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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 and Continuous Dynamical Systems - S, 2016, 9 (1) : 185-214. doi: 10.3934/dcdss.2016.9.185
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.

[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.

[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.

[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.

[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.

[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.

[7]

A. Bossavit, On the geometry of electromagnetism. (2): 'Geometrical objects', J. Japan Soc. Appl. Electromagn. and Mech., 6 (1998), 114-123.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[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.

[16]

P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333.

[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.

[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.

[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.

[20]

K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418. doi: 10.1002/cpa.3160110306.

[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.

[22]

H. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616945.

[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.

[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.

[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.

[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.

[27]

R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), 237-339. doi: 10.1017/S0962492902000041.

[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.

[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.

[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.

[31]

J. M. Lee, Introduction to Smooth Manifolds, $2^{nd}$ edition, Graduate Texts in Mathematics, Volume 218, Springer, New York, 2013.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, Lecture Notes in Mathematics, 1607, Springer-Verlag, Berlin, 1995.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

A. Bossavit, On the geometry of electromagnetism. (2): 'Geometrical objects', J. Japan Soc. Appl. Electromagn. and Mech., 6 (1998), 114-123.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[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.

[16]

P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333.

[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.

[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.

[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.

[20]

K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418. doi: 10.1002/cpa.3160110306.

[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.

[22]

H. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616945.

[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.

[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.

[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.

[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.

[27]

R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), 237-339. doi: 10.1017/S0962492902000041.

[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.

[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.

[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.

[31]

J. M. Lee, Introduction to Smooth Manifolds, $2^{nd}$ edition, Graduate Texts in Mathematics, Volume 218, Springer, New York, 2013.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, Lecture Notes in Mathematics, 1607, Springer-Verlag, Berlin, 1995.

[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.

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