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Stable localized orthogonal decomposition in Raviart-Thomas spaces

Patrick Henning, Hao Li, Timo Sprekeler

TL;DR

The work develops a stable localized orthogonal decomposition (LOD) in Raviart-Thomas spaces for the mixed finite element approximation of a divergence-form elliptic equation with highly heterogeneous diffusion $\mathbf{A}$ under Neumann boundary conditions. By constructing a stable quasi-interpolation $\pi_H$ and an ideal correction operator, it forms an ideal multiscale space $V_H^{k,\mathrm{ms}}$ that preserves the coarse problem size while capturing fine-scale effects; a localized version with exponential decay yields a practical, robust method in dimensions $2$ and $3$ without pollution terms. Theoretical results establish energy-norm and pressure-error convergence that are independent of $\mathbf{A}$’s oscillations, with explicit decay rates and conditions under which optimal rates are achieved when $f$ is sufficiently smooth. Numerical experiments on a checkerboard and the SPE10-85 data confirm the predicted convergence and localization efficiency, illustrating the method’s capacity to reproduce fine-scale flux patterns on coarse meshes. Overall, the approach provides a pollution-free, scalable multiscale framework for mixed finite element discretizations of heterogeneous elliptic problems with strong practical impact in subsurface flow and related applications.

Abstract

This work proposes a computational multiscale method for the mixed formulation of a second-order linear elliptic equation subject to a homogeneous Neumann boundary condition, based on a stable localized orthogonal decomposition (LOD) in Raviart-Thomas finite element spaces. In the spirit of numerical homogenization, the construction provides low-dimensional coarse approximation spaces that incorporate fine-scale information from the heterogeneous coefficients by solving local patch problems on a fine mesh. The resulting numerical scheme is accompanied by a rigorous error analysis, and it is applicable beyond periodicity and scale-separation in spatial dimensions two and three. In particular, this novel realization circumvents the presence of pollution terms observed in a previous LOD construction for elliptic problems in mixed formulation. Finally, various numerical experiments are provided that demonstrate the performance of the method.

Stable localized orthogonal decomposition in Raviart-Thomas spaces

TL;DR

The work develops a stable localized orthogonal decomposition (LOD) in Raviart-Thomas spaces for the mixed finite element approximation of a divergence-form elliptic equation with highly heterogeneous diffusion under Neumann boundary conditions. By constructing a stable quasi-interpolation and an ideal correction operator, it forms an ideal multiscale space that preserves the coarse problem size while capturing fine-scale effects; a localized version with exponential decay yields a practical, robust method in dimensions and without pollution terms. Theoretical results establish energy-norm and pressure-error convergence that are independent of ’s oscillations, with explicit decay rates and conditions under which optimal rates are achieved when is sufficiently smooth. Numerical experiments on a checkerboard and the SPE10-85 data confirm the predicted convergence and localization efficiency, illustrating the method’s capacity to reproduce fine-scale flux patterns on coarse meshes. Overall, the approach provides a pollution-free, scalable multiscale framework for mixed finite element discretizations of heterogeneous elliptic problems with strong practical impact in subsurface flow and related applications.

Abstract

This work proposes a computational multiscale method for the mixed formulation of a second-order linear elliptic equation subject to a homogeneous Neumann boundary condition, based on a stable localized orthogonal decomposition (LOD) in Raviart-Thomas finite element spaces. In the spirit of numerical homogenization, the construction provides low-dimensional coarse approximation spaces that incorporate fine-scale information from the heterogeneous coefficients by solving local patch problems on a fine mesh. The resulting numerical scheme is accompanied by a rigorous error analysis, and it is applicable beyond periodicity and scale-separation in spatial dimensions two and three. In particular, this novel realization circumvents the presence of pollution terms observed in a previous LOD construction for elliptic problems in mixed formulation. Finally, various numerical experiments are provided that demonstrate the performance of the method.

Paper Structure

This paper contains 20 sections, 12 theorems, 181 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Continuous problem
  5. Discretization with the Raviart--Thomas element
  6. Stable quasi-interpolation
  7. Construction of a stable quasi-interpolation operator
  8. Implementation of a stable quasi-interpolation operator
  9. Ideal multiscale method
  10. Correction operator and ideal multiscale space
  11. Error estimate for ideal multiscale problem
  12. Localized multiscale method
  13. Localized multiscale approximation and main result
  14. Exponential decay of element correctors for $d=3$
  15. Proof of Theorem \ref{['main-thm']}
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $\mathcal{V} \subset \mathbf{H}_0(\operatorname{div},\Omega)$ and $\mathcal{Q} \subset L^2_0(\Omega)$ denote respective closed subspaces. Introducing $\mathcal{V}_{\operatorname{div}\space0}:=\{\mathbf{v} \in \mathcal{V}: b(\mathbf{v}, q)=0 \;\;\mathrm{for}\; \mathrm{all}\; q \in \mathcal{Q}\}, Then, there exists a unique $(\mathbf{u}_{\ast}, p_{\ast}) \in \mathcal{V} \times \mathcal{Q}$ such

Figures (4)

  • Figure 1: Plots of the coefficient and the magnitude of the reference flux for Experiment 1.
  • Figure 2: Convergence Test. Relative errors for the LOD approximations of Experiment 1.
  • Figure 3: Plots of the coefficient and the magnitude of the reference flux for Experiment 2.
  • Figure 4: SPE10-85 test. Figure \ref{['spe10-ref-sol']} shows the magnitude of the reference flux solution. Figures \ref{['spe10-m2-l3', 'spe10-m3-l4', 'spe10-m4-l5']} display the magnitudes of the multiscale flux solutions for $m=2,3,4$ with $\ell=m+1$, respectively.

Theorems & Definitions (30)

  • Lemma 2.1: Well-posedness of the mixed formulation
  • Remark 2.2
  • Definition 2.5
  • Lemma 2.6
  • Remark 2.7: Local interpolation error bound for $\pi_H$
  • Definition 3.1: Ideal correction operators
  • Corollary 3.1.1
  • proof
  • Definition 3.2: Ideal multiscale approximation
  • Lemma 3.3: Well-posedness of the ideal multiscale problem
  • ...and 20 more