Abstract
This paper analyzes the discrete energy laws associated with first-order system least-squares (FOSLS) discretizations of time-dependent partial differential equations. Using the heat equation and the time-dependent Stokes’ equation as examples, we discuss how accurately a FOSLS finite-element formulation adheres to the underlying energy law associated with the physical system. Using regularity arguments involving the initial condition of the system, we are able to give bounds on the convergence of the discrete energy law to its expected value (zero in the examples presented here). Numerical experiments are performed, showing that the discrete energy laws hold with order \(\mathcal O\left( h^{2p}\right) \), where h is the mesh spacing and p is the order of the finite-element space. Thus, the energy law conformance is held with a higher order than the expected, \(\mathcal {O}\left( h^p\right) \), convergence of the finite-element approximation. Finally, we introduce an abstract framework for analyzing the energy laws of general FOSLS discretizations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adler, J.H., Manteuffel, T.A., McCormick, S.F., Nolting, J.W., Ruge, J.W., Tang, L.: Efficiency based adaptive local refinement for first-order system least-squares formulations. SIAM J. Sci. Comput. 33(1), 1–24 (2011). http://dx.doi.org/10.1137/100786897
Adler, J.H., Manteuffel, T.A., McCormick, S.F., Ruge, J.W.: First-order system least squares for incompressible resistive magnetohydrodynamics. SIAM J. Sci. Comput. 32(1), 229–248 (2010). http://dx.doi.org/10.1137/080727282
Adler, J.H., Manteuffel, T.A., McCormick, S.F., Ruge, J.W., Sanders, G.D.: Nested iteration and first-order system least squares for incompressible, resistive magnetohydrodynamics. SIAM J. Sci. Comput. 32(3), 1506–1526 (2010). http://dx.doi.org/10.1137/090766905
Bochev, P., Cai, Z., Manteuffel, T.A., McCormick, S.F.: Analysis of velocity-flux first-order system least-squares principles for the Navier-Stokes equations. I. SIAM J. Numer. Anal. 35(3), 990–1009 (1998). http://dx.doi.org/10.1137/S0036142996313592
Bochev, P., Gunzburger, M.: Analysis of least-squares finite element mehtods for the Stokes equations. Math. Comput. 63(208), 479–506 (1994)
Bochev, P., Manteuffel, T.A., McCormick, S.F.: Analysis of velocity-flux least-squares principles for the Navier-Stokes equations. II. SIAM J. Numer. Anal. 36(4), 1125–1144 (1999). (Electronic). http://dx.doi.org/10.1137/S0036142997324976
Bramble, J.H., Kolev, T.V., Pasciak, J.: A least-squares approximation method for the time-harmonic Maxwell equations. J. Numer. Math. 13, 237–263 (2005)
Cai, Z., Lazarov, R., Manteuffel, T.A., McCormick, S.F.: First-order system least squares for second-order partial differential equations. I. SIAM J. Numer. Anal. 31(6), 1785–1799 (1994). http://dx.doi.org/10.1137/0731091
Cai, Z., Manteuffel, T.A., McCormick, S.F.: First-order system least squares for second-order partial differential equations. II. SIAM J. Numer. Anal. 34(2), 425–454 (1997). http://dx.doi.org/10.1137/S0036142994266066
Davis, T.A.: Algorithm 832: Umfpack v4.3–an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2), 196–199 (2004). http://doi.acm.org/10.1145/992200.992206
Feng, J., Liu, C., Shen, J., Yue, P.: A energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges. In: Calderer, M.C.T., Terentjev, E.M. (eds.) Modeling of Soft Matter. The IMA Volumes in Mathematics and its Applications, vol. 141, pp. 1–26. Springer, New York (2005). https://doi.org/10.1007/0-387-32153-5_1
Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice-Hall Inc., Englewood Cliffs (1963). Revised English edition translated and edited by R.A. Silverman
Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. LNM, vol. 749. Springer, Berlin (1979). https://doi.org/10.1007/BFb0063447
Heys, J.J., Lee, E., Manteuffel, T.A., McCormick, S.F.: An alternative least-squares formulation of the Navier-Stokes equations with improved mass conservation. J. Comput. Phys. 226(1), 994–1006 (2007). http://dx.doi.org/10.1016/j.jcp.2007.05.005
Hyon, Y., Kwak, D.Y., Liu, C.: Energetic variational approach in complex fluids: maximum dissipation principle. Discret. Contin. Dyn. Syst. 26(4), 1291–1304 (2010). http://dx.doi.org/10.3934/dcds.2010.26.1291
Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover Publications Inc., Mineola (2009). Reprint of the 1987 edition
Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45(1–3), 285–312 (1984). http://dx.doi.org/10.1016/0045-7825(84)90158-0
Langer, U., Moore, S.E., Neumüller, M.: Space-time isogeometric analysis of parabolic evolution problems. Comput. Methods Appl. Mech. Eng. 306, 342–363 (2016). http://dx.doi.org/10.1016/j.cma.2016.03.042
Masud, A., Hughes, T.J.R.: A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems. Comput. Methods Appl. Mech. Eng. 146(1–2), 91–126 (1997). http://dx.doi.org/10.1016/S0045-7825(96)01222-4
MFEM: Modular finite element methods library (2016). http://mfem.org
Solonnikov, V.A.: Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations. Trudy Mat. Inst. Steklov. 70, 213–317 (1964)
Solonnikov, V.A.: On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov. 83, 3–163 (1965)
Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992). http://dx.doi.org/10.1137/1034116
Acknowledgements
The work of J. H. Adler was supported in part by NSF DMS-1216972. I. V. Lashuk was supported in part by NSF DMS-1216972 (Tufts University) and DMS-1418843 (Penn State). S. P. MacLachlan was partially supported by an NSERC Discovery Grant. The research of L. T. Zikatanov was supported in part by NSF DMS-1720114 and the Department of Mathematics at Tufts University.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Adler, J.H., Lashuk, I., MacLachlan, S.P., Zikatanov, L.T. (2018). Discrete Energy Laws for the First-Order System Least-Squares Finite-Element Approach. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-73441-5_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73440-8
Online ISBN: 978-3-319-73441-5
eBook Packages: Computer ScienceComputer Science (R0)