# American Institute of Mathematical Sciences

October  2011, 29(4): 1471-1495. doi: 10.3934/dcds.2011.29.1471

## Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms

 1 Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland, Switzerland

Received  January 2010 Revised  September 2010 Published  December 2010

We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Our focus is on derivations of the schemes, details of implementation, as well as on application to the discretization of eddy current equations in moving media.
Citation: Holger Heumann, Ralf Hiptmair. Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1471-1495. doi: 10.3934/dcds.2011.29.1471
##### References:
 [1] R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," Addison-Wesley, London, 1983. [2] D. A. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47 (2010), 281-354. doi: 10.1090/S0273-0979-10-01278-4. [3] D. N. Arnold, R. S. Falk and R. Winther, Multigrid in H(div) and H(curl), Numer. Math., 85 (2000), 197-217. doi: 10.1007/PL00005386. [4] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, 15 (2006), 1-155. doi: 10.1017/S0962492906210018. [5] A. Bossavit, Extrusion, contraction: Their discretization via Whitney forms, COMPEL, 22 (2004), 470-480. [6] A. Bossavit, Discretization of electromagnetic problems: The "generalized finite differences," in "Numerical Methods in Electromagnetics," volume XIII of Handbook of Numerical Analysis (eds. W. H. A. Schilders and W. J. W. ter Maten), Elsevier, Amsterdam, (2005), 443-522. [7] A. Bossavit and L. Kettunen, Yee-like schemes on staggered cellular grids: A synthesis between FIT and FEM approaches, contribution to COMPUMAG '99., 1999. [8] F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems, Math. Model. Numer. Anal., 43 (2009), 277-295. doi: 10.1051/m2an:2008046. [9] F. Brezzi, L. D. Marini and E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14 (2004), 1893-1903. doi: 10.1142/S0218202504003866. [10] M. A. Celia, T. F. Russell, I. Herrera and R. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Adv. Wat. Resour., 13 (1990), 187-206. doi: 10.1016/0309-1708(90)90041-2. [11] H. De Sterk, Multi-dimensional upwind constrained transport on unstructured grids for "shallow water" magnetohydrodynamics, in "Proceedings of the 15th AIAA CFD Conference, Jun 11-14, Anaheim, CA," AIAA, (2001). [12] M. Desbrun, A. N. Hirani, J. E. Marsden and M. Leok, Discrete exterior calculus,, preprint, (). [13] V. Dolejsi, M. Feistauer and J. Hozman, Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 2813-2827. doi: 10.1016/j.cma.2006.09.025. [14] M. E. Gurtin, "An Introduction to Continuum Mechanics," volume 158 of Mathematics in Science and Engineering, Academic Press, New York, 1981. [15] J. Harrison, Ravello lecture notes on geometric calculus - Part I,, preprint, (). [16] H. Heumann and R. Hiptmair, "Extrusion Contraction Upwind Schemes for Convection-Diffusion Problems," Report 2008-30, SAM, ETH Zürich, Zürich, Switzerland, 2008. http://www.sam.math.ethz.ch/reports/2008/30 [17] H. Heumann, R. Hiptmair and J. Xu, "A Semi-Lagrangian Method for Convection of Differential Forms," Report 2009-09, SAM, ETH Zürich, Zürich, Switzerland, 2009. http://www.sam.math.ethz.ch/reports/2009/09 [18] R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36(1) (1999), 204-225. doi: 10.1137/S0036142997326203. [19] R. Hiptmair, Discrete Hodge operators, Numer. Math., 90 (2001), 265-289. doi: 10.1007/s002110100295. [20] R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237-339. doi: 10.1017/CBO9780511550140.004. [21] R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM J. Numer. Anal., 45 (2007), 2483-2509. doi: 10.1137/060660588. [22] A. N. Hirani, "Discrete Exterior Calculus," Ph.D. Thesis, California Institute of Technology, 2003. http://resolver.caltech.edu/CaltechETD:etd-05202003-095403. [23] J. M. Hyman and M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, International Journal of Computers & Mathematics with Applications, 33 (1997), 81-104. doi: 10.1016/S0898-1221(97)00009-6. [24] J. M. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations, J. Comp. Phys., 151 (1999), 881-909. doi: 10.1006/jcph.1999.6225. [25] J. M. Hyman and M. Shashkov, Mimetic finite difference methods for Maxwell's equations and the equations of magnetic diffusion, in "Geometric Methods for Computational Electromagnetics," volume 32 of PIER (ed. F. L. Teixeira), EMW Publishing, Cambridge, MA, (2001), 89-121. [26] S. Lang, "Differential and Riemannian Manifolds in Mathematics," Springer, New York, 1995. [27] S. Lang, "Fundamentals of Differential Geometry," Springer, 1999. [28] K. W. Morton, A. Priestley and E. Süli, Convergence of the Lagrange-Galerkin method with non-exact integration, Mathematical Modelling and Numerical Analysis, 22 (1988), 625-653. [29] P. Mullen, A. McKenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Discrete Lie advection and differential forms,, preprint, (). [30] R. N. Rieben, D. A. White, B. K. Wallin and J. M. Solberg, An arbitrary Lagrangian-Eulerian discretization of MHD on 3D unstructured grids, J. Comp. Phys., 226 (2007), 534-570. doi: 10.1016/j.jcp.2007.04.031. [31] G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory, J. Elasticity, 71 (2003), 183-203. doi: 10.1023/B:ELAS.0000005545.46932.08. [32] H.-G. Roos, M. Stynes and L. Tobiska, "Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems," volume 24 of Springer Series in Computational Mathematics, Springer, Berlin, 1996. [33] J.-C. Xu, Optimal algorithms for discretized partial differential equations, in "Proceedings of the ICIAM'07 conference, Zürich, Switzerland, July 2007," EMS Publishing House, 2008.

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##### References:
 [1] R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," Addison-Wesley, London, 1983. [2] D. A. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47 (2010), 281-354. doi: 10.1090/S0273-0979-10-01278-4. [3] D. N. Arnold, R. S. Falk and R. Winther, Multigrid in H(div) and H(curl), Numer. Math., 85 (2000), 197-217. doi: 10.1007/PL00005386. [4] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, 15 (2006), 1-155. doi: 10.1017/S0962492906210018. [5] A. Bossavit, Extrusion, contraction: Their discretization via Whitney forms, COMPEL, 22 (2004), 470-480. [6] A. Bossavit, Discretization of electromagnetic problems: The "generalized finite differences," in "Numerical Methods in Electromagnetics," volume XIII of Handbook of Numerical Analysis (eds. W. H. A. Schilders and W. J. W. ter Maten), Elsevier, Amsterdam, (2005), 443-522. [7] A. Bossavit and L. Kettunen, Yee-like schemes on staggered cellular grids: A synthesis between FIT and FEM approaches, contribution to COMPUMAG '99., 1999. [8] F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems, Math. Model. Numer. Anal., 43 (2009), 277-295. doi: 10.1051/m2an:2008046. [9] F. Brezzi, L. D. Marini and E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14 (2004), 1893-1903. doi: 10.1142/S0218202504003866. [10] M. A. Celia, T. F. Russell, I. Herrera and R. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Adv. Wat. Resour., 13 (1990), 187-206. doi: 10.1016/0309-1708(90)90041-2. [11] H. De Sterk, Multi-dimensional upwind constrained transport on unstructured grids for "shallow water" magnetohydrodynamics, in "Proceedings of the 15th AIAA CFD Conference, Jun 11-14, Anaheim, CA," AIAA, (2001). [12] M. Desbrun, A. N. Hirani, J. E. Marsden and M. Leok, Discrete exterior calculus,, preprint, (). [13] V. Dolejsi, M. Feistauer and J. Hozman, Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 2813-2827. doi: 10.1016/j.cma.2006.09.025. [14] M. E. Gurtin, "An Introduction to Continuum Mechanics," volume 158 of Mathematics in Science and Engineering, Academic Press, New York, 1981. [15] J. Harrison, Ravello lecture notes on geometric calculus - Part I,, preprint, (). [16] H. Heumann and R. Hiptmair, "Extrusion Contraction Upwind Schemes for Convection-Diffusion Problems," Report 2008-30, SAM, ETH Zürich, Zürich, Switzerland, 2008. http://www.sam.math.ethz.ch/reports/2008/30 [17] H. Heumann, R. Hiptmair and J. Xu, "A Semi-Lagrangian Method for Convection of Differential Forms," Report 2009-09, SAM, ETH Zürich, Zürich, Switzerland, 2009. http://www.sam.math.ethz.ch/reports/2009/09 [18] R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36(1) (1999), 204-225. doi: 10.1137/S0036142997326203. [19] R. Hiptmair, Discrete Hodge operators, Numer. Math., 90 (2001), 265-289. doi: 10.1007/s002110100295. [20] R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237-339. doi: 10.1017/CBO9780511550140.004. [21] R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM J. Numer. Anal., 45 (2007), 2483-2509. doi: 10.1137/060660588. [22] A. N. Hirani, "Discrete Exterior Calculus," Ph.D. Thesis, California Institute of Technology, 2003. http://resolver.caltech.edu/CaltechETD:etd-05202003-095403. [23] J. M. Hyman and M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, International Journal of Computers & Mathematics with Applications, 33 (1997), 81-104. doi: 10.1016/S0898-1221(97)00009-6. [24] J. M. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations, J. Comp. Phys., 151 (1999), 881-909. doi: 10.1006/jcph.1999.6225. [25] J. M. Hyman and M. Shashkov, Mimetic finite difference methods for Maxwell's equations and the equations of magnetic diffusion, in "Geometric Methods for Computational Electromagnetics," volume 32 of PIER (ed. F. L. Teixeira), EMW Publishing, Cambridge, MA, (2001), 89-121. [26] S. Lang, "Differential and Riemannian Manifolds in Mathematics," Springer, New York, 1995. [27] S. Lang, "Fundamentals of Differential Geometry," Springer, 1999. [28] K. W. Morton, A. Priestley and E. Süli, Convergence of the Lagrange-Galerkin method with non-exact integration, Mathematical Modelling and Numerical Analysis, 22 (1988), 625-653. [29] P. Mullen, A. McKenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Discrete Lie advection and differential forms,, preprint, (). [30] R. N. Rieben, D. A. White, B. K. Wallin and J. M. Solberg, An arbitrary Lagrangian-Eulerian discretization of MHD on 3D unstructured grids, J. Comp. Phys., 226 (2007), 534-570. doi: 10.1016/j.jcp.2007.04.031. [31] G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory, J. Elasticity, 71 (2003), 183-203. doi: 10.1023/B:ELAS.0000005545.46932.08. [32] H.-G. Roos, M. Stynes and L. Tobiska, "Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems," volume 24 of Springer Series in Computational Mathematics, Springer, Berlin, 1996. [33] J.-C. Xu, Optimal algorithms for discretized partial differential equations, in "Proceedings of the ICIAM'07 conference, Zürich, Switzerland, July 2007," EMS Publishing House, 2008.
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