TY - JOUR

T1 - A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity

AU - Cockburn, Bernardo

AU - Qiu, Weifeng

AU - Solano, Manuel

PY - 2014

Y1 - 2014

N2 - 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 Ω by using finite element methods defined in polyhedral subdomains Dh ⊂ Ω. For a wide variety of hybridizable discontinuous Galerkin and mixed methods, we prove that the order of convergence in the L2-norm of the approximate 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 Γh is of order h. We also prove that the L2-norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between Γ and Γh is of order h5/4 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 Γh is of order h, the above-mentioned projection of the error of the scalar variable can still superconverge with a full additional order.

AB - 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 Ω by using finite element methods defined in polyhedral subdomains Dh ⊂ Ω. For a wide variety of hybridizable discontinuous Galerkin and mixed methods, we prove that the order of convergence in the L2-norm of the approximate 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 Γh is of order h. We also prove that the L2-norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between Γ and Γh is of order h5/4 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 Γh 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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U2 - 10.1090/S0025-5718-2013-02747-0

DO - 10.1090/S0025-5718-2013-02747-0

M3 - Article

AN - SCOPUS:84891790049

SN - 0025-5718

VL - 83

SP - 665

EP - 699

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 286

ER -