A Multiscale Finite Element Method for reaction-diffusion eigenproblems arising from neutronics
Abstract
We consider reaction-diffusion eigenproblems with oscillatory diffusion and reaction coefficients. The reaction coefficient magnitude is large: the corrector equation identified by periodic homogenization involves both the diffusion and the reaction operators. We study the numerical approximation of this problem using the Multiscale Finite Element Method (MsFEM). This now classical method is a finite element type method that performs a Galerkin approximation of the oscillatory problem on a specific, problem dependent, basis set. The basis functions are precomputed in an offline stage. Inspired by homogenization theory and using some filtering ideas, we show how to define these basis functions in order to obtain an efficient method. The comprehensive set of numerical experiments that we present, in periodic and non-periodic cases, for the scalar-valued version of the problem (which is then self-adjoint) and for the vector-valued version of the problem (which is then in general non self-adjoint), demonstrates the performance of the approach. Some theoretical arguments complement the numerical observations.
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BibTeX
@misc{bris2026multiscale,
title = {{A Multiscale Finite Element Method for reaction-diffusion eigenproblems arising from neutronics}},
author = {Claude Le Bris and Albéric Lefort and Frédéric Legoll},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02420},
eprint = {2609.02420},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05737113v1},
}
RIS
TY - UNPB
TI - A Multiscale Finite Element Method for reaction-diffusion eigenproblems arising from neutronics
AU - Claude Le Bris
AU - Albéric Lefort
AU - Frédéric Legoll
PY - 2026
DA - 2026/09/02
PB - HAL
DO - 10.48550/arxiv.2609.02420
UR - https://hal.science/hal-05737113v1
AB - We consider reaction-diffusion eigenproblems with oscillatory diffusion and reaction coefficients. The reaction coefficient magnitude is large: the corrector equation identified by periodic homogenization involves both the diffusion and the reaction operators. We study the numerical approximation of this problem using the Multiscale Finite Element Method (MsFEM). This now classical method is a finite element type method that performs a Galerkin approximation of the oscillatory problem on a specific, problem dependent, basis set. The basis functions are precomputed in an offline stage. Inspired by homogenization theory and using some filtering ideas, we show how to define these basis functions in order to obtain an efficient method. The comprehensive set of numerical experiments that we present, in periodic and non-periodic cases, for the scalar-valued version of the problem (which is then self-adjoint) and for the vector-valued version of the problem (which is then in general non self-adjoint), demonstrates the performance of the approach. Some theoretical arguments complement the numerical observations.
ER -