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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 1/2021

01.04.2021

Adaptive HDG Methods for the Steady-State Incompressible Navier–Stokes Equations

verfasst von: Haitao Leng

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2021

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Abstract

We consider a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier–Stokes equations. We use polynomials of degree \(k+1\), k, k and k for approximations of the velocity, the velocity gradient, the pressure and the boundary traces. Some stability results for approximate solutions and some relationships between norms are provided. Moreover an a posteriori error estimator is introduced. By \(L^2\)-projection and inf-sup condition, we prove that the error estimator is robust for the global \(L^2\) errors in the velocity, the velocity gradient and the pressure. Finally, a Picard iteration method and an adaptive HDG algorithm are presented. Furthermore, several numerical examples are shown to validate the theoretical analysis.
Vorheriger Artikel Metrics for Performance Quantification of Adaptive Mesh Refinement
Nächster Artikel An Efficient Iterative Method for Reconstructing Surface from Point Clouds

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Literatur
1.
Araya, R., Poza, A.H., Valentin, F.: An adaptive residual local projection finite element method for the Navier-Stokes equations. Adv. Comput. Math. 40, 1093–1119 (2014)MathSciNetMATHCrossRef
2.
Araya, R., Solano, M., Vega, P.: Analysis of an adaptive HDG method for the Brinkman problem. IMA J. Numer. Anal. 39, 1502–1528 (2019)MathSciNetMATHCrossRef
3.
Araya, R., Solano, M., Vega, P.: A posteriori error analysis of an HDG method for the Oseen problem. Appl. Numer. Math. 146, 291–308 (2019)MathSciNetMATHCrossRef
4.
5.
Berrone, S.: Adaptive discretization of stationary and incompressible Navier-Stokes equations by stabilized finite element methods. Comput. Methods Appl. Mech. Eng. 190, 4435–4455 (2001)MathSciNetCrossRef
6.
7.
Bonito, A., Nochetto, R.H.: Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method. SIAM J. Numer. Anal. 48, 734–771 (2010)MathSciNetMATHCrossRef
8.
Cesmelioglu, A., Cockburn, B., Qiu, W.: Analysis of a hybridizable discontinuous Galerkin method for steady-state incompressible Navier-Stokes equations. Math. Comput. 86, 1643–1670 (2017)MathSciNetMATHCrossRef
9.
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATH
10.
Cai, Z., Kanschat, G., Wang, C., Zhang, S.: Mixed finite element methods for incompressible flow: stationary Navier–Stokes equations. SIAM J. Numer. Anal. 48, 79–94 (2010)MathSciNetMATHCrossRef
11.
Cai, Z., Kanschat, G., Wang, C., Zhang, S.: Mixed finite element methods for stationary Navier-Stokes equations based on pseudostree- pressure-velocity formulation. Math. Comput. 81, 1903–1927 (2012)CrossRef
12.
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, 1319–1365 (2009)MathSciNetMATHCrossRef
13.
Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47, 1092–1125 (2009)MathSciNetMATHCrossRef
14.
Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.-J.: Analysis of HDG methods for Stokes flow. Math. Comput. 80, 723–760 (2011)MathSciNetMATHCrossRef
15.
16.
Cockburn, B., Zhang, W.: A posteriori error analysis for bybridizable discontinuous Galerkine methods for second order elliptic problems. SIAM J. Numer. Anal. 51, 676–693 (2013)MathSciNetMATHCrossRef
17.
Chen, H., Li, J., Qiu, W.: Robust a posteriori error estimates for HDG method for convection diffusion equations. IMA J. Numer. Anal. 36, 437–462 (2016)MathSciNetMATH
18.
Chen, H., Qiu, W., Shi, K.: A priori and computable a posteriori error estimates for an HDG method for the coercive Maxwell equations. Comput. Methods Appl. Mech. Eng. 333, 287–310 (2018)MathSciNetMATHCrossRef
19.
Chen, G., Hu, W., Shen, J., Singler, J.R., Zhang, Y., Zheng, X.: An HDG method for distributed control of convection diffusion PDEs. J. Comput. Appl. Math. 343, 643–661 (2018)MathSciNetMATHCrossRef
20.
Chalmers, N., Agbaglah, G., Chrust, M., Mavriplis, C.: A parallel hp-adaptive high order discontinuous Galerkin method for the incompressible Navier-Stokes equations. J. Comput. Phys. 2, 100023 (2019)MathSciNet
21.
Dauge, M.:: Stationary Stokes and Navier–Stokes systems on two- and three-dimensional domains with corners. Part I. Linearized equatons. SIAM J. Math. Anal. 20, 27–52 (1989)CrossRef
22.
Durango, F., Novo, J.: A posteriori error estimations for mixed finite element approximations to the Navier–Stokes equations based on Newton-type linearization. J. Comput. Appl. Math. 367, 112429 (2020)MathSciNetMATHCrossRef
23.
Farhloul, M., Nicaise, S., Paquet, L.: A refined mixed finite element method for Boussinesq equations in polygonal domains. IMA J. Numer. Anal. 21, 525–551 (2001)MathSciNetMATHCrossRef
24.
Farhloul, M., Nicaise, S., Paquet, L.: A priori and a posteriori error estimations for the dual mixed fintie element method of the Navier–Stokes problem. Numer. Methods Part. Differ. Equ. 25, 843–869 (2009)MATHCrossRef
25.
Frutos, J.G., Archilla, B., Novo, J.: A posteriori error estimations for mixed finite-element approximations to the Navier–Stokes equations. J. Comput. Appl. Math. 236, 1103–1122 (2011)MathSciNetMATHCrossRef
26.
Fu, G., Qiu, W., Zhang, W.: An analysis of HDG methods for convection-dominated diffusion problems. ESIAM: M2AN, 49, 225-256 (2015)
27.
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5, Springer (1986)
28.
Gong, W., Hu, W., Mateos, M., Singler, J., Zhang, X., Zhang, Y.: A new HDG method for Dirichlet boundary control of convection diffusion PDEs II: Low regularity. SIAM J. Numer. Anal. 56, 2262–2287 (2018)MathSciNetMATHCrossRef
29.
Gatica, L.F., Sequeira, F.A.: A priori and a posteriori error analyses of an HDG method for the Brinkman problem. Comput. Math. Appl. 75, 1191–1212 (2018)MathSciNetMATHCrossRef
30.
Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982)MATHCrossRef
31.
Huynh, H.T., Wang, Z.J., Vincent, P.E.: High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grid. Comput. Fluids 98, 209–220 (2014)MathSciNetMATHCrossRef
32.
Hoppe, R.H.W., Sharma, N.: Convergence analysis of an adaptive interiori penalty discontinuous Galerkin method for Helmholtz equation. IMA J. Numer. Anal. 33, 898–921 (2013)MathSciNetMATHCrossRef
33.
Kirk, K.L.A., Rhebergen, S.: Analysis of a pressure-robust hybridized discontinuous Galerkin method for the stationary Navier-Stokes equations. J. Sci. Comput. 81, 881–897 (2019)MathSciNetMATHCrossRef
34.
Karakashian, O.A., Jureidini, W.N.: A nonconforming finite element method for the stationary Navier–Stokes equations. SIAM J. Numer. Anal. 35, 93–120 (1998)MathSciNetMATHCrossRef
35.
Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003)MathSciNetMATHCrossRef
36.
Kanschat, G., Schötzau, D.: Energy norm a posteriori error estimation for divergence-free discontinuous Galerkin approximations of the Navier-Stokes equations. Int. J. Numer. Meth. Fluids 57, 1093–1113 (2008)MathSciNetMATHCrossRef
37.
Larson, M.G., Moalqvist, A.: A posteriori error estimate for mixed finite element approximation of elliptic problems. Numer. Math. 108, 487–500 (2008)MathSciNetCrossRef
38.
Lu, P., Chen, H., Qiu, W.: An absolutely stable hp-HDG method for the time-harmonic Maxwell equations with high wave number. Math. Comput. 86, 1553–1577 (2017)MathSciNetMATHCrossRef
39.
Leng, H., Chen, Y.: Adaptive hybridizable discontinuous Galerkin methods for nonstationary convection diffusion problems. Adv. Comput. Math. 46, 50 (2020)MathSciNetMATHCrossRef
40.
Leng, H., Chen, Y.: Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems. arXiv:​ 2004.​09319, (2020)
41.
Montlaur, A.de, Fernández-Méndez, S., Huerta, A.: Discontinuous Galerkin methods for the Stokes equations using divergence-free approximations. International Journal for Numerical Methods in Fluids, 579, 1071-1092 (2008)
42.
Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations. J. Comput. Phys. 230, 7151–7175 (2011)MathSciNetMATHCrossRef
43.
Petzoldt, M.: A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16, 47–75 (2002)MathSciNetMATHCrossRef
44.
Panourgias, K.T., Ekaterinaris, J.A.: A discontinuous Galerkin approach for high-resolution simulations of three-dimensional flows. Comput. Methods Appl. Mech. Eng. 299, 245–282 (2016)MathSciNetMATHCrossRef
45.
Qiu, W., Shi, K.: A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedrel meshes. IMA J. Numer. Anal. 36, 1943–1967 (2016)MathSciNetMATHCrossRef
46.
Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87, 69–93 (2018)MathSciNetMATHCrossRef
47.
Schwab, C.: p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. The Clarendon Press, Oxford University Press (1998)
48.
Soon, S.-C., Cockburn, B., Stolarski, H.K.: A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Meth. Eng. 80, 1058–1092 (2009)MathSciNetMATHCrossRef
49.
Temam, R.: Navier-Stokes equations, 3rd edn. Elsevier Science Publishers B. V, Amsterdam (1984)
50.
Verfürth, R.: Robust a posteriori error estimates for stationary convection diffusion equations. SIAM J. Numer. Anal. 43, 1766–1782 (2005)MathSciNetMATHCrossRef
Metadaten
Titel
Adaptive HDG Methods for the Steady-State Incompressible Navier–Stokes Equations
verfasst von
Haitao Leng
Publikationsdatum
01.04.2021
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 1/2021
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-021-01456-5

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