| Citation: |
Iury Igreja. STABILIZED PRIMAL AND DUAL HYBRID MIXED DGFEM FOR DARCY FLOW[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1269-1301. doi: 10.11948/20230087
|
STABILIZED PRIMAL AND DUAL HYBRID MIXED DGFEM FOR DARCY FLOW
-
Abstract
This work presents and analyzes two stabilized mixed hybrid Discontinuous Galerkin Finite Element Methods (DGFEM) for Darcy flows. The difference between the methodologies lies in the choice of Lagrange multipliers that aim to weakly enforce continuity on the edge/face of the elements. Thus, we study methods with multipliers associated with the normal component of the velocity field and the trace of the pressure field that naturally gives rise, respectively, to primal and dual formulations. However, despite the difference between the formulations, both methods can be associated with the same discontinuous Galerkin formulation. In this sense, the analysis of consistency, existence, uniqueness, and error estimates is similar for both methods, even when continuous interpolations are employed to approximate the Lagrange multipliers. Moreover, stability and convergence rate are improved by adding the least-squares residual forms of the governing equations. Besides, for specific edge/face stabilization parameter choices, the methods become locally conservative, allowing the use of non-conforming Raviart-Thomas spaces. Finally, to illustrate the accuracy, convergence rates, local mass conservation, computational efficiency, and flexibility of the methods, several two and three-dimensional numerical experiments are performed considering homogeneous and heterogeneous porous media.
-
-
References
[1] P. F. Antonietti and L. Heltai, Numerical validation of a class of mixed discontinuous Galerkin methods for Darcy flow, Computer Methods in Applied Mechanics and Engineering, 2007, 196(45), 4505–4520. [2] T. Arbogast and M. R. Correa, Two families of H(div) mixed finite elements on quadrilaterals of minimal dimension, SIAM Journal on Numerical Analysis, 2016, 54(6), 3332–3356. doi: 10.1137/15M1013705 [3] D. Arndt, W. Bangerth, M. Feder, et al., The deal. Ⅱ library, version 9.4, Journal of Numerical Mathematics, 2022, 30(3), 231–246. doi: 10.1515/jnma-2022-0054 [4] D. Arndt, W. Bangerth, M. Feder, et al., An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 1982, 19(4), 742–760. doi: 10.1137/0719052 [5] D. N. Arnold, D. Boffi and R. S. Falk, Quadrilateral H(div) finite elements, SIAM Journal on Numerical Analysis, 2005, 42(6), 2429–2451. doi: 10.1137/S0036142903431924 [6] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, ESAIM: M2AN, 1985, 19(1), 7–32. doi: 10.1051/m2an/1985190100071 [7] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 2002, 39(5), 1749–1779. doi: 10.1137/S0036142901384162 [8] N. C. Arruda, A. F. Loula and R. C. Almeida, Locally discontinuous but globally continuous Galerkin methods for elliptic problems, Computer Methods in Applied Mechanics and Engineering, 2013, 255, 104–120. doi: 10.1016/j.cma.2012.11.013 [9] I. Babuška, Error-bounds for finite element method, Numerische Mathematik, 1971, 16(4), 322–333. doi: 10.1007/BF02165003 [10] G. R. Barrenechea, L. P. Franca and F. Valentin, A Petrov–Galerkin enriched method: A mass conservative finite element method for the Darcy equation, Computer Methods in Applied Mechanics and Engineering, 2007, 196(21), 2449–2464. [11] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Berlin, Heidelberg, 2013. [12] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, Revue Française d'Automatique Informatique et Recherche Opérationnelle, Séries Rouge, 1974, 8(R–2), 129–151. [13] F. Brezzi, J. Douglas and L. Marini, Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, 1985, 47, 217–235. doi: 10.1007/BF01389710 [14] F. Brezzi and M. Fortin, A minimal stabilisation procedure for mixed finite element methods, Numerische Mathematik, 2001, 89(3), 457–491. doi: 10.1007/PL00005475 [15] F. Brezzi, T. J. R. Hughes, L. D. Marini and A. Masud, Mixed discontinuous Galerkin methods for Darcy flow, Journal of Scientific Computing, 2005, 22(1), 119–145. [16] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, SIAM Journal on Numerical Analysis, 2009, 47(2), 1319–1365. doi: 10.1137/070706616 [17] B. Cockburn, J. Guzmán, S. -C. Soon and H. K. Stolarski, An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems, SIAM Journal on Numerical Analysis, 2009, 47(4), 2686–2707. doi: 10.1137/080726914 [18] B. Cockburn and W. Zhang, A posteriori error estimates for HDG methods, Journal of Scientific Computing, 2012, 51(3), 582–607. doi: 10.1007/s10915-011-9522-2 [19] M. Correa and A. Loula, Unconditionally stable mixed finite element methods for Darcy flow, Computer Methods in Applied Mechanics and Engineering, 2008, 197(17–18), 1525–1540. doi: 10.1016/j.cma.2007.11.025 [20] P. Crumpton, G. Shaw and A. Ware, Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients, Journal of Computational Physics, 1995, 116(2), 343–358. doi: 10.1006/jcph.1995.1032 [21] R. Griesmaier and P. Monk, Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation, Journal of Scientific Computing, 2011, 49(3), 291–310. doi: 10.1007/s10915-011-9460-z [22] T. J. Hughes, A. Masud and J. Wan, A stabilized mixed discontinuous Galerkin method for Darcy flow, Computer Methods in Applied Mechanics and Engineering, 2006, 195(25), 3347–3381. [23] I. Igreja and G. de Miranda, Hybrid mixed methods applied to miscible displacements with adverse mobility ratio, in Computational Science – ICCS 2020 (Edited by V. V. Krzhizhanovskaya, G. Závodszky, M. H. Lees, et al. ), Springer International Publishing, 2020, 257–268. [24] I. Igreja and A. F. D. Loula, Stabilized velocity and pressure mixed hybrid DGFEM for the Stokes problem, International Journal for Numerical Methods in Engineering, 2017, 112(7), 603–628. doi: 10.1002/nme.5527 [25] I. Igreja and A. F. D. Loula, A stabilized hybrid mixed DGFEM naturally coupling Stokes–Darcy flows, Computer Methods in Applied Mechanics and Engi- neering, 2018, 339, 739–768. doi: 10.1016/j.cma.2018.05.026 [26] A. Masud and T. J. Hughes, A stabilized mixed finite element method for Darcy flow, Computer Methods in Applied Mechanics and Engineering, 2002, 191(39), 4341–4370. [27] E. Y. Medina, E. M. Toledo, I. Igreja and B. M. Rocha, A stabilized hybrid discontinuous Galerkin method for the Cahn–Hilliard equation, Journal of Computational and Applied Mathematics, 2022, 406, 114025. doi: 10.1016/j.cam.2021.114025 [28] Y. R. Núñez, C. O. Faria, A. F. D. Loula and S. M. C. Malta, Um método híbrido de elementos finitos aplicado a deslocamentos miscíveis em meios porosos heterogêneos, Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 2017, 33(1), 45–51. [29] P. A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the FEM. Lecture Notes in Mathematics, 1977, 606, 292–315. [30] B. M. Rocha, R. W. dos Santos, I. Igreja and A. F. D. Loula, Stabilized hybrid discontinuous Galerkin finite element method for the cardiac monodomain equation, International Journal for Numerical Methods in Biomedical Engineering, 2020, 36(7), e3341. doi: 10.1002/cnm.3341 [31] A. Samii, C. Michoski and C. Dawson, A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion, Computer Methods in Applied Mechanics and Engineering, 2016, 304, 118–139. doi: 10.1016/j.cma.2016.02.009 -
-
Figures(9) / Tables(2)
Export File
Citation
Iury Igreja. STABILIZED PRIMAL AND DUAL HYBRID MIXED DGFEM FOR DARCY FLOW[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1269-1301. doi: 10.11948/20230087
Format
Content
DownLoad:
-
Figure 1.
Numerical test case 1: h-convergence studies of the velocity (top), divergence (middle), and pressure (bottom) comparing SPHM-D, SPHM-RT, and SDHM-C in
$ L^2 $ -
Figure 2.
Numerical test case 2: h-convergence studies of the velocity (top), divergence (middle), and pressure (bottom) comparing SPHM-D, SPHM-RT, and SDHM-C in
$ L^2 $ -
Figure 3.
Numerical test case 3: h-convergence studies of the velocity (top), divergence (middle), and pressure (bottom) comparing SPHM-D, SPHM-RT, and SDHM-C in
$ L^2 $ -
Figure 4.
Numerical test case 4: h-convergence studies of the velocity (top), divergence (middle), and pressure (bottom) comparing SPHM-D, SPHM-RT, and SDHM-C in
$ L^2 $ -
Figure 5.
Numerical test case 5: h-convergence studies of the velocity (top), divergence (middle), and pressure (bottom) comparing SPHM-D, SPHM-RT, and SDHM-C in
$ L^2 $ -
Figure 6.
Numerical test case 6: h-convergence studies of the velocity (top), divergence (middle), and pressure (bottom) comparing SPHM-D, SPHM-RT, and SDHM-C in
$ L^2 $ $ \#dof $ -
Figure 7.
p-Convergence studies of the velocity field in two (top) and three (bottom) dimensions comparing SPHM-D, SPHM-RT, and SDHM-C in
$ L^2 $ -
Figure 8.
Influence of the stabilization parameter
$ \beta_0 $ -
Figure 9.
Three-dimensional computational cost of assembly, linear system solver, and post-processing comparing SPHM and SDHM-C formulations increasing the polynomial order and mesh refinement.