Unified framework for an a posteriori error analysis of non-standard finite element approximations of H(curl)-elliptic problems

Abstract

Abstract A unified framework for a residual-based a posteriori error analysis of standard conforming finite element methods as well as non-standard techniques such as nonconforming and mixed methods has been developed in [Carstensen, Numer. Math. 100: 617 – 637, 2005, Carstensen, Gudi, and Jensen, A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Department of Mathematics, Humboldt University of Berlin, 2008, Carstensen and Hoppe, J. Numer. Math. 13: 19 – 32, 2005, Carstensen and Hu, Numer. Math. 107: 473 – 502, 2007, Carstensen, Hu, and Orlando, SIAM J. Numer. Anal. 45: 68 – 82, 2007]. This paper provides such a framework for an a posteriori error control of nonconforming finite element discretizations of H(curl)-elliptic problems as they arise from low-frequency electromagnetics. These nonconforming approximations include the interior penalty discontinuous Galerkin (IPDG) approach considered in [Houston, Perugia, and Schötzau, SIAM J. Numer. Anal. 42: 434 – 459, 2004, Houston, Perugia, and Schötzau, IMA J. Numer. Anal. 27: 122 – 150, 2007], and mortar edge element approximations studied in [Belgacem, Buffa, and Maday, SIAM J. Numer. Anal. 39: 880 – 901, 2001, Hoppe, East-West J. Numer. Math. 7: 159 – 173, 1999, Hoppe, Adaptive domain decomposition techniques in electromagnetic field computation and electrothermomechanical coupling problems: Springer, 2002, Hoppe, J. Comp. Appl. Math. 168: 245 – 254, 2004, Hoppe, Contemporary Math. 383, 63 – 111, 2005, Rapetti, Buffa, Maday, and Bouillault, COMPEL 19: 332 – 340, 2000, Xu and Hoppe, SIAM J. Numer. Anal. 43: 1276 – 1294, 2005].