Abstract
For elliptic interface problems in two- and three-dimensions with a possible very low regularity, this paper establishes a priori error estimates for the Raviart–Thomas and Brezzi–Douglas–Marini mixed finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with respect to the local regularity of the solution. Several versions of the robust best approximations of the flux and the potential approximations are obtained. These robust and local optimal a priori estimates provide guidance for constructing robust a posteriori error estimates and adaptive methods for the mixed approximations.
Similar content being viewed by others
References
Ainsworth, M.: A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements. SIAM J. Sci. Comput. 30(1), 189–204 (2007)
Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comp. 64(211), 943–972 (1995)
Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comp. 35, 1039–1062 (1980)
Bernardi, C., Hecht, F.: Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comp. 71, 1371–1403 (2001)
Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608 (2000)
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Berlin (2013)
Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)
Cai, D., Cai, Z., Zhang, S.: Robust equilibrated error estimator for diffusion problems: mixed finite elements in two dimensions. J. Sci. Comput. 83, 22 (2020). https://doi.org/10.1007/s10915-020-01199-9
Cai, Z., He, C., Zhang, S.: Discontinuous finite element methods for interface problems: robust a priori and a posteriori error estimates. SIAM J. Numer. Anal. 55(1), 400–418 (2017)
Cai, Z., He, C., Zhang, S.: Improved ZZ a posteriori error estimators for diffusion problems: conforming linear elements. Comput. Methods Appl. Mech. Eng. 313, 433–449 (2017)
Cai, Z., Ye, X., Zhang, S.: Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations. SIAM J. Numer. Anal. 49(5), 1761–1787 (2011)
Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: conforming linear elements. SIAM J. Numer. Anal. 47(3), 2132–2156 (2009)
Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: mixed and nonconforming elements. SIAM J. Numer. Anal 48(1), 30–52 (2010)
Cai, Z., Zhang, S.: Robust equilibrated residual error estimator for diffusion problems: conforming elements. SIAM J. Numer. Anal 50(1), 151–170 (2012)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978, Reprinted as “SIAM Classics in Applied Mathematics”, No. 40, SIAM, Philadelphia (2002)
Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 2, 77–84 (1975)
Douglas, J., Roberts, J.E.: Mixed finite element methods for second order elliptic problems. Mat. Appli. Comput. 1(1), 91–103 (1982)
Dryja, M., Sarkis, M.V., Widlund, O.B.: Multilevel Schwartz method for elliptic problems with discontinuous in three dimensions. Numer. Math. 72, 313–348 (1996)
Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34(150), 441–463 (1980)
Gatica, G.: A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, SpringerBriefs in Mathematics. Springer, Berlin (2014)
Girault, G., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations Theory and Algorithms. Springer, Berlin (1986)
Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4, 101–129 (1975)
Kim, K.Y.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76(257), 43–66 (2007)
Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comp. 75(256), 1659–1674 (2006)
Nochetto, R.H., Veeser, A.: Primer of adaptive finite element methods. In: Bertoluzza, S., Nochetto, R.H., Quarteroni, A., Siebert, K.G., Veeser, A. (eds.) Multiscale and Adaptivity: Modeling, Numerics and Applications. Lecture Notes in Mathematics, vol. 2040. Springer, Berlin, pp. 125–225 (2012)
Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: Lions, J.-L., Ciarlet, P.G. (eds.) Handbook of Numerical Analysis Vol. II, Finite Element Methods (Part 1). Elsevier Science Publishers B.V. (North Holland), Amsterdam, pp. 523–639 (1991)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)
Vohralik, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection–diffusion–reaction equations. SIAM J. Numer. Anal. 45(4), 1570–1599 (2007)
Vohralik, M.: Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. Math. Comp. 79(272), 2001–2032 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported in part by Hong Kong Research Grants Council under the GRF Grant Project No. CityU 11305319.
Rights and permissions
About this article
Cite this article
Zhang, S. Robust and Local Optimal A Priori Error Estimates for Interface Problems with Low Regularity: Mixed Finite Element Approximations. J Sci Comput 84, 40 (2020). https://doi.org/10.1007/s10915-020-01284-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01284-z