A posteriori error estimates for mixed finite element discretization of the multigroup Neutron Simplified Transport equations with Robin boundary condition

TL;DR

This paper develops guaranteed and efficient a posteriori error estimators for mixed finite element discretizations of multigroup neutron transport equations with Robin boundary conditions, including adaptive mesh refinement strategies.

Contribution

It introduces a novel estimator specifically designed for Robin boundary conditions and extends the theory to mixed boundary conditions and domain decomposition methods.

Findings

The estimators are guaranteed and locally efficient.

Numerical experiments confirm the theoretical results.

Adaptive mesh refinement improves solution accuracy.

Abstract

We analyse a posteriori error estimates for the discretization with mixed finite elements on simplicial or Cartesian meshes of the multigroup neutron simplified transport (SPN ) equations, in the case where a Robin (or Fourier type) boundary condition is imposed on the boundary. This boundary condition is of particular importance in neutronics, since it corresponds to the well-known vacuum boundary condition. We provide guaranteed and locally efficient estimators. In particular, a specific estimator is designed to handle the Robin boundary condition. We also develop the theory in the case of mixed imposed boundary conditions, of Dirichlet, Neumann or Fourier type. The approach is further extended to a Domain Decomposition Method, the so-called DD+L 2 jumps method. In this framework, the adaptive mesh refinement strategy is implemented for a discretization using Cartesian meshes on each subdomain. Numerical experiments illustrate the theory.