# American Institute of Mathematical Sciences

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

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 and Related Models, 2019, 12 (1) : 105-131. doi: 10.3934/krm.2019005
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##### References:
Magnetic ($B$) and electric ($E_1$, $E_2$) energy for case~1 (left) and case 2 (right).
Kinetic energy for case 1 (left) and case 2 (right).
$L_1$ 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
$L_1$ 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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