergence of AFEM for semilinear problems with inexact solvers. Preprint, 2011.
L. Chen, M. Holst, J. Xu, and Y. Zhu. Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids. Arxiv preprint arXiv:1006.3277, 2010.
L. Chen, M. Holst, and J. Xu. The finite element approximation of the nonlinear Poisson-Boltzmann equation. SIAM Journal on Numerical Analysis, 45(6):2298–2320, 2007.
I-L. Chern, J.-G. Liu, and W.-C. Wan. Accurate evaluation of electrostatics for macromolecules in solution. Methods and Applications of Analysis, 10:309–328, 2003.
W. Dörfler. A convergent adaptive algorithm for Poisson’s equation. SIAM Journal on Numerical Analysis, 33:1106–1124, 1996.
FETK. The Finite Element ToolKit. http://www.FETK.org.
M. Holst, J.A. McCammon, Z. Yu, Y.C. Zhou, and Y. Zhu. Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation. Accepted for publication in Communications in Computational Physics, 2009.
M. Holst, G. Tsogtgerel, and Y. Zhu. Local Convergence of Adaptive Methods for Nonlinear Partial Differential Equations. arXiv, (1001.1382v1), 2010.
R.H. Nochetto, K.G. Siebert, and A. Veeser. Theory of adaptive finite element methods: An introduction. In R.A. DeVore and A. Kunoth, editors, Multiscale, Nonlinear and Adaptive Approximation, pages 409–542. Springer, 2009.
R. Stevenson. Optimality of a standard adaptive finite element method. Found. Comput. Math., 7(2):245–269, 2007.
J. Xu. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM Journal on Numerical Analysis, 33(5):1759–1777, 1996.
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Holst, M., Szypowski, R., Zhu, Y. (2013). Adaptive Finite Element Methods with Inexact Solvers for the Nonlinear Poisson-Boltzmann Equation. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_18
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DOI: https://doi.org/10.1007/978-3-642-35275-1_18
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