A Posteriori Error Estimation for Parabolic Problems with Dynamic Boundary Conditions

Robert Altmann; Christoph Zimmer

doi:10.52825/dae-p.v2i.181

Authors

Robert Altmann Otto-von-Guericke University Magdeburg https://orcid.org/0000-0002-4161-6704
Christoph Zimmer University of Augsburg https://orcid.org/0000-0002-4589-9000

DOI:

https://doi.org/10.52825/dae-p.v2i.181

Keywords:

Dynamic Boundary Conditions, Adaptivity, PDAE, Parabolic PDE

Abstract

This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf–sup stable discretization schemes for a stationary model problem as a preliminary step. Based on an alternative formulation of the system as a partial differential–algebraic equation, we introduce a posteriori error estimators which allow local refinements as well as a special treatment of the boundary. We prove reliability and efficiency of the estimators and illustrate their performance in several numerical experiments.

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G. Acosta, J. P. Borthagaray. A fractional Laplace equation: Regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 2017;55(2):472–495. DOI: https://doi.org/10.1137/15M1033952

M. Ainsworth, J. T. Oden. A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 1997;142(1):1–88. DOI: https://doi.org/10.1016/S0045-7825(96)01107-3

R. Altmann. A PDAE formulation of parabolic problems with dynamic boundary conditions. Appl. Math. Lett. 2019;90:202–208. DOI: https://doi.org/10.1016/j.aml.2018.11.010

R. Altmann. “Regularization and Simulation of Constrained Partial Differential Equations”. PhD thesis. Technische Universität Berlin, 2015.

R. Altmann, B. Kovács, C. Zimmer. Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions. IMA J. Numer. Anal. 2023;2(43):950–975. DOI: https://doi.org/10.1093/imanum/drac002

R. Altmann, B. Verfürth. A multiscale method for heterogeneous bulk–surface coupling. Multiscale Model. Simul. 2021;19(1):374–400. DOI: https://doi.org/10.1137/20M1338290

R. Altmann, C. Zimmer. Dissipation-preserving discretization of the Cahn–Hilliard equation with dynamic boundary conditions. Appl. Numer. Math. 2023:254–269. DOI: https://doi.org/10.1016/j.apnum.2023.04.012

R. Altmann, C. Zimmer. Second-order bulk–surface splitting for parabolic problems with dynamic boundary conditions. IMA J. Numer. Anal. 2023. published online. DOI: https://doi.org/10.1093/imanum/drad062

S. Bartels. Numerical Approximation of Partial Differential Equations. Cham: Springer, 2016. DOI: https://doi.org/10.1007/978-3-319-32354-1

D. Braess. Finite Elements - Theory, Fast Solvers, and Applications in Solid Mechanics. Third. New York, NY: Cambridge University Press, 2007. DOI: https://doi.org/10.1017/CBO9780511618635

J. H. Bramble, J. E. Pasciak, P. S. Vassilevski. Computational scales of Sobolev norms with application to preconditioning. Math. Comp. 2000;69(230):463–480. DOI: https://doi.org/10.1090/S0025-5718-99-01106-0

F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods. New York, NY: Springer, 1991. DOI: https://doi.org/10.1007/978-1-4612-3172-1

C. Carstensen. Clément interpolation and its role in adaptive finite element error control. Partial Differential Equations and Functional Analysis: The Philippe Clément Festschrift. Ed. by E. Koelink, J. van Neerven, B. de Pagter, G. Sweers, A. Luger, H. Woracek. Basel: Birkhäuser Verlag, 2006:27–43. DOI: https://doi.org/10.1007/3-7643-7601-5_2

C. Carstensen, M. Feischl, M. Page, D. Praetorius. Axioms of adaptivity. Comput. Math. with Appl. 2014;67(6):1195–1253. DOI: https://doi.org/10.1016/j.camwa.2013.12.003

P. Clément. Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 1975;2:77–84. DOI: https://doi.org/10.1051/m2an/197509R200771

P. Csomós, B. Farkas, B. Kovács. Error estimates for a splitting integrator for abstract semilinear boundary coupled systems. IMA J. Numer. Anal. 2023;published online. DOI: https://doi.org/10.1093/imanum/drac079

A. Demlow, R. Stevenson. Convergence and quasi-optimality of an adaptive finite element method for controlling L2 errors. Numer. Math. 2011;117:185–218. DOI: https://doi.org/10.1007/s00211-010-0349-9

W. Dörfler. A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 1996;33(3):1106–1124. DOI: https://doi.org/10.1137/0733054

E. Emmrich, V. Mehrmann. Operator differential-algebraic equations arising in fluid dynamics. Comp. Methods Appl. Math. 2013;13(4):443–470. DOI: https://doi.org/10.1515/cmam-2013-0018

J. Escher. Quasilinear parabolic systems with dynamical boundary conditions. Commun. Part. Diff. Eq. 1993;18(7-8):1309–1364. DOI: https://doi.org/10.1080/03605309308820976

G. Fairweather. On the approximate solution of a diffusion problem by Galerkin methods. J. Inst. Math. Appl. 1979;24(2):121–137. DOI: https://doi.org/10.1093/imamat/24.2.121

A. Favini, G. R. Goldstein, J. A. Goldstein, S. Romanelli. The heat equation with generalized Wentzell boundary condition. J. Evol. Equ. 2002;2(1):1–19. DOI: https://doi.org/10.1007/s00028-002-8077-y

S. Funken, D. Praetorius, P. Wissgott. Efficient implementation of adaptive P1-FEM in Matlab. Comput. Methods Appl. Math. 2011;11(4):460–490. DOI: https://doi.org/10.2478/cmam-2011-0026

H. Gajewski, K. Gröger, K. Zacharias. Nichtlineare Operatorgleichungen und Operatordifferential-Gleichungen. Berlin: Akademie-Verlag, 1974. DOI: https://doi.org/10.1002/mana.19750672207

D. Gilbarg, N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 2001. DOI: https://doi.org/10.1007/978-3-642-61798-0

G. R. Goldstein. Derivation and physical interpretation of general boundary conditions. Adv. Differential Equ. 2006;11(4):457–480. DOI: https://doi.org/10.57262/ade/1355867704

E. Hairer, C. Lubich, M. Roche. The Numerical Solution of Differential-Algebraic Systems by Runge–Kutta Methods. Berlin: Springer-Verlag, 1989. DOI: https://doi.org/10.1007/BFb0093947

E. Hairer, G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Second. Berlin: Springer-Verlag, 1996. DOI: https://doi.org/10.1007/978-3-642-05221-7

R. Hiptmair, S. Mao. Stable multilevel splittings of boundary edge element spaces. BIT Numer. Math. 2012;52:661–685. DOI: https://doi.org/10.1007/s10543-012-0369-1

J. Kaˇcur. Method of Rothe in evolution equations. Leipzig: BSB B. G. Teubner Verlagsgesellschaft, 1985.

B. Kovács, C. Lubich. Numerical analysis of parabolic problems with dynamic boundary conditions. IMA J. Numer. Anal. 2017;37(1):1–39. DOI: https://doi.org/10.1093/imanum/drw015

R. Lamour, R. März, C. Tischendorf. Differential-Algebraic Equations: A Projector Based Analysis. Berlin, Heidelberg: Springer-Verlag, 2013. DOI: https://doi.org/10.1007/978-3-642-27555-5

P. Lancaster, M. Tismenetsky. The Theory of Matrices: With Applications. Second edition. San Diego, CA: Academic Press, Inc., 1985.

I. Lasiecka. Mathematical Control Theory of Coupled PDEs. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2002. DOI: https://doi.org/10.1137/1.9780898717099

J.-L. Lions, E. Magenes. Non-Homogeneous Boundary Value Problems and Applications I. New York, NY: Springer-Verlag, 1972. DOI: https://doi.org/10.1007/978-3-642-65217-2

M. K. Lipinski. “A posteriori Fehlerschätzer für Sattelpunktsformulierungen nicht-homogener Randwertprobleme”. PhD thesis. Ruhr Universität Bochum, 2004.

C. Liu, H. Wu. An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: Model derivation and mathematical analysis. Arch. Rational Mech. Anal. 2019;233:167–247. DOI: https://doi.org/10.1007/s00205-019-01356-x

W. Mitchell. Adaptive refinement for arbitrary finite-element spaces with hierarchical bases. J. Comp. Appl. Math. 1991;36(1):65–78. DOI: https://doi.org/10.1016/0377-0427(91)90226-A

L. R. Scott, S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 1990;54(190):483–493. DOI: https://doi.org/10.1090/S0025-5718-1990-1011446-7

L. Tartar. An Introduction to Sobolev Spaces and Interpolation Spaces. New York, NY: Springer-Verlag, 2007.

M. E. Taylor. Partial Differential Equations I: Basic Theory. Second edition. New York, NY: Springer, 2011. DOI: https://doi.org/10.1007/978-1-4419-7055-8

J. L. Vázquez, E. Vitillaro. Heat equation with dynamical boundary conditions of reactive type. Commun. Part. Diff. Eq. 2008;33(4):561–612. DOI: https://doi.org/10.1080/03605300801970960

R. Verfürth. A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford: Oxford University Press, 2013. DOI: https://doi.org/10.1093/acprof:oso/9780199679423.001.0001

E. Vitillaro. On the the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and source. J. Differ. Equations. 2018;265(10):4873–4941. DOI: https://doi.org/10.1016/j.jde.2018.06.022

V. Vrábel’, M. Slodiˇcka. Nonlinear parabolic equation with a dynamical boundary condition of diffusive type. Appl. Math. Comput. 2013;222:372–380. DOI: https://doi.org/10.1016/j.amc.2013.07.057

C. Zimmer. “Temporal Discretization of Constrained Partial Differential Equations”. PhD thesis. Technische Universität Berlin, 2021.

A Posteriori Error Estimation for Parabolic Problems with Dynamic Boundary Conditions

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