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A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity

Authors: Bernardo Cockburn, Weifeng Qiu and Manuel Solano
Journal: Math. Comp. 83 (2014), 665-699
MSC (2010): Primary 65N30, 65M60
Published electronically: July 18, 2013
MathSciNet review: 3143688
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Abstract: We present the first a priori error analysis of a technique that allows us to numerically solve steady-state diffusion problems defined on curved domains $ \Omega $ by using finite element methods defined in polyhedral subdomains $ \mathsf {D}_h\subset \Omega $. 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 approximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain $ \Gamma $ and that of the computational domain $ \Gamma _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 distance between $ \Gamma $ and $ \Gamma _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 $ \Gamma $ and $ \Gamma _h$ is of order $ h$, the above-mentioned projection of the error of the scalar variable can still superconverge with a full additional order.

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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Weifeng Qiu
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong

Manuel Solano
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716

Keywords: Curved domains, discontinuous Galerkin methods, hybridization, superconvergence, elliptic problems
Received by editor(s): March 15, 2012
Received by editor(s) in revised form: July 6, 2012
Published electronically: July 18, 2013
Additional Notes: The first author was partially supported by the National Science Foundation (Grant DMS-1115331) and by the Minnesota Supercomputing Institute. The second author gratefully acknowledges the collaboration opportunities provided by the IMA during their 2011–12 program
Corresponding author: Weifeng Qiu
Article copyright: © Copyright 2013 American Mathematical Society