Abstract
We propose a hybridizable discontinuous Galerkin (HDG) method to numerically solve the Oseen equations which can be seen as the linearized version of the incompressible Navier-Stokes equations. We use same polynomial degree to approximate the velocity, its gradient and the pressure. With a special projection and postprocessing, we obtain optimal convergence for the velocity gradient and pressure and superconvergence for the velocity. Numerical results supporting our theoretical results are provided.
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Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, San Diego (1975) [A subsidiary of Harcourt Brace Jovanovich, Publishers]
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02) (electronic)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)
Brezzi, F., Douglas, J. Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)
Burman, E., Fernández, M.A., Hansbo, P.: Continuous interior penalty finite element method for Oseen’s equations. SIAM J. Numer. Anal. 44(3), 1248–1274 (2006). (electronic)
Castillo, P.: Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J. Sci. Comput. 24(2), 524–547 (2002)
Chen, Y., Cockburn, B.: Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: general nonconforming meshes. IMA J. Numer. Anal. (2012). doi:10.1093/imanum/drr058
Cockburn, B., Dong, B.: An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems. J. Sci. Comput. 32(2), 233–262 (2007)
Cockburn, B., Sayas, F.J.: Divergence-conforming {HDG} methods for Stokes flow (2012, submitted)
Cockburn, B., Shi, K.: Conditions for superconvergence of HDG methods for Stokes flow. Math. Comput. (2012, accepted)
Cockburn, B., Kanschat, G., Schötzau, D.: The local discontinuous Galerkin method for the Oseen equations. Math. Comput. 73(246), 569–593 (2004). (electronic)
Cockburn, B., Kanschat, G., Schotzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74(251), 1067–1095 (2005). (electronic)
Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77(264), 1887–1916 (2008)
Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)
Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79(271), 1331–1349 (2010)
Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010)
Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.-J.: Analysis of an HDG methods for Stokes flow. Math. Comput. 80, 723–760 (2011)
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986). Theory and algorithms
Kovasznay, L.I.G.: Laminar flow behind two-dimensional grid. Proc. Camb. Philos. Soc. 44, 58–62 (1948)
Nédélec, J.-C.: Mixed finite elements in R 3. Numer. Math. 35(3), 315–341 (1980)
Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977)
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B. Cockburn supported in part by the National Science Foundation (Grant DMS-0712955).
N.C. Nguyen, J. Peraire supported in part by the Singapore-MIT Alliance.
Appendix: Approximation Properties of the Auxiliary Projection \(\varPi^{*}_{h}\)
Appendix: Approximation Properties of the Auxiliary Projection \(\varPi^{*}_{h}\)
The proofs in this appendix are quite similar to the proofs for the approximation properties of the projection Π h given in Sect. 3.3. We only need to recall that and that \(\widetilde{\varLambda}_{K}^{\max}\) is defined to be the maximum eigenvalue of \(\widetilde{\mathrm{S}}_{\boldsymbol{\beta}}\) over all faces of K.
1.1 A.1 Approximation Properties of Π ∗ ϕ
Proposition A.1
(Characterization of Π ∗ ϕ)
Proof
First equation follows from the definition of \(\varPi^{*}_{h}\). The second follows from (2.13d). Indeed if we take μ∈P k (K)⊥ and using integration by parts,
Then, using integration by parts on the right hand side,
where the last equality holds by (2.13a) and (2.13c) and as . □
Let δ ϕ=Π ∗ ϕ−ϕ k where ϕ k is the L 2-projection onto P k (K). Then, from (A.1a), δ ϕ∈P k (K)⊥. Then, from the second characterization,
Now let μ=δ ϕ. Then, as , we have,
From Lemma 3.12,
Then, as S β is positive definite, for any face F of K,
Therefore,
∥b ϕ ∥ is bounded exactly as in Sect. 3.3, with one difference in the outcome. Rather than \(\varLambda_{K}^{\max}\), we have \(\widetilde{\varLambda}_{K}^{\max}\) and ∥b Φ ∥ is also bounded the same way except that we have \(|\nabla\cdot(\nu\varPhi +\phi\mathrm{I} )|_{k_{\sigma}}\) rather than \(|\nabla\cdot(\nu\varPhi -\phi\mathrm{I} )|_{k_{\sigma}}\).
1.2 A.2 Approximation Properties of νΠ ∗ Φ+Π ∗ ϕI
As in Sect. 3.3, we need two additional projections. We introduce a projection \(\widetilde{\mathrm{P}}^{1}\) similar to P1 as defined in Sect. 3.3 and we define \(\widetilde{\mathrm {P}}^{2}\) to suit to the form of the projection \(\varPi_{h}^{\star}\). Let \(\widetilde {\mathrm{P} }^{1}\varPhi \in\mathrm{P}_{k}(K)\) be such that
for all faces F of the simplex K except for an arbitrary one and let \(\widetilde{\mathrm{P}}^{2}\varPhi \in\mathrm{P}_{k}(K)\) be such that
for all faces F of the simplex K except for an arbitrary one.
Proposition A.2
(Characterization of νΠ ∗ Φ+Π ∗ ϕI)
for all faces F of K.
Proof
The first equation follows directly from (2.13a)–(2.13c). For the second pick an arbitrary face F of K and let w∈P k (F). Then, there exists μ∈P k (K)⊥ such that μ=w on F. Therefore, in a similar fashion to the proof of the result for (ΠL−L)−(Πp−p)I, splitting the integral over ∂K to F and ∂K∖F and using (2.13d) with this μ,
where
But by (A.1b) and integration by parts,
The first, third and the last terms vanish by cancellation and from the fact that w∈P k (K)⊥. The second, fourth and fifth terms vanish by (A.2a). □
The proof of the estimate for ν(Π ∗ Φ−Φ) is very similar to the one for ν(ΠL−L). In short, from the representation of ν(Π ∗ Φ−Φ) using (3.24), it boils down to bounding ν(Π ∗ Φ−Φ)n F ⋅t for all faces F of K and for all . Using the projections we write . It is easy to show from the properties of \(\widetilde{\mathrm {P}}^{2}_{F}\), P F and (A.2b) that
Therefore,
The terms on the right hand side are bounded exactly the same way with the only difference being the equations defining . Now they are given by
for all faces F of K except for an arbitrarily chosen one.
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Cesmelioglu, A., Cockburn, B., Nguyen, N.C. et al. Analysis of HDG Methods for Oseen Equations. J Sci Comput 55, 392–431 (2013). https://doi.org/10.1007/s10915-012-9639-y
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DOI: https://doi.org/10.1007/s10915-012-9639-y