TY - JOUR

T1 - Analysis of hdg methods for stokes flow

AU - Cockburn, Bernardo

AU - Gopalakrishnan, Jayadeep

AU - Nguyen, Ngoc Cuong

AU - Peraire, Jaume

AU - Sayas, Francisco Javier

PY - 2011

Y1 - 2011

N2 - In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree k for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of k+1 in L2 for any k ≥ 0. Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k ≥ 1 and with order 1 for k = 0. Numerical experiments are presented which validate the theoretical results.

AB - In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree k for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of k+1 in L2 for any k ≥ 0. Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k ≥ 1 and with order 1 for k = 0. Numerical experiments are presented which validate the theoretical results.

KW - Discontinuous galerkin methods

KW - Hybridized methods

KW - Lagrange multipliers

KW - Mixed methods

KW - Stokes flow

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U2 - 10.1090/S0025-5718-2010-02410-X

DO - 10.1090/S0025-5718-2010-02410-X

M3 - Article

AN - SCOPUS:78651536264

VL - 80

SP - 723

EP - 760

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 274

ER -