# On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system Journal article, 2019

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 HS+1(Omega), we derive global a priori error bound of order 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) = diam K being the diameter of the phase-space-time element K and pR-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 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].

Streamline diffusion

Nitsche scheme

hp-method

discontinuous Galerkin

Vlasov-Maxwell system

## Author

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

### Piotr Kowalczyk

University of Warsaw

### Christoffer Standar

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

University of Gothenburg

#### Kinetic and Related Models

1937-5093 (ISSN) 1937-5077 (eISSN)

Vol. 12 1 105-131

### Subject Categories

Computational Mathematics

Control Engineering

Mathematical Analysis

### DOI

10.3934/krm.2019005