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Calcolo
Published in: Calcolo 2/2023

01-06-2023

An adaptive immersed finite element method for linear parabolic interface problems with nonzero flux jump

Authors: Tanushree Ray, Rajen Kumar Sinha

Published in: Calcolo | Issue 2/2023

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Abstract

We study an adaptive immersed finite element method for solving parabolic interface problems with nonzero flux jump in a two-dimensional convex polygonal domain. We use unfitted finite element meshes to discretize the spatial domain where the grid points do not need to fit the interface. New error indicators are introduced to control the error due to unfitted meshes. We derive a global upper bound as well as a local lower bound for the error using energy method. An adaptive algorithm for immersed finite element method is provided using the error indicators. Numerical experiment is presented to demonstrate the behavior of the adaptive algorithm for the proposed method.
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Literature
1.
Babuška, I., Rheinboldt, W.C.: A posteriori error estimates for the finite element method. Int. J. Numer. Methods Eng. 12, 1597–1615 (1978)CrossRefMATH
2.
Chen, Z., Dai, S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24, 443–462 (2002)MathSciNetCrossRefMATH
3.
Chen, Z., Feng, J.: An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comp. 73, 1167–1193 (2004)MathSciNetCrossRefMATH
4.
Chen, Z., Wu, Z., Xiao, Y.: An adaptive immersed finite element method with arbitrary Lagrangian–Eulerian scheme for parabolic equations in time variable domains. Int. J. Numer. Anal. Model. 12, 567–591 (2015)MathSciNetMATH
5.
Chen, Z., Xiao, Y., Zhang, L.: The adaptive immersed interface finite element method for elliptic and Maxwell interface problems. J. Comput. Phys. 228, 5000–5019 (2009)MathSciNetCrossRefMATH
6.
Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79, 175–202 (1998)MathSciNetCrossRefMATH
7.
Clément, P.: Approximation by finite element functions using local regularization. RAIRO. Anal. Numer. 9, 77–84 (1975)MathSciNetMATH
8.
9.
Dörfler, W.: A time–space adaptive algorithm for the linear time-dependent Schrödinger equation. Numer. Math. 73, 419–448 (1996)MathSciNetCrossRefMATH
10.
Feng, W., He, X., Lin, Y., Zhang, X.: Immersed finite element method for interface problems with algebraic multigrid solver. Commun. Comput. Phys. 15, 1045–1067 (2014)MathSciNetCrossRefMATH
11.
Gao, T., Zhang, W.H., Zhu, J.H., Xu, Y.J., Bassir, D.H.: Topology optimization of heat conduction problem involving design-dependent heat load effect. Finite Elements Anal. Design 44, 805–813 (2008)CrossRef
12.
Gong, Y., Li, B., Li, Z.: Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions. SIAM J. Numer. Anal. 46, 472–495 (2008)MathSciNetCrossRefMATH
13.
Gulrajani, R.M.: The forward and inverse problems of electrocardiography. IEEE Eng. Med. BioMag. 17, 84–101 (1998)CrossRef
14.
Guzmán, J., Sánchez, M., Sakis, M.: On the accuracy of finite element approximation to a class of interface problems. Math. Comput. 85, 2071–2098 (2016)MathSciNetCrossRefMATH
15.
Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191, 5537–5552 (2002)MathSciNetCrossRefMATH
16.
He, X., Lin, T., Lin, Y., Zhang, X.: Immersed finite element methods for parabolic equations with moving interface. Numer. Methods Partial Differ. Equ. 29, 619–646 (2013)MathSciNetCrossRefMATH
17.
18.
Ladyzenskaja, O.A., Solonnikov, V.A.: Numerical Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith, Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence, RI (1967)
19.
Leveque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)MathSciNetCrossRefMATH
20.
Li, Z., Ito, K.: The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. Frontiers in Applied Mathematics, vol. 33. SIAM, Philadaelphia, PA (2006)CrossRefMATH
21.
Li, Z., Lin, T., Wu, X.: New cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96, 61–98 (2003)MathSciNetCrossRefMATH
22.
Lin, T., Yang, Q., Zhang, X.: Partially penalized immersed finite element methods for parabolic interface problems. Numer. Methods Partial Differ. Equ. 31, 1925–1947 (2015)MathSciNetCrossRefMATH
23.
Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive fem. SIAM J. Numer. Anal. 38, 466–488 (2000)MathSciNetCrossRefMATH
24.
Pegoretti, A., Fambri, L., Zappini, G., Bianchetti, M.: Finite element analysis of a glass fibre reinforced composite endodontic post. Biomaterials 23, 2667–2682 (2002)CrossRef
25.
Schmidt, A., Siebert, K.G.: ALBERT: An Adaptive Hierarchical Finite Element Toolbox. University of Freiburg, Germany (2000)
26.
Sen Gupta, J., Sinha, R.K., Reddy, G.M.M., Jain, J.: A posteriori error analysis of two-step backward differentiation formula finite element approximation for parabolic interface problems. J. Sci. Comput. 69, 406–429 (2016)MathSciNetCrossRefMATH
27.
Sen Gupta, J., Sinha, R.K., Reddy, G.M.M., Jain, J.: New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems. Numer. Methods Part. Differ. Equ. 33, 570–598 (2017)MathSciNetCrossRefMATH
28.
Sinha, R.K., Deka, B.: Optimal error estimates for linear parabolic problems with discontinuous coefficients. SIAM J. Numer. Anal. 43, 733–749 (2005)MathSciNetCrossRefMATH
29.
Sinha, R.K., Deka, B.: An unfitted finite-element method for elliptic and parabolic interface problems. IMA J. Numer. Anal. 27, 529–549 (2007)MathSciNetCrossRefMATH
30.
Sinha, R.K., Deka, B.: \({L^{\infty }({L}^2)}\) and \({L^{\infty }({H}^1)}\) norms error estimates in finite element method for linear parabolic interface problems. Numer. Funct. Anal. Optim. 32, 267–285 (2011)MathSciNetCrossRefMATH
31.
Verfürth, R.: A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Eng. 176, 419–440 (1999)MathSciNetCrossRefMATH
Metadata
Title
An adaptive immersed finite element method for linear parabolic interface problems with nonzero flux jump
Authors
Tanushree Ray
Rajen Kumar Sinha
Publication date
01-06-2023
Publisher
Springer International Publishing
Published in
Calcolo / Issue 2/2023
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-023-00515-7

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