February  2019, 12(1): 105-131. doi: 10.3934/krm.2019005

On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

1. 

Department of Mathematics, Chalmers University of Technology and Göteborg University, SE-412 96, Göteborg, Sweden

2. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

3. 

Department of Mathematics, Chalmers University of Technology and Göteborg University, SE-412 96, Göteborg, Sweden

* Corresponding author: Mohammad Asadzadeh

Received  November 2017 Revised  May 2018 Published  July 2018

Fund Project: The research of the first author was supported by the Swedish Research Council VR.

We study stability and convergence of $hp$-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the $hp$ scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space $H^{s+1}(Ω)$, we derive global a priori error bound of order ${\mathcal O}(h/p)^{s+1/2}$, where $h ( = \max_K h_K)$ is the mesh parameter and $p ( = \max_K p_K)$ is the spectral order. This estimate is based on the local version with $h_K = \mbox{ diam } K$ being the diameter of the phase-space-time element $K$ and $p_K$ is the spectral order (the degree of approximating finite element polynomial) for $K$. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of ${\mathcal O}(h^2+k^2)$, where $h$ is the spatial mesh size and $k$ is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [22].

Citation: Mohammad Asadzadeh, Piotr Kowalczyk, Christoffer Standar. On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system. Kinetic & Related Models, 2019, 12 (1) : 105-131. doi: 10.3934/krm.2019005
References:
[1]

M. Asadzadeh, Streamline diffusion methods for the Vlasov-Poisson equation, ESAIM: Math. Model. Numer. Anal., 24 (1990), 177-196.  doi: 10.1051/m2an/1990240201771.  Google Scholar

[2]

M. Asadzadeh and P. Kowalczyk, Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov-Fokker-Planck system, Numer. Methods Partial Differential Equations, 21 (2005), 472-495.  doi: 10.1002/num.20044.  Google Scholar

[3]

M. Asadzadeh and P. Kowalczyk, Convergence analysis for backward-Euler and mixed discontinuous Galerkin methods for the Vlasov-Poisson system, Adv. Comput. Math., 41 (2015), 833-852.  doi: 10.1007/s10444-014-9388-6.  Google Scholar

[4]

M. Asadzadeh and A. Sopasakis, Convergence of a hp-streamline diffusion scheme for Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 17 (2007), 1159-1182.  doi: 10.1142/S0218202507002236.  Google Scholar

[5]

F. Assous and M. Michaeli, Solving Maxwell's equations in singular domains with a Nitsche type method, J. Comput. Phys., 230 (2011), 4922-4939.  doi: 10.1016/j.jcp.2011.03.013.  Google Scholar

[6]

N. Besse, Convergence of a high-order semi-Lagrangian scheme with propagation of gradients for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 46 (2008), 639-670.  doi: 10.1137/050635171.  Google Scholar

[7]

F. BrezziL. 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

[8]

E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: Ⅱ. A stabilized Nitsche method, Appl. Numer. Math., 62 (2012), 328-341.  doi: 10.1016/j.apnum.2011.01.008.  Google Scholar

[9]

F. CharlesB. Després and M. Mehrenberger, Enhanced convergence estimates for semi-Lagrangian schemes: Application to the Vlasov-Poisson equation, SIAM J. Numer. Anal., 51 (2013), 840-863.  doi: 10.1137/110851511.  Google Scholar

[10]

Y. ChengI. M. GambaF. Li and P. J. Morrison, Discontinuous Galerkin methods for the Vlasov-Maxwell equations, SIAM J. Numer. Anal., 52 (2014), 1017-1049.  doi: 10.1137/130915091.  Google Scholar

[11]

P. DegondF. Deluzet and D. Doyen, Asymptotic-preserving particle-in-cell methods for the Vlasov-Maxwell system in the quasi-neutral limit, J. Comput. Phys., 330 (2017), 467-492.  doi: 10.1016/j.jcp.2016.11.018.  Google Scholar

[12]

B. A. de Dios, J. Carrillo and C. -W. Shu, Discontinuous Galerkin methods for the multidimensional Vlasov-Poisson problem, Math. Models Methods Appl. Sci., 22 (2012), 1250042, 45pp. doi: 10.1142/S021820251250042X.  Google Scholar

[13]

R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.  doi: 10.1002/cpa.3160420603.  Google Scholar

[14]

F. Filbet, Convergence of a finite volume scheme for the Vlasov-Poisson system, SIAM J. Numer. Anal., 39 (2001), 1146-1169.  doi: 10.1137/S003614290037321X.  Google Scholar

[15]

F. FilbetY. Guo and C.-W. Shu, Analysis of the relativistic Vlasov-Maxwell model in an interval, Quart. Appl. Math., 63 (2005), 691-714.  doi: 10.1090/S0033-569X-05-00977-9.  Google Scholar

[16]

R. GlasseyS. Pankavich and J. Schaeffer, Separated characteristics and global solvability for the one and one-half dimensional Vlasov-Maxwell system, Kinet. Relat. Models, 9 (2016), 455-467.  doi: 10.3934/krm.2016003.  Google Scholar

[17]

R. Glassey and J. Schaeffer, On the 'one and one-half' dimensional relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179.  doi: 10.1002/mma.1670130207.  Google Scholar

[18]

P. HoustonC. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163.  doi: 10.1137/S0036142900374111.  Google Scholar

[19]

P. HoustonC. Schwab and E. Süli, Stabilized hp-finite element methods for first-order hyperbolic problems, SIAM J. Numer. Anal., 37 (2000), 1618-1643.  doi: 10.1137/S0036142998348777.  Google Scholar

[20]

T. J. R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion, in Finite element methods for convection dominated flows, ASME Winter Annual Meeting (ed. T. J. R. Hughes), Amer. Soc. Mech. Engrs., 34 (1979), 19–35.  Google Scholar

[21]

C. Johnson and J. Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp., 47 (1986), 1-18.  doi: 10.1090/S0025-5718-1986-0842120-4.  Google Scholar

[22]

J. B. Malmberg and C. Standar, Computational aspects of the streamline diffusion schemes for the one and one-half dimensional Vlasov-Maxwell system, in preparation. Google Scholar

[23]

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1971), 9-15.  doi: 10.1007/BF02995904.  Google Scholar

[24]

C. Standar, On streamline diffusion schemes for the one and one-half dimensional relativistic Vlasov-Maxwell system, Calcolo, 53 (2016), 147-169.  doi: 10.1007/s10092-015-0141-4.  Google Scholar

[25]

S. Sticko and G. Kreiss, A stabilized Nitsche cut element method for the wave equation, Comput. Methods Appl. Mech. Engrg., 309 (2016), 364-387.  doi: 10.1016/j.cma.2016.06.001.  Google Scholar

[26]

A. Szepessy, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, ESAIM: Math. Model. Numer. Anal., 25 (1991), 749-782.  doi: 10.1051/m2an/1991250607491.  Google Scholar

[27]

S. I. ZakiL. R. T. Gardner and T. J. M. Boyd, A finite element code for the simulation of one-dimensional Vlasov plasmas. I. Theory, J. Comput. Phys., 79 (1988), 184-199.  doi: 10.1016/0021-9991(88)90010-1.  Google Scholar

show all references

References:
[1]

M. Asadzadeh, Streamline diffusion methods for the Vlasov-Poisson equation, ESAIM: Math. Model. Numer. Anal., 24 (1990), 177-196.  doi: 10.1051/m2an/1990240201771.  Google Scholar

[2]

M. Asadzadeh and P. Kowalczyk, Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov-Fokker-Planck system, Numer. Methods Partial Differential Equations, 21 (2005), 472-495.  doi: 10.1002/num.20044.  Google Scholar

[3]

M. Asadzadeh and P. Kowalczyk, Convergence analysis for backward-Euler and mixed discontinuous Galerkin methods for the Vlasov-Poisson system, Adv. Comput. Math., 41 (2015), 833-852.  doi: 10.1007/s10444-014-9388-6.  Google Scholar

[4]

M. Asadzadeh and A. Sopasakis, Convergence of a hp-streamline diffusion scheme for Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 17 (2007), 1159-1182.  doi: 10.1142/S0218202507002236.  Google Scholar

[5]

F. Assous and M. Michaeli, Solving Maxwell's equations in singular domains with a Nitsche type method, J. Comput. Phys., 230 (2011), 4922-4939.  doi: 10.1016/j.jcp.2011.03.013.  Google Scholar

[6]

N. Besse, Convergence of a high-order semi-Lagrangian scheme with propagation of gradients for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 46 (2008), 639-670.  doi: 10.1137/050635171.  Google Scholar

[7]

F. BrezziL. 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

[8]

E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: Ⅱ. A stabilized Nitsche method, Appl. Numer. Math., 62 (2012), 328-341.  doi: 10.1016/j.apnum.2011.01.008.  Google Scholar

[9]

F. CharlesB. Després and M. Mehrenberger, Enhanced convergence estimates for semi-Lagrangian schemes: Application to the Vlasov-Poisson equation, SIAM J. Numer. Anal., 51 (2013), 840-863.  doi: 10.1137/110851511.  Google Scholar

[10]

Y. ChengI. M. GambaF. Li and P. J. Morrison, Discontinuous Galerkin methods for the Vlasov-Maxwell equations, SIAM J. Numer. Anal., 52 (2014), 1017-1049.  doi: 10.1137/130915091.  Google Scholar

[11]

P. DegondF. Deluzet and D. Doyen, Asymptotic-preserving particle-in-cell methods for the Vlasov-Maxwell system in the quasi-neutral limit, J. Comput. Phys., 330 (2017), 467-492.  doi: 10.1016/j.jcp.2016.11.018.  Google Scholar

[12]

B. A. de Dios, J. Carrillo and C. -W. Shu, Discontinuous Galerkin methods for the multidimensional Vlasov-Poisson problem, Math. Models Methods Appl. Sci., 22 (2012), 1250042, 45pp. doi: 10.1142/S021820251250042X.  Google Scholar

[13]

R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.  doi: 10.1002/cpa.3160420603.  Google Scholar

[14]

F. Filbet, Convergence of a finite volume scheme for the Vlasov-Poisson system, SIAM J. Numer. Anal., 39 (2001), 1146-1169.  doi: 10.1137/S003614290037321X.  Google Scholar

[15]

F. FilbetY. Guo and C.-W. Shu, Analysis of the relativistic Vlasov-Maxwell model in an interval, Quart. Appl. Math., 63 (2005), 691-714.  doi: 10.1090/S0033-569X-05-00977-9.  Google Scholar

[16]

R. GlasseyS. Pankavich and J. Schaeffer, Separated characteristics and global solvability for the one and one-half dimensional Vlasov-Maxwell system, Kinet. Relat. Models, 9 (2016), 455-467.  doi: 10.3934/krm.2016003.  Google Scholar

[17]

R. Glassey and J. Schaeffer, On the 'one and one-half' dimensional relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179.  doi: 10.1002/mma.1670130207.  Google Scholar

[18]

P. HoustonC. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163.  doi: 10.1137/S0036142900374111.  Google Scholar

[19]

P. HoustonC. Schwab and E. Süli, Stabilized hp-finite element methods for first-order hyperbolic problems, SIAM J. Numer. Anal., 37 (2000), 1618-1643.  doi: 10.1137/S0036142998348777.  Google Scholar

[20]

T. J. R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion, in Finite element methods for convection dominated flows, ASME Winter Annual Meeting (ed. T. J. R. Hughes), Amer. Soc. Mech. Engrs., 34 (1979), 19–35.  Google Scholar

[21]

C. Johnson and J. Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp., 47 (1986), 1-18.  doi: 10.1090/S0025-5718-1986-0842120-4.  Google Scholar

[22]

J. B. Malmberg and C. Standar, Computational aspects of the streamline diffusion schemes for the one and one-half dimensional Vlasov-Maxwell system, in preparation. Google Scholar

[23]

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1971), 9-15.  doi: 10.1007/BF02995904.  Google Scholar

[24]

C. Standar, On streamline diffusion schemes for the one and one-half dimensional relativistic Vlasov-Maxwell system, Calcolo, 53 (2016), 147-169.  doi: 10.1007/s10092-015-0141-4.  Google Scholar

[25]

S. Sticko and G. Kreiss, A stabilized Nitsche cut element method for the wave equation, Comput. Methods Appl. Mech. Engrg., 309 (2016), 364-387.  doi: 10.1016/j.cma.2016.06.001.  Google Scholar

[26]

A. Szepessy, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, ESAIM: Math. Model. Numer. Anal., 25 (1991), 749-782.  doi: 10.1051/m2an/1991250607491.  Google Scholar

[27]

S. I. ZakiL. R. T. Gardner and T. J. M. Boyd, A finite element code for the simulation of one-dimensional Vlasov plasmas. I. Theory, J. Comput. Phys., 79 (1988), 184-199.  doi: 10.1016/0021-9991(88)90010-1.  Google Scholar

Figure 1.  Magnetic ($B$) and electric ($E_1$, $E_2$) energy for case~1 (left) and case 2 (right).
Figure 2.  Kinetic energy for case 1 (left) and case 2 (right).
Table 1.  $L_1$ and $L_2$ errors for different polynomial degrees and fixed mesh sizes set $H_1$
Error Degree $f$ $E_1$ $E_2$ $B$
$L_1$ $p=1$ 3.801e-1 7.086e-4 1.599e-6 1.645e-5
$p=2$ 1.614e-1 3.248e-9 1.770e-7 9.092e-7
$p=3$ 1.891e-2 2.295e-10 9.753e-9 3.321e-8
$L_2$ $p=1$ 7.302e-1 6.204e-7 3.517e-12 4.303e-10
$p=2$ 1.632e-1 1.498e-17 4.113e-14 1.070e-12
$p=3$ 2.833e-3 6.648e-20 1.185e-16 2.186e-15
Error Degree $f$ $E_1$ $E_2$ $B$
$L_1$ $p=1$ 3.801e-1 7.086e-4 1.599e-6 1.645e-5
$p=2$ 1.614e-1 3.248e-9 1.770e-7 9.092e-7
$p=3$ 1.891e-2 2.295e-10 9.753e-9 3.321e-8
$L_2$ $p=1$ 7.302e-1 6.204e-7 3.517e-12 4.303e-10
$p=2$ 1.632e-1 1.498e-17 4.113e-14 1.070e-12
$p=3$ 2.833e-3 6.648e-20 1.185e-16 2.186e-15
Table 2.  $L_1$ and $L_2$ errors for different mesh sizes and fixed polynomial degree $p = 1$
Error Degree $f$ $E_1$ $E_2$ $B$
$L_1$ $H_1$ 3.801e-1 7.086e-4 1.599e-6 1.645e-5
$H_2$ 1.629e-1 8.304e-10 1.791e-7 8.387e-6
$H_3$ 4.324e-2 2.016e-10 4.750e-8 2.099e-6
$L_2$ $H_1$ 7.302e-1 6.204e-7 3.517e-12 4.303e-10
$H_2$ 1.939e-1 8.520e-19 3.956e-14 9.298e-11
$H_3$ 1.444e-2 5.014e-20 2.784e-15 5.850e-12
Error Degree $f$ $E_1$ $E_2$ $B$
$L_1$ $H_1$ 3.801e-1 7.086e-4 1.599e-6 1.645e-5
$H_2$ 1.629e-1 8.304e-10 1.791e-7 8.387e-6
$H_3$ 4.324e-2 2.016e-10 4.750e-8 2.099e-6
$L_2$ $H_1$ 7.302e-1 6.204e-7 3.517e-12 4.303e-10
$H_2$ 1.939e-1 8.520e-19 3.956e-14 9.298e-11
$H_3$ 1.444e-2 5.014e-20 2.784e-15 5.850e-12
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