Error estimate of a fully discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations. (English) Zbl 1397.76029
Summary: In this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuation of our formal paper [the last author et al., Math. Methods Appl. Sci. 40, No. 11, 4179–4196 (2017; Zbl 1368.76016)]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [loc. cit.] show that this method is highly efficient for the unsteady incompressible MHD problems.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35Q30 | Navier-Stokes equations |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
unsteady incompressible MHD equations; finite element method; fully discrete defect correction method; stability analysis; error analysisCitations:
Zbl 1368.76016References:
| [1] | Gunzburger, MD; Meir, AJ; Peterson, JS, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math Comput, 56, 523-563, (1991) · Zbl 0731.76094 |
| [2] | Hughes, WF; Young, FJ, The electromagnetodynamics of fluids, (1966), Wiley, New York (NY) |
| [3] | Walker, JS; Buckmaster, J., Fluid mechanics in energy conversion, Large interaction parameter magnetohydrodynamics and applications in fusion reactor technology, (1980), SIAM, Philadelphia |
| [4] | Winowich, NS; Hughes, W.; Branover, H.; Lykoudis, PS; Yakhot, A., Liquid-metal flows and magnetohydrodynamics, A finite element analysis of two dimensional MHD flow, (1983), AIAA, New York (NY) |
| [5] | Greif, C.; Li, D.; Schötzau, D., A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics, Comput Methods Appl Mech Eng, 199, 2840-2855, (2010) · Zbl 1231.76146 |
| [6] | Hasler, U.; Schneebeli, A.; Schötzau, D., Mixed finite element approximation of incompressible MHD problems based on weighted regularization, Appl Numer Math, 51, 19-45, (2004) · Zbl 1126.76341 |
| [7] | Zhang, GD; He, Y.; Zhang, Y., Streamline diffusion finite element method for stationary incompressible magnetohydrodynamics, Numer Methods Partial Differ Equ, 30, 1877-1901, (2014) · Zbl 1446.76188 |
| [8] | Dong, XJ; He, YN; Zhang, Y., Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics, Comput Method Appl Math, 276, 287-311, (2014) · Zbl 1423.76226 |
| [9] | He, YN, Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations, IMA J Numer Anal, 25, 8, 1912-1923, (2014) · Zbl 1356.76451 |
| [10] | Dong, XJ; He, Y., Two-level Newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics, J Sci Comput, 63, 2, 426-451, (2015) · Zbl 06456931 |
| [11] | Zhang, GD; He, Y.; Yang, D., Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain, Comput Math Appl, 68, 770-788, (2014) · Zbl 1362.76035 |
| [12] | Shi, D.; Yu, Z., Low-order nonconforming mixed finite element methods for stationary incompressible magnetohydrodynamics equations, J Appl Math, 2012, 331-353, (2012) |
| [13] | Ervin, V.; Layton, W., An analysis of a defect-correction method for a model convection-diffusion equation, SIAM J Numer Anal, 26, 1, 169-179, (1989) · Zbl 0672.65063 |
| [14] | Axelsson, O.; Nikolova, M., Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method, Computing, 58, 1-30, (1997) · Zbl 0876.35009 |
| [15] | Ervin, V.; Layton, W.; Maubach, J., Adaptive defect correction methods for viscous incompressible flow problems, SIAM J Numer Anal, 37, 1165-185, (2000) · Zbl 1049.76038 |
| [16] | Ervin, VJ; Howell, JS; Lee, H., A two-parameter defect-correction method for computation of steady-state viscoelastic fluid flow, Appl Math Comput, 196, 818-834, (2008) · Zbl 1132.76033 |
| [17] | Frank, R.; Hertling, J.; Monnet, JP, The application of iterated defect correction to variational methods for elliptic boundary value problems, Computing, 30, 121-135, (1983) · Zbl 0498.65051 |
| [18] | Gracia, JL; O’Riordan, E., A defect-correction parameter-uniform numerical method for a singularly perturbed convection-diffusion problem in one dimension, Numer Algorithms, 41, 359-385, (2006) · Zbl 1098.65083 |
| [19] | Kramer, W.; Minero, R.; Clercx, HJH, A finite volume local defect correction method for solving the transport equation, Comput Fluids, 38, 533-543, (2009) · Zbl 1193.76089 |
| [20] | Koch, O.; Weinmüller, EB, Iterated defect correction for the solution of singular initial value problems, SIAM J Numer Anal, 38, 6, 1784-1799, (2001) · Zbl 0989.65068 |
| [21] | Labovschii, A., A defect correction method for the time-dependent Navier-Stokes equations, Numer Methods Partial Differ. Equ, 25, 1-25, (2009) · Zbl 1394.76087 |
| [22] | Labovschii, A., Technical Report, A defect correction method for the time-dependent Navier-Stokes equations, (2007), University of Pittsburgh, Pennsylvania · Zbl 1394.76087 |
| [23] | Minero, R.; Anthonissen, MJH; Mattheij, RMM, A local defect correction technique for time-dependent problems, Numer Methods Partial Differ Equ, 23, 128-144, (2006) · Zbl 1084.65082 |
| [24] | Layton, W.; Lee, HK; Peterson, J., A defect-correction method for the incompressible Navier-Stokes equations, Appl Math Comput, 129, 1-19, (2002) · Zbl 1074.76033 |
| [25] | Martin, R.; Guillard, H., A second order defect correction scheme for unsteady problems, Comput. Fluids, 25, 1, 9-21, (1996) · Zbl 0865.76054 |
| [26] | Si, ZY; He, YN, A defect-correction mixed finite element method for stationary conduction-convection problems, Math Prob Eng, (2011) · Zbl 1204.76021 · doi:10.1155/2011/370192 |
| [27] | Si, ZY; He, YN; Wang, K., A defect-correction method for unsteady conduction convection problems I: spatial discretization, Sci China-Math, 54, 185-204, (2011) · Zbl 1227.76040 |
| [28] | Si, ZY; He, YN; Zhang, T., A defect-correction method for unsteady conduction convection problems II: time discretization, J Comput Appl Math, 236, 2553-2573, (2012) · Zbl 1427.76146 |
| [29] | Si, ZY, Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier-Stokes problems, Numer Algorithms, 59, 271-300, (2012) · Zbl 1432.76084 |
| [30] | Si, Z.; He, Y.; Wang, Y., Modified characteristics mixed defect-correction finite element method for the time-dependent Navier-Stokes problems, Appl Anal, 94, 701-724, (2015) · Zbl 1446.76096 |
| [31] | Si, ZY; Jing, SJ; Wang, YX, Defect correction finite element method for the stationary incompressible magenetohydrodynamics equation, Appl Math Comput, 285, 184-194, (2016) · Zbl 1410.76198 |
| [32] | Si, ZY; Liu, C.; Wang, YX, A semi-discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations, Math Method Appl Sci, 40, 4179-4196, (2017) · Zbl 1368.76016 |
| [33] | Girault, V.; Raviart, PA, Finite element method for Navier-Stokes equations: theory and algorithms, (1987), Springer-Verlag, Berlin |
| [34] | He, YN, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J Numer Anal, 41, 4, 1263-1285, (2003) · Zbl 1130.76365 |
| [35] | He, YN; Liu, KM, A multilevel finite element method in space-time for the Navier-Stokes problem, Numer Methods Partial Differ Equ, 21, 6, 1052-1078, (2005) · Zbl 1081.76044 |
| [36] | Adams, RA, Sobolev space, 65, (1975), Academic Press, New York (NY) |
| [37] | Sermange, M.; Temam, R., Some mathematical questions related to the MHD equations, Commun Pure Appl Math, 36, 635-664, (1983) · Zbl 0524.76099 |
| [38] | Temam, R., Navier-Stokes equations, (1977), North-Holland, Amsterdam · Zbl 0335.35077 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
