Abstract
In this paper, we consider the div-curl problem posed on nonconvex polyhedral domains. We propose a least-squares method based on discontinuous elements with normal and tangential continuity across interior faces, as well as boundary conditions, weakly enforced through a properly designed least-squares functional. Discontinuous elements make it possible to take advantage of regularity of given data (divergence and curl of the solution) and obtain convergence also on nonconvex domains. In general, this is not possible in the least-squares method with standard continuous elements. We show that our method is stable, derive a priori error estimates, and present numerical examples illustrating the method.
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Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)
Apel, T., Nicaise, S., Schöberl, J.: Crouzeix-Raviart type finite elements on anisotropic meshes. Numer. Math., 89(2), 193–223 (2001)
Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5), 1749–1779 (electronic), 2001/2002
Bergström, R.: Least-Squares Finite Element Method with Applications in Electromagnetics. Preprint 10, Chalmers Finite Element Center, Chalmers University of Technology. To appear in Math. Models Methods Appl. Sci. 2002
Bergström, R., Bondeson, A., Johnson, C., Larson, M.G., Liu, Y., Samuelsson, K.: Adaptive Finite Element Methods in Electromagnetics. Swedish Institute of Applied Mathematics (ITM) 2 (1999)
Bergström, R., Larson, M.G.: Discontinuous/Continuous Least-Squares Finite Element Methods for Elliptic Problems. Preprint 11, Chalmers Finite Element Center, Chalmers University of Technology. To appear in Math. Models Methods Appl. Sci. 2002
Bergström, R., Larson, M.G., Samuelsson, K.: The
Finite Element Method and Multigrid for the Magnetostatic Problem. Chalmers Finite Element Center Chalmers University of Technology Preprint 2 2001Bochev, P.B., Gunzburger, M.D.: Finite Element Methods of Least-Squares Type. SIAM Rev. 40(4), 789–837 (1998)
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(6), 2028–2044 (1999)
Bramble, J. H., Pasciak, J. E.: A new approximation technique for div-curl systems. Math. Comp., 73(248), 1739–1762 (electronic), 2004
Bramble, J.H., Lazarov, R.D., Pasciak, J.E.: A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comp. 66(219), 935–955 (1997)
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)
Cao, Y., Gunzberger, M.D.: Least-Squares Finite Element Approximations to Solutions of Interface Problems. SIAM J. Numer. Anal. 35(1), 393–405 (1998)
Carey, G.F., Pehlivanov, A.I.: Local error estimation and adaptive remeshing scheme for least-squares mixed finite elements. Comput. Methods Appl. Mech. Engrg. 150, 125–131 (1997)
Chen, Z.M., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37, 1542–1570 (2000)
Costabel, M.: A coercive bilinear form for Maxwell's equations. J. Math. Anal. Appl. 157(2), 527–541 (1991)
Costabel, M., Dauge, M.: Maxwell and Lamé eigenvalues on polyhedra. Math. Methods Appl. Sci. 22(3), 243–258 (1999)
Costabel, M., Dauge, M.: Weigthed Regularization of Maxwell Equations in Polyhedral Domains. Technical Report 01-26, IRMAR, Universitè de Rennes, France, 2001. To appear in Numer. Math.
Cox, C. L., Fix, G. J.: On the accuracy of least squares methods in the presence of corner singularities. Comput. Math. Appl. 10(6), 463–475 (1985), (1984)
Formaggia, L., Perotto, S.: New anisotropic a priori estimates. Numer. Math. 89, 641–667 (2001)
Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, 1986
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Publishing Inc., 1985
Hansbo, P., Larson, M. G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method. Comput. Methods Appl. Mech. Engrg. 191(17–18), 1895–1908 (2002)
Jackson, J.D.: Classical Electrodynamics. John Wiley & Sons, 2nd edition, 1975
Manteuffel, T.A., McCormick, S.F., Starke, G.: First-order system of least-squares for second order elliptic problems with discontinuous coefficients. In: Proc. 7th Copper Mountain Conferrence on Multigrid Methods, NASA Conference Pub.No. 3339, 551 (1995)
Pasciak J. E., Zhao J.: Overlapping Schwarz methods in H(curl) on polyhedral domains. J. Numer. Math. 10(3), 221–234 (2002)
Pehlivanov, A.I., Carey, G.F., Vassilevski, P.S.: Least-squares mixed finite element methods for non-selfadjoint problems: I. Error estimates. Numer. Math. 72, 501–522 (1996)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)
Siebert, K.G.: An a posteriori error estimator for anisotropic refinement. Numer. Math. 73(3), 373–398 (1996)
Finite Element Method and Multigrid for the Magnetostatic Problem. Chalmers Finite Element Center Chalmers University of Technology Preprint 2 2001Author information
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Bensow, R., Larson, M. Discontinuous Least-Squares finite element method for the Div-Curl problem. Numer. Math. 101, 601–617 (2005). https://doi.org/10.1007/s00211-005-0600-y
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DOI: https://doi.org/10.1007/s00211-005-0600-y