Stable finite element methods preserving $$\nabla \cdot \varvec{B}=0$$ ∇ · B = 0 exactly for MHD models - Numerische Mathematik
- ️Xu, Jinchao
- ️Mon Mar 28 2016
References
Armero, F., Simo, J.: Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 131, 41–90 (1996)
Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1 (2006)
Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010)
Badia, S., Codina, R., Planas, R.: On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics. J. Comput. Phys. 234, 399–416 (2013)
Balsara, D., Kim, J.: A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics. Astrophys. J. 602, 1079–1090 (2004)
Balsara, D., Spicer, D.: A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149, 270–292 (1999)
Bandaru, V., Boeck, T., Krasnov, D., Schumacher, J.: Numerical computation of liquid metal MHD duct flows at finite magnetic Reynolds number. pamir.sal.lv (1999)
Ba\(\breve{\rm n}\)as, L., Prohl, A.: Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations. Math. Comput. 79(272), 1957–1999 (2010)
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)
Bossavit, A.: Discretization of Electromagnetic Problems: The “Generalized Finite Differences” Approach. Handbook of Numerical Analysis, vol. XIII(04) (2005)
Brackbill, J.: Fluid modeling of magnetized plasmas. Space Plasma Simul. 42, 153–167 (1985)
Brackbill, J.U., Barnes, D.C.: The effect of nonzero \(\nabla \cdot B\) on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 430, 426–430 (1980)
Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. ESAIM. Math. Model. Numer. Anal. 8, 129–151 (1974)
Cai, W., Wu, J., Xin, J.: Divergence-free H(div)-conforming hierarchical bases for magnetohydrodynamics (MHD). Commun. Math. Stat. 1, 19–35 (2013)
Cockburn, B., Li, F., Shu, C.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194(2), 588–610 (2004)
Conraths, H.: Eddy current and temperature simulation in thin moving metal strips. Int. J. Numer. Methods Eng. 39, 141–163 (1996)
Cyr, E., Shadid, J., Tuminaro, R., Pawlowski, R., Chacón, L.: A new approximate block factorization preconditioner for two-dimensional incompressible (reduced) resistive MHD. SIAM J. Sci. Comput. 35(3), 701–730 (2013)
Dai, W., Woodward, P.: On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows. Astrophys. J. 494(1) (1998)
Davidson, P.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)
Dedner, A., Kemm, F., Kröner, D., Munz, C., Schnitzer, T., Wessenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175(2), 645–673 (2002)
Demkowicz, L., Vardapetyan, L.: Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements. Comput. Methods Appl. Mech. Eng. 152, 103–124 (1998)
de Dios, B., Brezzi, F., Marini, L., Xu, J., Zikatanov, L.: A simple preconditioner for a discontinuous Galerkin method for the Stokes problem. J. Sci. Comput. 58(3), 517–547 (2014)
Elman, H., Howle, V., Shadid, J., Shuttleworth, R., Tuminaro, R.: Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27, 1651–1668 (2006)
Evans, C., Hawley, J.: Simulation of magnetohydrodynamic flows—a constrained transport method. Astrophys. J. 332, 659–677 (1988)
Fey, M., Torrilhon, M.: A constrained transport upwind scheme for divergence-free advection. Hyperbolic Problems: Theory, Numerics, Applications, pp. 529–538 (2003)
Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)
Guermond, J.L., Minev, P.D.: Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case. Numer. Methods Partial Differ. Equ. 19(6), 709–731 (2003). doi:10.1002/num.10067
Gunzburger, M., Meir, A., Peterson, J.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Math. Comput. 56(194), 523–563 (1991)
Helzel, C., Rossmanith, J.A., Taetz, B.: An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations. J. Comput. Phys. 230, 3803–3829 (2011)
Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)
Hiptmair, R.: Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40(1), 41–65 (2002)
Hiptmair, R., Heumann, H., Mishra, S., Pagliantini, C.: Discretizing the advection of differential forms. In: ICERM Topical Workshop, Robust Discretization and Fast Solvers for Computable Multi-physics Models. ICERM (2014)
Jardin, S.: Computational Methods in Plasma Physics. CRC Press, Boca Raton (2010)
Li, F., Shu, C.: Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comput. 22(1–3), 413–442 (2005)
Li, F., Xu, L.: Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations. J. Comput. Phys. 231, 2655–2675 (2012)
Lin, P., Sala, M., Shadid, J., Tuminaro, R.: Performance of fully-coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport. Int. J. Numer. Methods Eng. 67(9), 208–225 (2006)
Liu, C.: Energetic variational approaches in complex fluids. In: Multi-Scale Phenomena in Complex Fluids: Modeling, Analysis and Numerical Simulations. World Scientific Publishing Company, Singapore (2009)
Liu, J., Wang, W.: An energy-preserving MAC-Yee scheme for the incompressible MHD equation. J. Comput. Phys. 174(1), 12–37 (2001)
Liu, J., Wang, W.: Energy and helicity preserving schemes for hydro- and magnetohydro-dynamics flows with symmetry. J. Comput. Phys. 200, 8–33 (2004)
Logg, A., Mardal, K., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)
Londrillo, P., Zanna, L.: On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method. J. Comput. Phys. 195, 17–48 (2004)
Ma, Y., Hu, K., Hu, X., Xu, J.: Robust preconditioners for the magnetohydrodynmaics models (2015). arXiv:1503.02553
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)
Moreau, R.: Magnetohydrodynamics. Kluwer, Dordrecht (1990)
Nédélec, J.: Mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 35, 315–341 (1980)
Nédélec, J.: A new family of mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 50, 57–81 (1986)
Pekmen, B., Tezer-Sezgin, M.: DRBEM solution of incompressible MHD flow with magnetic potential. Comput. Model. Eng. Sci. 96(4), 275–292 (2013)
Phillips, E., Elman, H., Cyr, E., Shadid, J., Pawlowski, R.: A block preconditioner for an exact penalty formulation for stationary MHD. SIAM J. Sci. Comput. 36(6), 930–951 (2014)
Powell, K.: An approximate Riemann solver for magnetohydrodynamics. Upwind and High-Resolution Schemes, pp. 570–583 (1997)
Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. Math. Model. Numer. Anal. 42, 1065–1087 (2008)
Raviart, P., Thomas, J.: A mixed finite element method for second order elliptic problems. Lecture Notes Math. 606, 292–315 (1977)
Rossmanith, J.A.: An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows. SIAM J. Sci. Comput. 28(5), 1766–1797 (2006)
Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)
Schötzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004)
Shadid, J., Cyr, E., Pawlowski, R., Tuminaro, R., Chacón, L., Lin, P.: Initial performance of fully-coupled AMG and approximate block factorization preconditioners for solution of implicit FE resistive MHD. In: Pereira, J., Sequeira, A. (eds.) V Europena Conference on Computational Fluid Dynamics, pp. 1–19 (2010)
Shadid, J., Pawlowski, R., Banks, J., Chacon, L., Lin, P., Tuminaro, R.: Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods. J. Comput. Phys. 229, 7649–7671 (2010)
Tóth, G.: The \(\nabla \cdot B= 0\) constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161, 605–652 (2000)
Tuminaro, R., Tong, C., Shadid, J., Devine, K.: On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz. Commun. Numer. Methods Eng. 18, 383–389 (2002)
Wiedmer, M.: Finite element approximation for equations of magnetohydrodynamics. Math. Comput. 69, 83–101 (2000)
Xu, J.: Fast Auxiliary Space Preconditioning (FASP) Software Package. http://fasp.sourceforge.net/
Ye, X., Hall, C.A.: A discrete divergence-free basis for finite element methods. Numer. Algorithms 16, 365–380 (1997)
Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. J. Comput. Phys. 14, 302 (1966)
Zhang, S.: Bases for C0-P1 divergence-free elements and for C1–P2 finite elements on union jack grids (2012). http://www.math.udel.edu/~szhang/research/p/uj.pdf