An interior penalty method for a two dimensional curl-curl and grad-div problem

Abstract

We study an interior penalty method for a two dimensional curl-curl and grad-div problem that appears in electromagnetics and in fluid-structure interactions. The method uses discontinuous $P_1$~vector fields on graded meshes and satisfies optimal convergence rates (up to an arbitrarily small parameter) in both the energy norm and the $L_2$~norm. These theoretical results are corroborated by results of numerical experiments. References

• Th. Apel. Anisotropic Finite Elements. Teubner, Stuttgart, 1999.
• Th. Apel, A.-M. Saendig, and J.R. Whiteman. Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci., 19:63--85, 1996.
• F. Assous, P. Ciarlet, Jr., S. Labrunie, and J. Segre. Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method. J. Comput. Phys., 191:147--176, 2003.
• F. Assous, P. Ciarlet, Jr., E. Garcia, and J. Segre. {Time-dependent Maxwell's equations with charges in singular geometries}. Comput. Methods Appl. Mech. Engrg., 196:665--681, 2006.
• F. Assous, P. Ciarlet Jr., S. Labrunie, and S. Lohrengel. The singular complement method. In N. Debit, M. Garbey, R. Hoppe, D. Keyes, Y. Kuznetsov, and J. Periaux, editors, Domain Decomposition Methods in Science and Engineering, pages 161--189. CIMNE, Barcelona, 2002.
• F. Assous, P. Ciarlet, Jr., and E. Sonnendruecker. Resolution of the Maxwell equation in a domain with reentrant corners. M2AN Math. Model. Numer. Anal., 32:359--389, 1998.
• C. Bacuta, V. Nistor, and L. T. Zikatanov. Improving the rate of convergence of `high order finite elements' on polygons and domains with cusps. Numer. Math., 100:165--184, 2005.
• K. J. Bathe, C. Nitikitpaiboon, and X. Wang. A mixed displacement-based finite element formulation for acoustic fluid-structure interaction. Comput. Struct., 56:225--237, 1995.
• A. Bermudez and R. Rodr\OT1\i guez. Finite element computation of the vibration modes of a fluid-solid system. Comput. Methods Appl. Mech. Engrg., 119:355--370, 1994.
• D. Boffi and L. Gastaldi. On the ``$-{grad div}+s {curl rot}$'' operator. In Computational fluid and solid mechanics, Vol. 1, 2 (Cambridge, MA, 2001), pages 1526--1529. Elsevier, Amsterdam, 2001.
• A.-S. Bonnet-Ben Dhia, C. Hazard, and S. Lohrengel. A singular field method for the solution of {M}axwell's equations in polyhedral domains. SIAM J. Appl. Math., 59:2028--2044, 1999.
• S. C. Brenner. Poincare--Friedrichs inequalities for piecewise $H^1$ functions. SIAM J. Numer. Anal., 41:306--324, 2003.
• S. C. Brenner. Korn's inequalities for piecewise $H^1$ vector fields. Math. Comp., 73:1067--1087, 2004.
• S. C. Brenner and C. Carstensen. Finite Element Methods. In E. Stein, R. de Borst, and T.J.R. Hughes, editors, Encyclopedia of Computational Mechanics, pages 73--118. Wiley, Weinheim, 2004.
• S. C. Brenner, J. Cui, F. Li, and L.-Y. Sung. A nonconforming finite element method for a two-dimensional curl-curl and grad-div problem. Numer. Math., 109:509--533, 2008.
• S. C. Brenner, F. Li, and L.-Y. Sung. A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations. Math. Comp., 76:573--595, 2007.
• S. C. Brenner, F. Li, and L.-Y. Sung. A nonconforming penalty method for a two dimensional curl-curl problem. M3AS, 19:651--668, 2009.
• S. C. Brenner, F. Li, and L.-Y. Sung. A locally divergence-free interior penalty method for two dimensional curl-curl problems. SIAM J. Numer. Anal., 46:1190--1211, 2008.
• S. C. Brenner, L. Owens, and L.-Y. Sung. A weakly over-penalized symmetric interior penalty method. ETNA, 30:107--127, 2008.
• S. C. Brenner and L.-Y. Sung. A quadratic nonconforming element for $H(\protect \unhbox \voidb@x \hbox {curl};\Omega )\cap H(\protect \unhbox \voidb@x \hbox {div};\Omega )$. Appl. Math. Lett., 22:892--896, 2009.
• P. Ciarlet, Jr. Augmented formulations for solving Maxwell equations. Comput. Methods Appl. Mech. Engrg., 194:559--586, 2005.
• M. Costabel. A coercive bilinear form for Maxwell's equations. J. Math. Anal. Appl., 157:527--541, 1991.
• M. Costabel and M. Dauge. Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal., 151:221--276, 2000.
• M. Costabel and M. Dauge. Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math., 93:239--277, 2002.
• M. Costabel, M. Dauge, and C. Schwab. Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci., 15:575--622, 2005.
• M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numer., 7:33--75, 1973.
• M. A. Hamdi, Y. Ousset, and G. Verchery. A displacement method for the analysis of vibrations of coupled fluid-structure systems. Internat. J. Numer. Methods Engrg., 13:139--150, 1978.
• C. Hazard and S. Lohrengel. A singular field method for Maxwell's equations: Numerical aspects for 2D magnetostatics. SIAM J. Numer. Anal., 40:1021--1040, 2002.
• J.-M. Jin. The Finite Element Method in Electromagnetics (Second Edition). John Wiley and Sons, Inc., New York, 2002.
• P. Monk. Finite Element Methods for Maxwell's Equations. Oxford University Press, New York, 2003.
• L. Owens. Multigrid Methods for Weakly Over-Penalized Interior Penalty Methods. PhD thesis, University of South Carolina, 2007.
• A. Schatz. An observation concerning Ritz--Galerkin methods with indefinite bilinear forms. Math. Comp., 28:959--962, 1974.