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

Patrick Ciarlet, Minh-Hieu Do, Mario Gervais, François Madiot

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.

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

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.

Paper Structure

This paper contains 24 sections, 12 theorems, 112 equations, 7 figures, 5 tables.

Table of Contents

  1. Notations
  2. The model
  3. Variational formulation and discretization
  4. Mixed variational formulation
  5. Discretization and a priori error analysis
  6. A posteriori studies for a mixed Finite element discretization
  7. Reconstruction of the discrete solution
  8. Averaging operator
  9. Post-processing approach
  10. A posteriori error estimates
  11. Extension to the Domain Decomposition+$L^2$-jumps method
  12. Numerical experiment
  13. Adaptive mesh refinement
  14. SOLVE module
  15. ESTIMATE module
  16. ...and 9 more sections

Key Result

Proposition 3.1

Let ${\mathbb T}_{o}$ and ${\mathbb T}_{e}$ satisfy Pos-SPN. The solution $({\bf p},\phi)\in{{\bf {\underline {\underline Q} } }(\Omega)}\times {{\underline {\underline H}}^1(\Omega)}$ to eq:diff-mixed is such that $\phi$ is a solution to eq:diff_primal with the same data. Conversely, the solution

Figures (7)

  • Figure 1: Description of the AMR process.
  • Figure 2: The benchmark geometry.
  • Figure 3: Subdomains and initial mesh for the multi-domain approach.
  • Figure 4: Relative error in the $\|\cdot\|_S$ norm (left) and maximum of the total estimator (right) as a function of the total number of mesh elements.
  • Figure 5: Final mesh for the mono-domain configurations.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.1
  • Remark 4.1
  • Remark 4.2
  • ...and 18 more