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February  2016, 9(1): 185-214. doi: 10.3934/dcdss.2016.9.185

## Stabilized Galerkin for transient advection of differential forms

 1 EP CASTOR, INRIA Méditerranée and University Nice-Sophia Antipolis, 2004 Route des Lucioles, Sophia Antipolis, France 2 Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, Zürich, Switzerland, Switzerland

Received  September 2014 Revised  February 2015 Published  December 2015

We deal with the discretization of generalized transient advection problems for differential forms on bounded spatial domains. We pursue an Eulerian method of lines approach with explicit timestepping. Concerning spatial discretization we extend the jump stabilized Galerkin discretization proposed in $[$ H. HEUMANN and R.HIPTMAIR, Stabilized Galerkin methods for magnetic advection, Math. Modelling Numer. Analysis, 47 (2013), pp.1713--1732$]$ to forms of any degree and, in particular, advection velocities that may have discontinuities resolved by the mesh. A rigorous a priori convergence theory is established for Lipschitz continuous velocities, conforming meshes and standard finite element spaces of discrete differential forms. However, numerical experiments furnish evidence of the good performance of the new method also in the presence of jumps of the advection velocity.
Citation: Holger Heumann, Ralf Hiptmair, Cecilia Pagliantini. Stabilized Galerkin for transient advection of differential forms. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 185-214. doi: 10.3934/dcdss.2016.9.185
##### References:
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Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), 1-155. doi: 10.1017/S0962492906210018.  Google Scholar [6] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 281-354. doi: 10.1090/S0273-0979-10-01278-4.  Google Scholar [7] A. Bossavit, On the geometry of electromagnetism. (2): 'Geometrical objects', J. Japan Soc. Appl. Electromagn. and Mech., 6 (1998), 114-123. Google Scholar [8] F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ., 10 (2013), 235-282. doi: 10.1142/S0219891613500100.  Google Scholar [9] F. Boyer, Analysis of the upwind finite volume method for general initial- and boundary-value transport problems, IMA J. Numer. Anal., 32 (2012), 1404-1439. doi: 10.1093/imanum/drr031.  Google Scholar [10] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, $3^{rd}$ edition, Texts in Applied Mathematics, Volume 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar [11] F. Brezzi, J. Douglas, Jr. and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235. doi: 10.1007/BF01389710.  Google Scholar [12] 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.  Google Scholar [13] E. Burman, A. Ern and M. A. Fernández, Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems, SIAM J. Numer. Anal., 48 (2010), 2019-2042. doi: 10.1137/090757940.  Google Scholar [14] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.  Google Scholar [15] G. Crippa and C. De Lellis, Regularity and compactness for the DiPerna-Lions flow, in Hyperbolic problems: Theory, numerics, applications, 4, Springer, Berlin, 2008, 423-430. doi: 10.1007/978-3-540-75712-2_39.  Google Scholar [16] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333.  Google Scholar [17] C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 277-382. doi: 10.1016/S1874-5717(07)80007-7.  Google Scholar [18] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar [19] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, Volume 159, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar [20] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418. doi: 10.1002/cpa.3160110306.  Google Scholar [21] F. G. Fuchs, A. D. McMurry, S. Mishra, N. H. Risebro and K. Waagan, Approximate Riemann solvers and robust high-order finite volume schemes for multi-dimensional ideal MHD equations, Commun. Comput. Phys., 9 (2011), 324-362. doi: 10.4208/cicp.171109.070510a.  Google Scholar [22] H. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616945.  Google Scholar [23] H. Heumann, Eulerian and Semi-Lagrangian Methods for Advection-Diffusion of Differential Forms, ETH dissertation 19608, ETH Zürich, 2011. doi: 10.3929/ethz-a-006506738.  Google Scholar [24] H. Heumann and R. Hiptmair, Stabilized Galerkin methods for magnetic advection, ESAIM Math. Model. Numer. Anal., 47 (2013), 1713-1732. doi: 10.1051/m2an/2013085.  Google Scholar [25] H. Heumann and R. Hiptmair, Convergence of Lowest Order Semi-Lagrangian Schemes, Found. Comput. Math., 13 (2013), 187-220. doi: 10.1007/s10208-012-9139-3.  Google Scholar [26] H. Heumann, R. Hiptmair and C. Pagliantini, Stabilized Galerkin for Transient Advection of Differential Forms, Technical Report 2015-06, Seminar for Applied Mathematics, ETH Zürich, Switzerland, 2015. Available from: http://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-06.pdf. Google Scholar [27] R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), 237-339. doi: 10.1017/S0962492902000041.  Google Scholar [28] P. Houston, I. Perugia, A. Schneebeli and D. Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math., 100 (2005), 485-518. doi: 10.1007/s00211-005-0604-7.  Google Scholar [29] O. A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., 41 (2003), 2374-2399. doi: 10.1137/S0036142902405217.  Google Scholar [30] T. Kato, Linear and quasi-linear equations of evolution of hyperbolic type, in Hyperbolicity, C.I.M.E. Summer Schools, Volume 72, Springer Berlin Heidelberg, 1976, 125-191. doi: 10.1007/978-3-642-11105-1_4.  Google Scholar [31] J. M. Lee, Introduction to Smooth Manifolds, $2^{nd}$ edition, Graduate Texts in Mathematics, Volume 218, Springer, New York, 2013.  Google Scholar [32] D. Levy and E. Tadmor, From semidiscrete to fully discrete: Stability of Runge-Kutta schemes by the energy method, SIAM Rev., 40 (1998), 40-73. doi: 10.1137/S0036144597316255.  Google Scholar [33] S. Mishra, Ch. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, J. Comput. Phys., 231 (2012), 3365-3388. doi: 10.1016/j.jcp.2012.01.011.  Google Scholar [34] S. A. Orszag and C. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90 (1979), 129-143. doi: 10.1017/S002211207900210X.  Google Scholar [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Volume 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [36] T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal., 28 (1991), 133-140. doi: 10.1137/0728006.  Google Scholar [37] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Lecture Notes in Math., Volume 606, Springer, Berlin, 1977, 292-315.  Google Scholar [38] G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, Lecture Notes in Mathematics, 1607, Springer-Verlag, Berlin, 1995.  Google Scholar [39] N. J. Walkington, Convergence of the discontinuous Galerkin method for discontinuous solutions, SIAM J. Numer. Anal., 42 (2005), 1801-1817. doi: 10.1137/S0036142902412233.  Google Scholar

show all references

##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2.  Google Scholar [2] L. Ambrosio, The flow associated to weakly differentiable vector fields: Recent results and open problems, in Nonlinear Conservation Laws and Applications, IMA Vol. Math. Appl., 153, Springer, New York, 2011, 181-193. doi: 10.1007/978-1-4419-9554-4_7.  Google Scholar [3] D. N. Arnold, Spaces of finite element differential forms, in Analysis and Numerics of Partial Differential Equations, Springer INdAM Ser., 4, Springer, Milan, 2013, 117-140. doi: 10.1007/978-88-470-2592-9_9.  Google Scholar [4] D. N. Arnold, D. Boffi and F. Bonizzoni, Finite element differential forms on curvilinear cubic meshes and their approximation properties, Numer. Math., 129 (2015), 1-20. doi: 10.1007/s00211-014-0631-3.  Google Scholar [5] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), 1-155. doi: 10.1017/S0962492906210018.  Google Scholar [6] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 281-354. doi: 10.1090/S0273-0979-10-01278-4.  Google Scholar [7] A. Bossavit, On the geometry of electromagnetism. (2): 'Geometrical objects', J. Japan Soc. Appl. Electromagn. and Mech., 6 (1998), 114-123. Google Scholar [8] F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ., 10 (2013), 235-282. doi: 10.1142/S0219891613500100.  Google Scholar [9] F. Boyer, Analysis of the upwind finite volume method for general initial- and boundary-value transport problems, IMA J. Numer. Anal., 32 (2012), 1404-1439. doi: 10.1093/imanum/drr031.  Google Scholar [10] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, $3^{rd}$ edition, Texts in Applied Mathematics, Volume 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar [11] F. Brezzi, J. Douglas, Jr. and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235. doi: 10.1007/BF01389710.  Google Scholar [12] 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.  Google Scholar [13] E. Burman, A. Ern and M. A. Fernández, Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems, SIAM J. Numer. Anal., 48 (2010), 2019-2042. doi: 10.1137/090757940.  Google Scholar [14] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.  Google Scholar [15] G. Crippa and C. De Lellis, Regularity and compactness for the DiPerna-Lions flow, in Hyperbolic problems: Theory, numerics, applications, 4, Springer, Berlin, 2008, 423-430. doi: 10.1007/978-3-540-75712-2_39.  Google Scholar [16] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333.  Google Scholar [17] C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 277-382. doi: 10.1016/S1874-5717(07)80007-7.  Google Scholar [18] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar [19] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, Volume 159, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar [20] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418. doi: 10.1002/cpa.3160110306.  Google Scholar [21] F. G. Fuchs, A. D. McMurry, S. Mishra, N. H. Risebro and K. Waagan, Approximate Riemann solvers and robust high-order finite volume schemes for multi-dimensional ideal MHD equations, Commun. Comput. Phys., 9 (2011), 324-362. doi: 10.4208/cicp.171109.070510a.  Google Scholar [22] H. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616945.  Google Scholar [23] H. Heumann, Eulerian and Semi-Lagrangian Methods for Advection-Diffusion of Differential Forms, ETH dissertation 19608, ETH Zürich, 2011. doi: 10.3929/ethz-a-006506738.  Google Scholar [24] H. Heumann and R. Hiptmair, Stabilized Galerkin methods for magnetic advection, ESAIM Math. Model. Numer. Anal., 47 (2013), 1713-1732. doi: 10.1051/m2an/2013085.  Google Scholar [25] H. Heumann and R. Hiptmair, Convergence of Lowest Order Semi-Lagrangian Schemes, Found. Comput. Math., 13 (2013), 187-220. doi: 10.1007/s10208-012-9139-3.  Google Scholar [26] H. Heumann, R. Hiptmair and C. Pagliantini, Stabilized Galerkin for Transient Advection of Differential Forms, Technical Report 2015-06, Seminar for Applied Mathematics, ETH Zürich, Switzerland, 2015. Available from: http://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-06.pdf. Google Scholar [27] R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), 237-339. doi: 10.1017/S0962492902000041.  Google Scholar [28] P. Houston, I. Perugia, A. Schneebeli and D. Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math., 100 (2005), 485-518. doi: 10.1007/s00211-005-0604-7.  Google Scholar [29] O. A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., 41 (2003), 2374-2399. doi: 10.1137/S0036142902405217.  Google Scholar [30] T. Kato, Linear and quasi-linear equations of evolution of hyperbolic type, in Hyperbolicity, C.I.M.E. Summer Schools, Volume 72, Springer Berlin Heidelberg, 1976, 125-191. doi: 10.1007/978-3-642-11105-1_4.  Google Scholar [31] J. M. Lee, Introduction to Smooth Manifolds, $2^{nd}$ edition, Graduate Texts in Mathematics, Volume 218, Springer, New York, 2013.  Google Scholar [32] D. Levy and E. Tadmor, From semidiscrete to fully discrete: Stability of Runge-Kutta schemes by the energy method, SIAM Rev., 40 (1998), 40-73. doi: 10.1137/S0036144597316255.  Google Scholar [33] S. Mishra, Ch. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, J. Comput. Phys., 231 (2012), 3365-3388. doi: 10.1016/j.jcp.2012.01.011.  Google Scholar [34] S. A. Orszag and C. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90 (1979), 129-143. doi: 10.1017/S002211207900210X.  Google Scholar [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Volume 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [36] T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal., 28 (1991), 133-140. doi: 10.1137/0728006.  Google Scholar [37] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Lecture Notes in Math., Volume 606, Springer, Berlin, 1977, 292-315.  Google Scholar [38] G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, Lecture Notes in Mathematics, 1607, Springer-Verlag, Berlin, 1995.  Google Scholar [39] N. J. Walkington, Convergence of the discontinuous Galerkin method for discontinuous solutions, SIAM J. Numer. Anal., 42 (2005), 1801-1817. doi: 10.1137/S0036142902412233.  Google Scholar
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