Abstract
A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive \({\epsilon}\)) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.
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References
Apel, Th.: Anisotropic Finite Elements. Teubner, Stuttgart (1999)
Apel, Th., Sändig, A.-M., Whiteman, J.R.: 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)
Assous, F., Ciarlet, P. Jr., Labrunie, S., Segré, J.: Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method. J. Comput. Phys. 191, 147–176 (2003)
Assous, F., Ciarlet, P. Jr., Garcia, E., Segré, J.: Time-dependent Maxwell’s equations with charges in singular geometries. Comput. Methods Appl. Mech. Eng. 196, 665–681 (2006)
Assous, F., Ciarlet, P. Jr., Labrunie, S., Lohrengel, S.: The singular complement method. In: Debit, N., Garbey, M., Hoppe, R., Keyes, D., Kuznetsov, Y., Périaux, J.(eds) Domain Decomposition Methods in Science and Engineering, pp. 161–189. CIMNE, Barcelona (2002)
Assous, F., Ciarlet, P. Jr., Sonnendrücker, E.: Resolution of the Maxwell equation in a domain with reentrant corners. M 2(N Math. Model. Numer. Anal. 32), 359–389 (1998)
Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L.(eds) Handbook of Numerical Analysis II, pp. 641–787. North-Holland, Amsterdam (1991)
Băcuţă, C., Nistor, V., Zikatanov, L.T.: Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numer. Math. 100, 165–184 (2005)
Bathe, K.J., Nitikitpaiboon, C., Wang, X.: A mixed displacement-based finite element formulation for acoustic fluid-structure interaction. Comput. Struct. 56, 225–237 (1995)
Bermúdez, A., Rodríguez, R.: Finite element computation of the vibration modes of a fluid-solid system. Comput. Methods Appl. Mech. Eng. 119, 355–370 (1994)
Birman, M., Solomyak, M.: L 2-theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42, 75–96 (1987)
Boffi, D., Gastaldi, L.: On the “-grad div s curl rot” operator. In: Computational fluid and solid mechanics, vol. 1, 2 (Cambridge, MA, 2001), pp. 1526–1529. Elsevier, Amsterdam (2001)
Bonnet-Ben Dhia, A.-S., Hazard, C., Lohrengel, S.: A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J. Appl. Math. 59, 2028–2044 (1999)
Bossavit, A.: Discretization of electromagnetic problems: the “generalized finite differences” approach. In: Ciarlet, P.G., Schilders, W.H.A., Ter Maten, E.J.W.(eds) Handbook of numerical analysis, vol XIII, Handb. Numer. Anal., XIII, pp. 105–197. North-Holland, Amsterdam (2005)
Brenner, S.C., Carstensen, C.: Finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R.(eds) Encyclopedia of Computational Mechanics, pp. 73–118. Wiley, Weinheim (2004)
Brenner, S.C., Li, F., Sung, L.-Y.: A locally divergence-free interior penalty method for two dimensional curl–curl problems. SIAM J. Numer. Anal. 46, 1190–1211 (2008)
Brenner, S.C., Li, F., Sung, L.-Y.: A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations. Math. Comp. 76, 573–595 (2007)
Brenner, S.C., Li, F., Sung, L.-Y.: A nonconforming penalty method for a two dimensional curl–curl problem. preprint, (2007)
Brenner, S.C., Li, F., Sung, L.-Y.: Parameter free nonconforming Maxwell eigensolvers without spurious eigenmodes. preprint, (2008)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, Heidelberg (2002)
Ciarlet, P. Jr.: Augmented formulations for solving Maxwell equations. Comput. Methods Appl. Mech. Eng. 194, 559–586 (2005)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Costabel, M.: A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Methods Appl. Sci. 12, 36–368 (1990)
Costabel, M.: A coercive bilinear form for Maxwell’s equations. J. Math. Anal. Appl. 157, 527–541 (1991)
Costabel, M., Dauge, M.: Maxwell and Lamé eigenvalues on polyhedra. Math. Methods Appl. Sci. 22, 243–258 (1999)
Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151, 221–276 (2000)
Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93, 239–277 (2002)
Costabel, M., Dauge, M., Schwab, C.: Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci. 15, 575–622 (2005)
Coulomb, J.L.: Finite element three dimensional magnetic field computation. IEEE Trans. Magnetics 17, 3241–3246 (1981)
Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 7, 33–75 (1973)
Cui, J.: Nonconforming Multigrid Methods for Maxwell’s Equations. PhD thesis, Louisiana State University (in preparation)
Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Heidelberg (1988)
Demkowicz, L.: Finite Element Methods for Maxwell Equations. In: Stein, E., de Borst, R., Hughes, T.J.R.(eds) Encyclopedia of Computational Mechanics, pp. 723–737. Wiley, Weinheim (2004)
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer, Berlin (1986)
Grisvard, P.: Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)
Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)
Hamdi, M.A., Ousset, Y., Verchery, G.: A displacement method for the analysis of vibrations of coupled fluid–structure systems. Int. J. Numer. Methods Eng. 13, 139–150 (1978)
Hazard, C., Lohrengel, S.: A singular field method for Maxwell’s equations: numerical aspects for 2D magnetostatics. SIAM J. Numer. Anal. 40, 1021–1040 (2002)
Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)
Kato, T.: Perturbation Theory of Linear Operators. Springer, Berlin (1966)
Leis, R.: Zur Theorie elektromagnetischer Schwingungen in anisotopen inhomgenen Medien. Math. Z. 106, 213–224 (1968)
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)
Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter, Berlin (1994)
Nédélec, J.-C.: Mixed finite elements in R3. Numer. Math. 35, 315–341 (1980)
Nédélec, J.-C.: A new family of mixed finite elements in R3. Numer. Math. 50, 57–81 (1986)
Neittaanmäki, P., Picard, R.: Error estimates for the finite element approximation to a Maxwell-type boundary value problem. Numer. Funct. Anal. Optimiz. 2, 267–285 (1980)
Rahman, B.M.A., Davies, J.B.: Finite element analysis of optical and microwave waveguide problems. IEEE Trans. Microwave Theory Tech. 32, 20–28 (1984)
Schatz, A.: An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp. 28, 959–962 (1974)
Weber, C.: A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci. 2, 12–25 (1980)
Weiland, T.: On the unique numerical solution of Maxwellian eigenvalue problems in three dimensions. Part. Accel. 17, 227–242 (1985)
Witsch, K.J.: A remark on a compactness result in electromagnetic theory. Math. Methods. Appl. Sci 16, 123–129 (1993)
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The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-03-11790 and by the Humboldt Foundation through her Humboldt Research Award. The work of the third author was supported in part by the National Science Foundation under Grant No. DMS-06-52481.
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Brenner, S.C., Cui, J., Li, F. et al. A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem. Numer. Math. 109, 509–533 (2008). https://doi.org/10.1007/s00211-008-0149-7
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DOI: https://doi.org/10.1007/s00211-008-0149-7