MATHEMATICS OF COMPUTATION Volume 83, Number 286, March 2014, Pages 665–699 S 0025-5718(2013)02747-0 Article electronically published on July 18, 2013 A PRIORI ERROR ANALYSIS FOR HDG METHODS USING EXTENSIONS FROM SUBDOMAINS TO ACHIEVE BOUNDARY CONFORMITY BERNARDO COCKBURN, WEIFENG QIU, AND MANUEL SOLANO Abstract. We present the first a priori error analysis of a technique that al- lows us to numerically solve steady-state diffusion problems defined on curved domains Ω by using finite element methods defined in polyhedral subdomains D h ⊂ Ω. For a wide variety of hybridizable discontinuous Galerkin and mixed methods, we prove that the order of convergence in the L 2 -norm of the ap- proximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain Γ and that of the computational domain Γ h is of order h. We also prove that the L 2 -norm of a projection of the error of the scalar variable superconverges with a full additional order when the dis- tance between Γ and Γ h is of order h 5/4 but with only half an additional order when such a distance is of order h. Finally, we present numerical experiments confirming the theoretical results and showing that even when the distance between Γ and Γ h is of order h, the above-mentioned projection of the error of the scalar variable can still superconverge with a full additional order. 1. Introduction In this paper, we present the first a priori error analysis of a technique [12,16,17] which allows us to use finite element methods using solely polyhedral elements to numerically solve problems in domains which are not necessarily polyhedral. We present the error analysis in the framework of the approximation of the solution of the model problem q + ∇u =0 in Ω, (1.1a) ∇· q = f in Ω, (1.1b) u = g on Γ, (1.1c) where Ω is a subdomain of R d with a piecewise C 2 , Lipschitz boundary Γ, g is a function lying in H 1/2 (Γ) and f in L 2 (Ω). We take the finite element method to be the so-called hybridizable discontinuous Galerkin (HDG) method; see [10]. Roughly speaking, the above-mentioned technique consists in approximating the curved domain Ω by polyhedral subdomains D h where a finite element method is used to obtain the approximate solution. To be able to achieve this, the Dirichlet Received by the editor March 15, 2012 and, in revised form, July 6, 2012. 2010 Mathematics Subject Classification. Primary 65N30, 65M60. Key words and phrases. Curved domains, discontinuous Galerkin methods, hybridization, su- perconvergence, elliptic problems. The first author was partially supported by the National Science Foundation (Grant DMS- 1115331) and by the Minnesota Supercomputing Institute. The second author gratefully acknowl- edges the collaboration opportunities provided by the IMA during their 2011–12 program. Corresponding author: Weifeng Qiu. c 2013 American Mathematical Society 665 Licensed to City University of Hong Kong. Prepared on Thu Jun 19 06:03:27 EDT 2014 for download from IP 144.214.75.247. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use