Abstract
The interior penalty methods using C0 Lagrange elements (C0IPG) developed in the recent decade for the fourth order problems are an interesting topic in academia at present. In this paper, we discuss the adaptive fashion of C0IPG method for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a C0IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.
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Ainsworth M, Oden J T. A Posterior Error Estimation in Finite Element Analysis. New York: Wiley-Inter Science, 2011
An J, Shen J. A spectral-element method for transmission eigenvalue problems. J Sci Comput, 2013, 57: 670–688
Babuska I, Osborn J E. Eigenvalue problems. In: Finite Element Methods (Part 1). Handbook of Numerical Analysis, vol. 2. North-Holand: Elsevier, 1991, 640–787
Babuska I, Rheinboldt W C. Error estimates for adaptive finite element computations. SIAM J Numer Anal, 1978, 15: 736–754
Brenner S C. C 0 interior penalty methods. In: Frontiers in Numerical Analysis-Durham 2010. Lecture Notes in Computational Science and Engineering, vol. 85. New York: Springer-Verlag, 2012, 79–147
Brenner S C, Gedicke J, Sung L-Y. Adaptive C 0 interior penalty method for biharmonic eigenvalue problems. Oberwolfach Rep, 2013, 10: 3265–3267
Brenner S C, Monk P, Sun J. C 0IPG Method for Biharmonic Eigenvalue Problems. Spectral and High Order Methods for Partial Differential Equations, ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol. 106. Switzerland: Springer, 2015
Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods, 2nd ed. New york: Springer-Verlag, 2002
Brenner S C, Sung L. C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J Sci Comput, 2005, 22/23: 83–118
Brenner S C, Wang K, Zhao J. Poincaré-Friedrichs inequalities for piecewise H 2 functions. Numer Funct Anal Optim, 2004, 25: 463–478
Cakoni F, Cayoren M, Colton D. Transmission eigenvalues and the nondestructive testing of dielectrics. Inverse Problems, 2009, 24: 065016
Cakoni F, Gintides D, Haddar H. The existence of an infinite discrete set of transmission eigenvalues. SIAM J Math Anal, 2010, 42: 237–255
Cakoni F, Haddar H. On the existence of transmission eigenvalues in an inhomogeneous medium. Appl Anal, 2009, 88: 475–493
Cakoni F, Monk P, Sun J. Error analysis for the finite element approximation of transmission eigenvalues. Comput Methods Appl Math, 2014, 14: 419–427
Chatelin F. Spectral Approximations of Linear Operators. New York: Academic Press, 1983
Chen L. iFEM: An Integrated Finite Element Method Package in MATLAB. Technical Report. Irvine: University of California at Irvine, 2009
Ciarlet P G. Basic error estimates for elliptic proplems. In: Finite Element Methods (Part1). Handbook of Numerical Analysis, vol. 2. North-Holand: Elsevier, 1991, 17–351
Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. Applied Mathematical Sciences, vol. 93. New York: Springer, 1998
Colton D, Monk P, Sun J. Analytical and computational methods for transmission eigenvalues. Inverse Problems, 2010, 26: 045011
Dai X, Xu J, Zhou A. Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer Math, 2008, 110: 313–355
Dörfler W. A convergent adaptive algorithm for Poisson’s equation, SIAM J Numer Anal, 1996, 33: 1106–1124
Engel G, Garikipati K, Hughes T, et al. Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput Methods Appl Mech Engrg, 2001, 191: 3669–3750
Geng H, Ji X, Sun J, et al. C 0IP methods for the transmission eigenvalue problem. J Sci Comput, 2016, 68: 326–338
Gudi T. A new error analysis for discontinuous finite element methods for the linear elliptic problems. Math Comp, 2010, 79: 2169–2189
Han J, Yang Y. An adaptive finite element method for the transmission eigenvalue problem. J Sci Comput, 2016, 69: 326–338
Han J, Yang Y. An H m-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues. Sci China Math, 2017, 60: 1529–1542
Ji X, Sun J, Turner T. Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues. ACM Trans Math Software, 2012, 38: 1–8
Ji X, Sun J, Xie H. A multigrid method for Helmholtz transmission eigenvalue problems. J Sci Comput, 2014, 60: 276–294
Ji X, Sun J, Yang Y. Optimal penalty parameter for C 0IPDG. Appl Math Lett, 2014, 37: 112–117
Kleefeld A. A numerical method to compute interior transmission eigenvalues. Inverse Problems, 2013, 29: 104012
Li H, Yang Y. C 0 IPG adaptive algorithms for biharmonic eigenvalue problem. Numer Algorithms, 2018, 78: 553–567
Monk P, Sun J. Finite element methods of Maxwell transmission eigenvalues. SIAM J Sci Comput, 2012, 34: 247–264
Morin P, Nochetto R H, Siebert K. Convergence of adaptive finite element methods. SIAM Rev, 2002, 44: 631–658
Oden J T, Reddy J N. An Introduction to the Mathematical Theory of Finite Elements. New York: Courier Dover Publications, 2012
Rynne B P, Sleeman B D. The interior transmission problem and inverse scattering from inhomogeneous media. SIAM J Math Anal, 1991, 22: 1755–1762
Shi Z, Wang M. Finite Element Methods. Beijing: Scientific Publishers, 2013
Sun J. Estimation of transmission eigenvalues and the index of refraction from Cauchy data. Inverse Problems, 2011, 27: 015009
Sun J. Iterative methods for transmission eigenvalues. SIAM J Numer Anal, 2014, 49: 1860–1874
Sun J, Xu L. Computation of Maxwell’s transmission eigenvalues and its applications in inverse medium problems. Inverse Problems, 2013, 29: 104013
Verfürth R. A Posteriori Error Estimation Techniques. Oxford: Oxford University Press, 2013
Wells G N, Dung N T. A C 0 discontinuous Galerkin formulation for Kirhhoff plates. Comput Methods Appl. Mech Engrg, 2007, 196: 3370–3380
Yang Y, Bi H, Li H, et al. Mixed method for the Helmholtz transmission eigenvalues. SIAM J Sci Comput, 2016, 38: 1383–1403
Yang Y, Bi H, Li H, et al. A C 0IPG method and its error estimates for the Helmholtz transmission eigenvalue problem. J Comput Appl Math, 2017, 326: 71–86
Yang Y, Han J, Bi H. Error estimates and a two grid scheme for approximating transmission eigenvalues. ArXiv: 1506.06486, 2016
Yang Y, Han J, Bi H. Non-conforming finite element methods for transmission eigenvalue problem. Comput Methods Appl Mech Engrg, 2016, 307: 144–163
Zeng F, Sun J, Xu L. A spectral projection method for transmission eigenvalues. Sci China Math, 2016, 59: 1613–1622
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11561014). The authors thank the referees for their valuable comments and suggestions that led to the large improvement of this paper.
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Li, H., Yang, Y. An adaptive C0IPG method for the Helmholtz transmission eigenvalue problem. Sci. China Math. 61, 1519–1542 (2018). https://doi.org/10.1007/s11425-017-9334-9
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DOI: https://doi.org/10.1007/s11425-017-9334-9
Keywords
- transmission eigenvalues
- interior penalty Galerkin method
- Lagrange elements
- a posteriori error estimates
- adaptive algorithm