# 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 & Continuous Dynamical Systems - A, 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, (1983). Google Scholar [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. doi: 10.1090/S0273-0979-10-01278-4. Google Scholar [3] D. N. Arnold, R. S. Falk and R. Winther, Multigrid in H(div) and H(curl),, Numer. Math., 85 (2000), 197. doi: 10.1007/PL00005386. Google Scholar [4] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications,, Acta Numerica, 15 (2006), 1. doi: 10.1017/S0962492906210018. Google Scholar [5] A. Bossavit, Extrusion, contraction: Their discretization via Whitney forms,, COMPEL, 22 (2004), 470. Google Scholar [6] A. Bossavit, Discretization of electromagnetic problems: The "generalized finite differences,", in, XIII (2005), 443. Google Scholar [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). Google Scholar [8] F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems,, Math. Model. Numer. Anal., 43 (2009), 277. doi: 10.1051/m2an:2008046. Google Scholar [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. doi: 10.1142/S0218202504003866. Google Scholar [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. doi: 10.1016/0309-1708(90)90041-2. Google Scholar [11] H. De Sterk, Multi-dimensional upwind constrained transport on unstructured grids for "shallow water" magnetohydrodynamics,, in, (2001), 11. Google Scholar [12] M. Desbrun, A. N. Hirani, J. E. Marsden and M. Leok, Discrete exterior calculus,, preprint, (). Google Scholar [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. doi: 10.1016/j.cma.2006.09.025. Google Scholar [14] M. E. Gurtin, "An Introduction to Continuum Mechanics,", volume \textbf{158} of Mathematics in Science and Engineering, 158 (1981). Google Scholar [15] J. Harrison, Ravello lecture notes on geometric calculus - Part I,, preprint, (). Google Scholar [16] H. Heumann and R. Hiptmair, "Extrusion Contraction Upwind Schemes for Convection-Diffusion Problems,", Report 2008-30, (2008), 2008. Google Scholar [17] H. Heumann, R. Hiptmair and J. Xu, "A Semi-Lagrangian Method for Convection of Differential Forms,", Report 2009-09, (2009), 2009. Google Scholar [18] R. Hiptmair, Multigrid method for Maxwell's equations,, SIAM J. Numer. Anal., 36(1) (1999), 204. doi: 10.1137/S0036142997326203. Google Scholar [19] R. Hiptmair, Discrete Hodge operators,, Numer. Math., 90 (2001), 265. doi: 10.1007/s002110100295. Google Scholar [20] R. Hiptmair, Finite elements in computational electromagnetism,, Acta Numerica, 11 (2002), 237. doi: 10.1017/CBO9780511550140.004. Google Scholar [21] R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces,, SIAM J. Numer. Anal., 45 (2007), 2483. doi: 10.1137/060660588. Google Scholar [22] A. N. Hirani, "Discrete Exterior Calculus,", Ph.D. Thesis, (2003), 05202003. Google Scholar [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. doi: 10.1016/S0898-1221(97)00009-6. Google Scholar [24] J. M. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations,, J. Comp. Phys., 151 (1999), 881. doi: 10.1006/jcph.1999.6225. Google Scholar [25] J. M. Hyman and M. Shashkov, Mimetic finite difference methods for Maxwell's equations and the equations of magnetic diffusion,, in, 32 (2001), 89. Google Scholar [26] S. Lang, "Differential and Riemannian Manifolds in Mathematics,", Springer, (1995). Google Scholar [27] S. Lang, "Fundamentals of Differential Geometry,", Springer, (1999). Google Scholar [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. Google Scholar [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, (). Google Scholar [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. doi: 10.1016/j.jcp.2007.04.031. Google Scholar [31] G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory,, J. Elasticity, 71 (2003), 183. doi: 10.1023/B:ELAS.0000005545.46932.08. Google Scholar [32] H.-G. Roos, M. Stynes and L. Tobiska, "Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems,", volume \textbf{24} of Springer Series in Computational Mathematics, 24 (1996). Google Scholar [33] J.-C. Xu, Optimal algorithms for discretized partial differential equations,, in, (2007). Google Scholar

show all references

##### References:
 [1] R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications,", Addison-Wesley, (1983). Google Scholar [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. doi: 10.1090/S0273-0979-10-01278-4. Google Scholar [3] D. N. Arnold, R. S. Falk and R. Winther, Multigrid in H(div) and H(curl),, Numer. Math., 85 (2000), 197. doi: 10.1007/PL00005386. Google Scholar [4] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications,, Acta Numerica, 15 (2006), 1. doi: 10.1017/S0962492906210018. Google Scholar [5] A. Bossavit, Extrusion, contraction: Their discretization via Whitney forms,, COMPEL, 22 (2004), 470. Google Scholar [6] A. Bossavit, Discretization of electromagnetic problems: The "generalized finite differences,", in, XIII (2005), 443. Google Scholar [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). Google Scholar [8] F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems,, Math. Model. Numer. Anal., 43 (2009), 277. doi: 10.1051/m2an:2008046. Google Scholar [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. doi: 10.1142/S0218202504003866. Google Scholar [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. doi: 10.1016/0309-1708(90)90041-2. Google Scholar [11] H. De Sterk, Multi-dimensional upwind constrained transport on unstructured grids for "shallow water" magnetohydrodynamics,, in, (2001), 11. Google Scholar [12] M. Desbrun, A. N. Hirani, J. E. Marsden and M. Leok, Discrete exterior calculus,, preprint, (). Google Scholar [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. doi: 10.1016/j.cma.2006.09.025. Google Scholar [14] M. E. Gurtin, "An Introduction to Continuum Mechanics,", volume \textbf{158} of Mathematics in Science and Engineering, 158 (1981). Google Scholar [15] J. Harrison, Ravello lecture notes on geometric calculus - Part I,, preprint, (). Google Scholar [16] H. Heumann and R. Hiptmair, "Extrusion Contraction Upwind Schemes for Convection-Diffusion Problems,", Report 2008-30, (2008), 2008. Google Scholar [17] H. Heumann, R. Hiptmair and J. Xu, "A Semi-Lagrangian Method for Convection of Differential Forms,", Report 2009-09, (2009), 2009. Google Scholar [18] R. Hiptmair, Multigrid method for Maxwell's equations,, SIAM J. Numer. Anal., 36(1) (1999), 204. doi: 10.1137/S0036142997326203. Google Scholar [19] R. Hiptmair, Discrete Hodge operators,, Numer. Math., 90 (2001), 265. doi: 10.1007/s002110100295. Google Scholar [20] R. Hiptmair, Finite elements in computational electromagnetism,, Acta Numerica, 11 (2002), 237. doi: 10.1017/CBO9780511550140.004. Google Scholar [21] R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces,, SIAM J. Numer. Anal., 45 (2007), 2483. doi: 10.1137/060660588. Google Scholar [22] A. N. Hirani, "Discrete Exterior Calculus,", Ph.D. Thesis, (2003), 05202003. Google Scholar [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. doi: 10.1016/S0898-1221(97)00009-6. Google Scholar [24] J. M. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations,, J. Comp. Phys., 151 (1999), 881. doi: 10.1006/jcph.1999.6225. Google Scholar [25] J. M. Hyman and M. Shashkov, Mimetic finite difference methods for Maxwell's equations and the equations of magnetic diffusion,, in, 32 (2001), 89. Google Scholar [26] S. Lang, "Differential and Riemannian Manifolds in Mathematics,", Springer, (1995). Google Scholar [27] S. Lang, "Fundamentals of Differential Geometry,", Springer, (1999). Google Scholar [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. Google Scholar [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, (). Google Scholar [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. doi: 10.1016/j.jcp.2007.04.031. Google Scholar [31] G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory,, J. Elasticity, 71 (2003), 183. doi: 10.1023/B:ELAS.0000005545.46932.08. Google Scholar [32] H.-G. Roos, M. Stynes and L. Tobiska, "Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems,", volume \textbf{24} of Springer Series in Computational Mathematics, 24 (1996). Google Scholar [33] J.-C. Xu, Optimal algorithms for discretized partial differential equations,, in, (2007). Google Scholar
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