### CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK

Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski

1. 1. Department of Mathematics, University of California, San Diego, CA 92123, USA;
2. Department of Mathematics and Statistics, Cal Poly Pomona, USA
• Received:2018-12-02 Revised:2019-04-03 Online:2020-09-15 Published:2021-03-11
• Supported by:
MS was partially supported by NSF Awards 1620366, 1262982, and 1217175. YL was partially supported by NSF Award 1620366. AM adn RS were partially supported by NSF Award 1217175.

Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski. CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK[J]. Journal of Computational Mathematics, 2020, 38(5): 748-767.

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poissonls equation on contractible domains in ${\Bbb R}$2, which can be viewed as a boundary problem on the de Rham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371) developed fundamental tools for a posteriori analysis on the de Rham complex. In this paper, we use tools in FEEC to construct convergence and complexity results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.

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 [1] M. Ainsworth and J. Oden, A Posteriori Error Estimation in Finite Element Analysis, John Wiley& Sons, Inc., 2000.[2] A. Alonso. Error estimators for a mixed method. Numer. Math., 74:4(1996), 385-395.[3] D.N. Arnold, R.S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15(2006), 1-155.[4] D.N. Arnold, R.S. Falk, and R. Winther, Finite element exterior calculus:from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.), 47:2(2010), 281-354.[28] P. Morin, R.H. Nochetto, and K.G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44:4(2002), 631-658(electronic). Revised reprint of "Data oscillation and convergence of adaptive FEM", SIAM J. Numer. Anal. 38:2(2000), 466-488.[29] P. Morin, K.G. Siebert, and A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 18:5(2008), 707-737.[30] J.C. Nédélec, Mixed finite elements in ${\Bbb R}$3, Numer. Math., 35:3(1980), 315-341.[31] J.C. Nédélec, A new family of mixed finite elements in ${\Bbb R}$3, Numer. Math., 50:1(1986), 57-81.[32] R.H. Nochetto and A. Veeser, Primer of adaptive finite element methods, Springer-Verlag, Heidelberg, 2012, Lecture Notes in Mathematics.[33] P.A. Raviart and J. Thomas, A mixed finite element method for 2nd order elliptic problems, Springer, Berlin, 1977, Lecture notes in Mathematics 606.[34] S. Repin, A posteriori estimates for partial differential equations, volume 4 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.[35] J. Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp., 77:262(2008), 633-649.[36] R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7:2(2007), 245-269.[37] R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77:261(2008), 227-241(electronic).[38] R. Verfürth, A review of a posteriori error estimation and adaptive mesh refinement tecniques, B. G. Teubner, 1996.[5] I. Babuška and W.C. Rheinboldt, A posteriori error estimates for the finite element method, International Journal for Numerical Methods in Engineering, 12(1978), 1597-1615.[6] I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems, Numer. Math., 44(1984), 75-102.[7] R. Becker and S. Mao, An optimally convergent adaptive mixed finite element method, Numer. Math., 111(2008), 35-54.[8] P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., 97:2(2004), 219-268.[9] A. Bossavit, Whitney forms:a class of finite elements for three-dimensional computations in electromagnetism, Science, Measurement and Technology, IEE Proceedings A, 135:8(1988), 493500.[10] F. Brezzi, J. Douglas, and L.D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47:2(1985), 217-235.[11] J. Brüning and M. Lesch, Hilbert complexes, J. Funct. Anal., 108:1(1992), 88-132.[12] J.M. Cascon, C. Kreuzer, R.H. Nochetto, and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46:5(2008), 2524-2550.[13] L. Chen, M. Holst, and J. Xu, Convergence and optimality of adaptive mixed finite element methods, Math. Comp., 78:265(2009), 35-53.[14] L. Chen and Y. Wu, Convergence of adaptive mixed finite element methods for the Hodge Laplacian equation:without harmonic forms, SIAM J. Numer. Anal., 15(2017), 2905-2929.[15] S.H. Christiansen and R. Winther, Smoothed projections in finite element exterior calculus, Math. Comp., 77:262(2008), 813-829.[16] A. Demlow, Convergence and quasi-optimality of adaptive finite element methods for harmonic forms, Numer. Math., 136(2017), 941-971.[17] A. Demlow and A.N. Hirani, A posteriori error estimates for finite element exterior calculus:The de Rham complex, Found. Comput. Math., 14(2014), 1337-1371.[18] W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33(1996), 1106-1124.[19] P.W. Gross and P.R. Kotiuga, Electromagnetic theory and computation:a topological approach, volume 48 of Mathematical Sciences Research Institute Publications, Cambridge University Press, Cambridge, 2004.[20] M. Holst, A. Mihalik, and R. Szypowski, Convergence and optimality of adaptive methods in the finite element exterior calculus framework, Preprint, Available as http://arxiv.org/abs/1306.1886 arXiv:1306.1886v2[math.NA].[21] M. Holst and A. Stern, Geometric variational crimes:Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces, Found. Comput. Math., 12:3(2012), 263-293.[22] M. Holst and A. Stern, Semilinear mixed problems on Hilbert complexes and their numerical approximation, Found. Comput. Math., 12:3(2012), 363-387.[23] M. Holst, G. Tsogtgerel, and Y. Zhu, Local convergence of adaptive methods for nonlinear partial differential equations, Preprint, Available as http://arxiv.org/abs/1001.1382 arXiv:1001.1382[math.NA].[24] J. Hu and G. Yu, A unified analysis of quasi-optimal convergence for adaptive mixed finite element methods, SIAM J. Numer. Anal., 56:1(2018), 296-316.[25] J. Huang and Y. Xu, Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation, Sci China Math, 55:5(2012), 1083-1098.[26] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, second edition, 1976, Grundlehren der Mathematischen Wissenschaften, Band 132.[27] Y. Li, Some convergence and optimality results of adaptive mixed methods in finite element exterior calculus, to appear in SIAM J. Numer. Anal., Available as http://arxiv.org/abs/1811.11143 arXiv:1811.11143[math.NA].[28] P. Morin, R.H. Nochetto, and K.G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44:4(2002), 631-658(electronic). Revised reprint of "Data oscillation and convergence of adaptive FEM", SIAM J. Numer. Anal. 38:2(2000), 466-488.[29] P. Morin, K.G. Siebert, and A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 18:5(2008), 707-737.[30] J.C. Nédélec, Mixed finite elements in ${\Bbb R}$3, Numer. Math., 35:3(1980), 315-341.[31] J.C. Nédélec, A new family of mixed finite elements in ${\Bbb R}$3, Numer. Math., 50:1(1986), 57-81.[32] R.H. Nochetto and A. Veeser, Primer of adaptive finite element methods, Springer-Verlag, Heidelberg, 2012, Lecture Notes in Mathematics.[33] P.A. Raviart and J. Thomas, A mixed finite element method for 2nd order elliptic problems, Springer, Berlin, 1977, Lecture notes in Mathematics 606.[34] S. Repin, A posteriori estimates for partial differential equations, volume 4 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.[35] J. Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp., 77:262(2008), 633-649.[36] R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7:2(2007), 245-269.[37] R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77:261(2008), 227-241(electronic).[38] R. Verfürth, A review of a posteriori error estimation and adaptive mesh refinement tecniques, B. G. Teubner, 1996.
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