Research article Special Issues

Immersed finite element methods for convection diffusion equations

  • Received: 07 November 2022 Revised: 09 January 2023 Accepted: 17 January 2023 Published: 31 January 2023
  • MSC : 65M08, 65M15, 65M60

  • In this work, we develop two IFEMs for convection-diffusion equations with interfaces. We first define bilinear forms by adding judiciously defined convection-related line integrals. By establishing Gårding's inequality, we prove the optimal error estimates both in $ L^2 $ and $ H^1 $-norms. The second method is devoted to the convection-dominated case, where test functions are piecewise constant functions on vertex-associated control volumes. We accompany the so-called upwinding concepts to make the control-volume based IFEM robust to the magnitude of convection terms. The $ H^1 $ optimal error estimate is proven for control-volume based IFEM. We document numerical experiments which confirm the analysis.

    Citation: Gwanghyun Jo, Do Y. Kwak. Immersed finite element methods for convection diffusion equations[J]. AIMS Mathematics, 2023, 8(4): 8034-8059. doi: 10.3934/math.2023407

    Related Papers:

  • In this work, we develop two IFEMs for convection-diffusion equations with interfaces. We first define bilinear forms by adding judiciously defined convection-related line integrals. By establishing Gårding's inequality, we prove the optimal error estimates both in $ L^2 $ and $ H^1 $-norms. The second method is devoted to the convection-dominated case, where test functions are piecewise constant functions on vertex-associated control volumes. We accompany the so-called upwinding concepts to make the control-volume based IFEM robust to the magnitude of convection terms. The $ H^1 $ optimal error estimate is proven for control-volume based IFEM. We document numerical experiments which confirm the analysis.



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    [1] J. Bear, Dynamics of fluids in porous media, Elsevier, New York, 1972.
    [2] P. Bastian, A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure, Computat. Geosci., 18 (2014), 779–796. https://doi.org/10.1007/s10596-014-9426-y doi: 10.1007/s10596-014-9426-y
    [3] I. L. Chern, J. G. Liu, W. C. Wang, Accurate evaluation of electrostatics for macromolecules in solution, Meth. Appl. Anal., 10 (2003), 309–328. https://dx.doi.org/10.4310/MAA.2003.v10.n2.a9 doi: 10.4310/MAA.2003.v10.n2.a9
    [4] L. Chen, M. J. Holst, J. Xu, The finite element approximation of the nonlinear Poisson–Boltzmann equation, SIAM J. Numer. Anal., 45 (2007), 2298–2320. https://doi.org/10.1137/060675514 doi: 10.1137/060675514
    [5] I. Babuška, The finite element method for elliptic equations with discontinuous coefficients Computing, 5 (1970), 207–213. https://doi.org/10.1007/BF02248021
    [6] N. Moës, T. Belytschko, Extended finite element method for cohesive crack growth, Eng. Fract. Mech., 69 (2002), 813–833. https://doi.org/10.1016/S0013-7944(01)00128-X doi: 10.1016/S0013-7944(01)00128-X
    [7] J. Chessa, T. Belytschko, An extended finite element method for two-phase fluids, J. Appl. Mech., 70 (2003), 10–17. https://doi.org/10.1115/1.1526599 doi: 10.1115/1.1526599
    [8] G. Legrain, N. Moës, E. Verron, Stress analysis around crack tips in finite strain problems using the extended finite element method, Int. J. Numer. Meth. Eng., 63 (2005), 290–314. https://doi.org/10.1002/nme.1291 doi: 10.1002/nme.1291
    [9] M. Cervera, G. B. Barbat, M. Chiumenti, J. Y. Wu, A comparative review of xfem, mixed fem and phase-field models for quasi-brittle cracking, Arch. Comput. Method. E., 29, (2022), 1009–1083. https://doi.org/10.1007/s11831-021-09604-8 doi: 10.1007/s11831-021-09604-8
    [10] G. Jo, D. Y. Kwak, Geometric multigrid algorithms for elliptic interface problems using structured grids, Numer. Algorithms, 81 (2019), 211–235. https://doi.org/10.1007/s11075-018-0544-9 doi: 10.1007/s11075-018-0544-9
    [11] Z. Li, T. Lin, Y. Lin, R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Meth. Part. D. E., 20 (2004), 338–367. https://doi.org/10.1002/num.10092 doi: 10.1002/num.10092
    [12] X. He, T. Lin, Y. Lin, Approximation capability of a bilinear immersed finite element space, Numer. Meth. Part. D. E., 24 (2008), 1265–1300. https://doi.org/10.1002/num.20318 doi: 10.1002/num.20318
    [13] S. H. Chou, D. Y. Kwak, K. T. Wee, Optimal convergence analysis of an immersed interface finite element method, Adv. Comput. Math., 33 (2010), 149–168. https://doi.org/10.1007/s10444-009-9122-y doi: 10.1007/s10444-009-9122-y
    [14] D. Y. Kwak, K. T. Wee, K. S. Chang, An analysis of a broken $P_1$-nonconforming finite element method for interface problems, SIAM J. Numer. Anal., 48 (2010), 2117–2134. https://doi.org/10.1137/080728056 doi: 10.1137/080728056
    [15] D. Y. Kwak, S. Jin, D. Kyeong, A stabilized $P_1$-nonconforming immersed finite element method for the interface elasticity problems, ESAIM: Math. Model. Num., 51 (2017), 187–207. https://doi.org/10.1051/m2an/2016011 doi: 10.1051/m2an/2016011
    [16] G. Jo, D. Y. Kwak, A reduced Crouzeix-Raviart immersed finite element method for elasticity problems with interfaces, Comput. Meth. Appl. Math., 20 (2020), 501–516 https://doi.org/10.1515/cmam-2019-0046 doi: 10.1515/cmam-2019-0046
    [17] G. Jo, D. Y. Kwak, An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid, Comput. Method. Appl. M., 317 (2017), 684–701. https://doi.org/10.1016/j.cma.2017.01.005 doi: 10.1016/j.cma.2017.01.005
    [18] I. Kwon, D. Y. Kwak, G. Jo, Discontinuous bubble immersed finite element method for Poisson-Boltzmann-Nernst-Planck model, J. Comput. Phys., 438 (2021), 110370. https://doi.org/10.1016/j.jcp.2021.110370 doi: 10.1016/j.jcp.2021.110370
    [19] Y. Choi, G. Jo, D. Y. Kwak, Y. J. Lee, Locally conservative discontinuous bubble scheme for Darcy flow and its application to Hele-Shaw equation based on structured grids, Numer. Algorithms, (2022), https://doi.org/10.1007/s11075-022-01333-8 doi: 10.1007/s11075-022-01333-8
    [20] R. E. Ewing, Z. Li, T. Lin, Y. Lin, The immersed finite volume element methods for the elliptic interface problems, Math. Comput. Simulat., 50 (1999), 63–76. https://doi.org/10.1016/S0378-4754(99)00061-0 doi: 10.1016/S0378-4754(99)00061-0
    [21] X. M. He, T. Lin, Y. Lin, A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficient, Commun. Comput. Phys., 6 (2009), 185–202. 10.4208/cicp.2009.v6.p185 doi: 10.4208/cicp.2009.v6.p185
    [22] Q. Wang, Z. Zhang, A stabilized immersed finite volume element method for elliptic interface problems, Appl. Numer. Math., 143 (2019), 75–87. https://doi.org/10.1016/j.apnum.2019.03.010 doi: 10.1016/j.apnum.2019.03.010
    [23] Q. Wang, Z. Zhang, L. Wang, New immersed finite volume element method for elliptic interface problems with non-homogeneous jump conditions, J. Comput. Phys., 427 (2021), 110075. https://doi.org/10.1016/j.jcp.2020.110075 doi: 10.1016/j.jcp.2020.110075
    [24] H. G. Roos, M. Stynes, L. Tobiska, Robust numerical methods for singularly perturbed differential equations: Convection-diffusion-reaction and flow problems, Springer Science and Business Media, 2008. https://doi.org/10.1007/978-3-540-34467-4
    [25] S. C. Brenner, L. R. Scott, The mathematical theory of finite element methods, New York: Springer, 2008. https://doi.org/10.1007/978-1-4757-4338-8
    [26] Ja A. Ro$\mathop {\rm{i}}\limits^ \vee$tberg, Z. G. Šeftel, A theorem on homeomorphisms for elliptic systems and its applications, Math. USSR-Sbornik, 7 (1969), 439–465. https://doi.org/10.1070/SM1969v007n03ABEH001099 doi: 10.1070/SM1969v007n03ABEH001099
    [27] J. H. Bramble, J. T. King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math., 6 (1996), 109–138. https://doi.org/10.1007/BF02127700 doi: 10.1007/BF02127700
    [28] M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), 152–161. https://doi.org/10.1137/0715010 doi: 10.1137/0715010
    [29] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742–760. https://doi.org/10.1137/0719052 doi: 10.1137/0719052
    [30] K. Ohmori, T. Ushijima, A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations, RAIRO Anal. Numérique, 18 (1984), 309–322. https://doi.org/10.1051/m2an/1984180303091 doi: 10.1051/m2an/1984180303091
    [31] R. E. Bank, J. F. Burgler, W. Fichner, R. K. Smith, Some upwinding techniques for finite element approximations of convection-diffusion equations, Numer. Math., 58 (1990), 185–202. https://doi.org/10.1007/BF01385618 doi: 10.1007/BF01385618
    [32] A. N. Brooks, T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Method. Appl. M., 32 (1982), 199–259. https://doi.org/10.1016/0045-7825(82)90071-8 doi: 10.1016/0045-7825(82)90071-8
    [33] N. Ahmed, V. John, G. Matthies, J. Novo, A local projection stabilization/continuous Galerkin–Petrov method for incompressible flow problems, Appl. Math. Comput., 333 (2018), 304–324. https://doi.org/10.1016/j.amc.2018.03.088 doi: 10.1016/j.amc.2018.03.088
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