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A Multiscale Finite Element Method for reaction-diffusion eigenproblems arising from neutronics

Claude Le Bris, Albéric Lefort, Frédéric Legoll

arXiv:2609.02420v1math.NA

TL;DR

The paper addresses the costly approximation of reaction-diffusion eigenproblems with fine-scale oscillatory coefficients, for which standard finite elements require prohibitively fine meshes. It develops a homogenization- and filtering-informed MsFEM with precomputed, problem-adapted basis functions. Numerical and theoretical results show accurate, robust performance across periodic and non-periodic scalar and vector-valued settings, subject to stated convergence and filtering limitations.

  • Problem

    Standard finite elements require meshes fine enough to capture heterogeneous coefficients, while existing multiscale methods were not adapted to these reaction-diffusion eigenproblems.

  • Method

    The paper constructs MsFEM basis functions offline using homogenization theory and filtering ideas, then solves the oscillatory eigenproblem through a Galerkin approximation.

  • Results

    The approach gives similar accuracy for periodic and quasi-periodic problems and applies to both scalar and vector-valued formulations, with accuracy improving as ε decreases.

  • Takeaways & Limitations

    The filtering-based MsFEM provides an efficient coarse-mesh approximation that is robust to small scales and microstructure in the tested settings.

  • Takeaways & Limitations

    Eigenvector convergence is only subsequential when the limiting eigenvalue may be multiple, and filter order k = 1 can be inaccurate in tested one-dimensional cases.

Abstract

from arXiv · show

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.

1 Introduction

The paper develops an MsFEM approximation for reaction-diffusion eigenproblems with fine-scale heterogeneous coefficients, extending multiscale methods beyond purely diffusive problems. Homogenization and filtering guide basis construction for periodic, non-periodic, scalar, and vector-valued settings.

  • Problem setting: Reaction-diffusion eigenproblems model heterogeneous media with coefficients oscillating on a scale ε much smaller than the domain.In neutronics, the eigenfunction represents neutron density; scalar and vector formulations correspond to one or multiple neutron-energy families.
  • Motivation: Standard P1 finite elements require a mesh fine enough to resolve heterogeneity, making the discrete problem prohibitively expensive.Existing dedicated methods address oscillatory diffusion, but the paper identifies a need for methods adapted to reaction-diffusion eigenproblems.
  • MsFEM framework: MsFEM uses precomputed basis functions generated by local oscillatory problems, then applies a Galerkin approximation on the resulting problem-adapted space.The fine-scale local computations form an offline stage, while the reduced global problem is solved online.
  • Computational role: The same MsFEM basis functions can be reused for multiple eigencouples, reducing the global discrete dimension after the offline cost is paid.This multi-query setting includes computing several eigenvalue-eigenvector pairs.
  • Method development: The actual MsFEM approach generalizes a preliminary homogenization-based method through filtering and does not require coefficient periodicity or geometric assumptions.The paper first develops periodic homogenization results pedagogically before introducing the general method.
  • Conclusions: The approach is accurate and robust across periodic, quasi-periodic, scalar, and vector-valued problems, with accuracy improving as ε decreases or the coarse mesh is refined.For refined coarse meshes, each coarse element must still contain a sufficient number of small-scale oscillations.

2 The scalar-valued case

The scalar-valued analysis assumes uniformly elliptic, bounded diffusion and positive bounded reaction coefficients, initially under periodicity. The periodic framework supports homogenization and motivates the later filtering-based method for general coefficients.

  • Scalar assumptions: In the scalar case, Aε is a symmetric d×d diffusion matrix, while Σε and σε are scalar coefficients.The diffusion matrix is assumed bounded and uniformly coercive and bounded in the associated bilinear form.
  • Scalar assumptions: The scalar reaction coefficients Σε and σε belong to L∞(Ω) and are bounded away from zero.The supplied assumptions state α ≤ Σε(x) and α ≤ σε(x) almost everywhere.
  • Periodic framework: Sections 2.1 and 2.2 impose periodicity on the coefficients as a preliminary framework for the analysis.The periodic coefficients are represented by Y-periodic functions A, Σ, and σ.
  • Periodic framework: The periodic coefficients satisfy the same coercivity, boundedness, and positivity conditions on the periodicity cell Y.These assumptions enable the homogenization results and the preliminary MsFEM construction.
  • Method scope: The periodicity assumption is relaxed in Section 2.3 when the actual filtering-based MsFEM approach is introduced.Numerical results for the scalar case are collected afterward.

2.1 Periodic homogenization results

Periodic homogenization transforms the reaction-diffusion eigenproblem into a purely diffusive generalized problem after dividing by a positive periodic cell eigenfunction. This enables convergence analysis and guides the multiscale construction.

  • Original problem: The scalar reaction-diffusion problem has a countable positive spectrum, with a simple smallest eigenvalue and a positive first eigenfunction.The self-adjoint structure follows from symmetry of the diffusion matrix.
  • Cell problem: Periodic homogenization introduces a cell eigenproblem for the positive periodic eigenfunction ψ associated with λ∞.The cell problem uses the periodic coefficients A, Σ, and σ on the periodicity cell.
  • Transformation: Dividing uε by ψ(·/ε) defines vε and converts the original problem into a purely diffusive generalized eigenvalue problem.For the first eigenfunction, ψ does not vanish and vε remains well-defined as the first transformed eigenfunction.
  • Homogenization: The transformed problem is easier to homogenize because standard energy estimates bound vε in H1 and the min-max principle bounds its eigenvalue.These bounds support identification of the homogenized limit.
  • Convergence: As ε → 0, each transformed eigencouple converges up to a subsequence to an eigencouple of the homogenized problem.The limit eigenvalue and eigenvector correspond to the same mode of the homogenized problem.
  • Convergence caveat: Eigenvector convergence requires a subsequence when the limiting eigenvalue may be multiple, but the full sequence converges when that eigenvalue is simple.The first limiting eigenvalue is given as an example where simplicity holds.

2.2 Description of a preliminary method

The preliminary MsFEM method uses a coarse finite element mesh and homogenization-informed basis functions for a transformed purely diffusive eigenproblem. Its approximation error is controlled by a remainder that vanishes as ε tends to zero.

  • Coarse discretization: The method uses a regular coarse triangular mesh with characteristic size H, which need not be smaller than ε.The numerical setting typically takes H ≥ ε.
  • Transformed problem: The transformed eigenproblem for vε is purely diffusive, so standard MsFEM basis functions can approximate it on a coarse mesh.The transformed variable is related to the original eigenfunction by vε = uε/ψ(·/ε).
  • Basis construction: MsFEM-lin basis functions are defined through local problems and used in a Galerkin approximation of the transformed eigenproblem.Their name reflects the boundary conditions imposed in the local construction.
  • Offline computation: The local problems are discretized on fine meshes of characteristic size h ≪ ε within each coarse element.This fine local resolution is used during basis precomputation rather than for the global problem.
  • Return to original problem: The preliminary approach approximates the original reaction-diffusion problem through the relation uε = ψ(·/ε)vε and corresponding MsFEM basis functions.The basis construction is therefore tied to the periodic homogenization corrector.
  • Error control: The approximation error is bounded by a remainder R(ε) that tends to zero as ε → 0.The paper refers to a later theorem for the precise statement and proof of this bound.

2.3 Description of the actual MsFEM method

The actual MsFEM constructs filtered local proxies on oversampling patches, then uses them to build coarse-scale basis functions without requiring periodic coefficients. The resulting nonconforming Galerkin method separates offline local computation from online global eigenproblem solution.

  • Proxy construction: The method constructs, for each coarse element K, a proxy eψε approximating ψ(·/ε)|K without assuming periodic coefficients.The proxy is designed to be reliable in periodic settings while remaining applicable to general microstructures.
  • Proxy construction: Oversampling patches SK surround each coarse element, mitigating discrepancies caused when a patch does not contain an integer number of periodic cells.In the periodic case, periodic-boundary-condition eigenvectors coincide with ψ(·/ε) when the patch size is an integer multiple of the periodic cell; filtering addresses the general case.
  • Filtering: A filter function τK is inserted into the local variational integrals to define the filtered proxy problem and its smallest-eigenvalue eigenvector.The approach uses filters of practical order k = 1 or 2; analysis reports convergence for k ≥2 on domains interior to SK.
  • Algorithm: The offline stage solves the filtered local problem on each SK and computes MsFEM-lin basis functions, while the online stage solves the global Galerkin eigenproblem.The oversampling ratio is the patch-to-element size ratio, with patches typically chosen as cubes of edge length 2H.
  • Algorithm: Because the proxy may jump across neighboring elements, the approximation space Vε,H is generally nonconforming and the global integrals are assembled as broken sums.The method nevertheless performs a Galerkin approximation of the reaction-diffusion eigenproblem on the adapted space.

2.4 Numerical results

Numerical experiments compare the MsFEM approach with coarse-grid P1 and, in periodic cases, a preliminary method, using fine-grid P1 solutions as references. The method remains accurate across periodic and quasi-periodic coefficients, while refinement improves accuracy until the coarse scale becomes comparable to the oscillation scale.

  • Experimental setup: Experiments use two-dimensional periodic and quasi-periodic coefficients on Ω = (0, 1)^2, with coarse mesh H = 1/8 and fine mesh h ≪ ε.Reference eigencouples are computed using P1 finite elements on a fine mesh over the whole domain.
  • Experimental setup: The tests compare MsFEM with coarse-mesh P1, and periodic tests additionally compare the preliminary MsFEM method as an ideal benchmark.The scalar tests set σε = 1; more general reaction coefficients are treated in the vector-valued variant.
  • Periodic case: For periodic eigenvalues, the actual MsFEM results are close to the preliminary method, with relative error of the order of or smaller than 10^-3 (0.1%).Errors are plotted against 2H/ε on a log–log scale for the eigenvalue experiment.
  • Quasi-periodic case: For quasi-periodic coefficients, the eigenvector error is of order 20% and the eigenvalue error is smaller than 0.1%, comparable to the periodic case.This comparable accuracy is reported as evidence of robustness despite the more complex microstructure.
  • Mesh-size dependence: Decreasing H reduces both eigenvector and eigenvalue errors, but accuracy deteriorates when H becomes comparable to ε.The limitation is attributed to insufficient oscillations in the oversampling patch and to the ε/H dependence in the preliminary-method estimate.

3 The vector-valued case

The vector-valued problem couples multiple neutron families and is generally non-self-adjoint. The proposed MsFEM uses homogenization-inspired proxies and filtering, achieving robust accuracy for non-symmetric and quasi-periodic coefficients.

  • Problem formulation: The vector-valued formulation uses κ-component eigenvectors, matrix-valued reaction terms, and a fourth-order diffusion tensor.Component coupling arises from non-diagonal, potentially non-symmetric matrices Σε and σε.
  • Periodic homogenization: The first eigenvalue is real and simple, and its eigenfunction can be chosen with positive real-valued components.This property holds under the stated well-posedness assumptions despite the possibility of complex higher eigenvalues.
  • Periodic homogenization: Periodic homogenization introduces common cell eigenvectors ψ and ψ⋆ with a shared first eigenvalue λ∞.The associated cell eigenfunctions have positive components, while the homogenized limit can be scalar-valued.
  • MsFEM construction: The actual MsFEM constructs local proxies on oversampling patches during an offline stage and uses them to define the multiscale basis.The proxies approximate ψ(·/ε), ψ⋆(·/ε), and λ∞ locally without requiring periodicity.
  • Numerical results: For periodic non-symmetric coefficients, the method achieves roughly 20% eigenvector error and 1%–0.1% eigenvalue error as ε varies.These results use fixed coarse-mesh size H = 1/8 and remain comparable to scalar-valued periodic cases.
  • Numerical results: For quasi-periodic coefficients, eigenvector error stays below 20% and eigenvalue error below 0.1%, decreasing to 0.01% for the largest 2H/ε ratios.The experiments fix H = 1/8 while varying ε.

4 Proof of convergence of the preliminary method

The convergence proof analyzes the preliminary MsFEM method by comparing multiscale basis functions with two-scale expansions and controlling elementwise boundary-layer terms. Under periodicity, regularity, and scale-separation assumptions, it establishes an error bound.

  • Theorem and assumptions: The proof establishes an error bound for the preliminary MsFEM method in the scalar-valued periodic setting.The analysis assumes periodic coefficients and sufficient regularity, including a homogenized eigenvector in H2(Ω).
  • Relation to the actual method: The preliminary method is analytical rather than the practically used method, but its theorem and numerical analysis indicate the efficiency of the actual MsFEM approach.The paper explicitly distinguishes the preliminary basis from the filtered method used in practice.
  • Theorem and assumptions: Theorem 14 compares the first eigenvectors uε and uε,ψ,H when the coarse scale satisfies H ≥ ε.The estimate holds for sufficiently small ε, with ε0 independent of H.
  • Proof strategy: The argument uses an equivalent of Céa’s lemma for elliptic eigenvalue problems to estimate the approximation error.The numerator is controlled through two-scale expansions of the exact solution and MsFEM basis functions.
  • Proof strategy: Boundary-layer contributions are localized near element boundaries using a cutoff function supported outside an interior subset Kδ.Choosing δ = ε yields estimates independent of the mesh element and of H.
  • Proof strategy: The proof lower-bounds the denominator using strong convergence of vε and the nonvanishing periodic gradient of ψ.The assumption ∥∇ψ∥L2(Y) > 0 is ensured when the ratio Σ/σ is not constant.

5 Partial analysis of the filtering method

The analysis establishes convergence of the filtering-based local eigenproblem toward the homogenized eigencouple in periodic settings, then examines perturbative expansions and numerical convergence for scalar and vector-valued problems.

  • Analytical framework: The filtering method is analyzed by comparing the first eigencouple of the filtered problem with that of the homogenized cell problem.The analysis focuses on identities and estimates for the eigenvalue, eigenvector, and Lagrange multiplier.
  • Analytical framework: The filtered eigenproblem is posed with a Lagrange multiplier and a weighted normalization involving the filter τ.The filter vanishes on the boundary, enabling the equivalence with an associated first-eigencouple problem through integration by parts.
  • Perturbative analysis: For small perturbations of the reaction coefficients, expansions in δ are inserted into the homogenized and filtered problems and matched order by order.The coefficients and solution are expanded around constant or leading-order terms, yielding equations for successive corrections.
  • Perturbative analysis: At leading order in δ, the homogenized and filtered eigencouples coincide after comparing their respective expansion equations.This comparison identifies equality between the leading-order solutions as ε tends to zero.
  • Convergence results: The eigenvector error in the H1 semi-norm converges at rate O(ε) for every filter order k ≥1, so increasing k does not improve this rate.Numerical tests report eigenvalue rates ε^3 for k = 2 and ε^2 for k = 1, while the eigenvector H1 rate remains O(ε).
  • Convergence results: For periodic non-perturbative examples, filter order k = 2 is preferred because k = 1 can produce inaccurate results, while the same convergence rates appear in dimension 2 and vector-valued problems.The vector-valued tests include comparisons with adjoint eigencouples.
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