Abstract
We study a hybridizable discontinuous Galerkin method for solving the vorticity-velocity formulation of the Stokes equations in three-space dimensions. We show how to hybridize the method to avoid the construction of the divergence-free approximate velocity spaces, recover an approximation for the pressure and implement the method efficiently. We prove that, when all the unknowns use polynomials of degree k≥0, the L 2 norm of the errors in the approximate vorticity and pressure converge with order k+1/2 and the error in the approximate velocity converges with order k+1. We achieve this by letting the normal stabilization function go to infinity in the error estimates previously obtained for a hybridizable discontinuous Galerkin method.
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Carrero, J., Cockburn, B., Schötzau, D.: Hybridized, globally divergence-free LDG methods. Part I: The Stokes problem. Math. Comput. 75, 533–563 (2006)
Cockburn, B., Cui, J.: An analysis of HDG methods for the vorticity-velocity-pressure formulation of the Stokes Problem in three dimensions. Math. Comp. (2011, to appear)
Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47, 1092–1125 (2009)
Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.-J.: Analysis of an HDG method for Stokes flow. Math. Comput. 80, 723–760 (2011)
Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010)
Cockburn, B., Sayas, F.-J.: Divergence–conforming HDG methods for Stokes flow. Submitted
Dauge, M.: Stationary Stokes and Navier-Stokes systems on two- and three-dimensional domains with corners. I. Linearized equations. SIAM J. Numer. Anal. 40, 319–343 (1989)
Nédélec, J.-C.: A new family of mixed finite elements in R 3. Numer. Math. 50, 57–81 (1986)
Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Eng. 199, 582–597 (2010)
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The first author was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute.
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Cockburn, B., Cui, J. Divergence-Free HDG Methods for the Vorticity-Velocity Formulation of the Stokes Problem. J Sci Comput 52, 256–270 (2012). https://doi.org/10.1007/s10915-011-9542-y
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DOI: https://doi.org/10.1007/s10915-011-9542-y