Robust a posteriori estimates for the stochastic Cahn-Hilliard equation
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Abstract:
We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation. The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.References
- Nicholas D. Alikakos, Peter W. Bates, and Xinfu Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal. 128 (1994), no. 2, 165–205. MR 1308851, DOI 10.1007/BF00375025
- Dimitra Antonopoulou, Ĺubomír Baňas, Robert Nürnberg, and Andreas Prohl, Numerical approximation of the stochastic Cahn-Hilliard equation near the sharp interface limit, Numer. Math. 147 (2021), no. 3, 505–551. MR 4224928, DOI 10.1007/s00211-021-01179-7
- Sören Bartels, Numerical methods for nonlinear partial differential equations, Springer Series in Computational Mathematics, vol. 47, Springer, Cham, 2015. MR 3309171, DOI 10.1007/978-3-319-13797-1
- Sören Bartels and Rüdiger Müller, Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential, Numer. Math. 119 (2011), no. 3, 409–435. MR 2845623, DOI 10.1007/s00211-011-0389-9
- Sören Bartels and Rüdiger Müller, Quasi-optimal and robust a posteriori error estimates in $L^\infty (L^2)$ for the approximation of Allen-Cahn equations past singularities, Math. Comp. 80 (2011), no. 274, 761–780. MR 2772095, DOI 10.1090/S0025-5718-2010-02444-5
- L’ubomír Baňas and André Wilke, A posteriori estimates for the stochastic total variation flow, SIAM J. Numer. Anal. 60 (2022), no. 5, 2657–2680. MR 4487905, DOI 10.1137/21M1447982
- Ľubomír Baňas, Zdzisław Brzeźniak, Mikhail Neklyudov, and Andreas Prohl, Stochastic ferromagnetism, De Gruyter Studies in Mathematics, vol. 58, De Gruyter, Berlin, 2014. Analysis and numerics. MR 3157451
- L. Baňas, B. Gess, and C. Vieth, Numerical approximation of singular-degenerate parabolic stochastic PDEs, arXiv:2012.12150, 2022.
- Ľubomír Baňas and Robert Nürnberg, A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy, M2AN Math. Model. Numer. Anal. 43 (2009), no. 5, 1003–1026. MR 2559742, DOI 10.1051/m2an/2009015
- L. Baňas, H. Yang, and R. Zhu, Sharp interface limit of stochastic Cahn-Hilliard equation with singular noise, Potential Anal. (2022), DOI 10.1007/s11118-021-09976-3.
- J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis, European J. Appl. Math. 2 (1991), no. 3, 233–280. MR 1123143, DOI 10.1017/S095679250000053X
- J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis, European J. Appl. Math. 3 (1992), no. 2, 147–179. MR 1166255, DOI 10.1017/S0956792500000759
- Giuseppe Da Prato and Arnaud Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Anal. 26 (1996), no. 2, 241–263. MR 1359472, DOI 10.1016/0362-546X(94)00277-O
- Qiang Du and Xiaobing Feng, The phase field method for geometric moving interfaces and their numerical approximations, Geometric partial differential equations. Part I, Handb. Numer. Anal., vol. 21, Elsevier/North-Holland, Amsterdam, [2020] ©2020, pp. 425–508. MR 4378430, DOI 10.1007/s
- Xiaobing Feng and Andreas Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math. 94 (2003), no. 1, 33–65. MR 1971212, DOI 10.1007/s00211-002-0413-1
- Xiaobing Feng and Andreas Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math. 99 (2004), no. 1, 47–84. MR 2101784, DOI 10.1007/s00211-004-0546-5
- Xiaobing Feng and Andreas Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation of the Hele-Shaw problem, Interfaces Free Bound. 7 (2005), no. 1, 1–28. MR 2126141, DOI 10.4171/IFB/111
- Wei Liu and Michael Röckner, Stochastic partial differential equations: an introduction, Universitext, Springer, Cham, 2015. MR 3410409, DOI 10.1007/978-3-319-22354-4
- Ananta K. Majee and Andreas Prohl, A posteriori error estimation and space-time adaptivity for a linear stochastic PDE with additive noise, IMA J. Numer. Anal. 42 (2022), no. 2, 1526–1567. MR 4410751, DOI 10.1093/imanum/drab013
- Andreas Prohl and Christian Schellnegger, Adaptive concepts for stochastic partial differential equations, J. Sci. Comput. 80 (2019), no. 1, 444–474. MR 3954450, DOI 10.1007/s10915-019-00944-z
- Luca Scarpa, On the stochastic Cahn-Hilliard equation with a singular double-well potential, Nonlinear Anal. 171 (2018), 102–133. MR 3778274, DOI 10.1016/j.na.2018.01.016
- Jacques Simon, Sobolev, Besov and Nikol′skiĭ fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl. (4) 157 (1990), 117–148. MR 1108473, DOI 10.1007/BF01765315
Additional Information
- Ľubomír Baňas
- Affiliation: Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
- Email: banas@math.uni-bielefeld.de
- Christian Vieth
- Affiliation: Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
- ORCID: 0000-0001-5190-9220
- Email: cvieth@math.uni-bielefeld.de
- Received by editor(s): January 21, 2022
- Received by editor(s) in revised form: November 25, 2022, and January 25, 2023
- Published electronically: April 19, 2023
- Additional Notes: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - SFB 1283/2 2021 - 317210226.
The first author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 2025-2063
- MSC (2020): Primary 65M15, 65M50, 65M60, 65C30, 35K91, 35R60, 60H15, 60H35
- DOI: https://doi.org/10.1090/mcom/3836