A posteriori error estimates for the one and one-half Dimensional Relativistic Vlasov-Maxwell system

TL;DR

This paper develops a posteriori error estimates for the streamline diffusion finite element method applied to the relativistic Vlasov-Maxwell system, enhancing error control and convergence analysis for hyperbolic PDEs.

Contribution

It introduces a dual problem-based a posteriori error estimation approach for the Vlasov-Maxwell system using streamline diffusion finite elements, including negative norm estimates.

Findings

Error estimates based on residuals are derived.

The method improves convergence for hyperbolic problems.

Computational implementation supports theoretical results.

Abstract

This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov-Maxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, and convection-dominated convection diffusion problems. A procedure that improves the convergence of finite elements for this type of problems. The a posteriori error estimates relay on a dual problem formulation and yields an error control based on the, computable, residual of the approximate solution. The lack of dissipativity enforces us considering negative norm estimates. To derive these estimates, the error term is split into an iteration and an approximation error where the iteration procedure is assumed to converge. The computational aspects and implementations, which justify the theoretical results of this part, are the subject of these studies and addressed in [5].