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A Spectral LOD Method for Multiscale Problems with High Contrast

Susanne C. Brenner, José C. Garay, Li-yeng Sung

TL;DR

This work develops a spectral coarse-space localized-orthogonal-decomposition (LOD) finite element method for diffusion problems with rough, high-contrast coefficients. The approach builds local auxiliary spaces from eigenfunctions on coarse elements and uses a nonconforming kernel projection to form an ideal multiscale space, subsequently localized via a few conjugate gradient iterations to yield a fast online solve. The authors prove explicit energy- and $L^2$-error estimates that depend on coarse mesh size $H$, the number of CG steps $k$, and the contrast through computable constants, while numerical experiments show near-FE performance and contrast-independence of errors. The method offers a practical bridge between coarse and fine scales, with offline basis construction and online inexpensive solves, applicable to general 2D/3D domains with simplicial meshes and $P_1$ elements.

Abstract

We present a multiscale finite element method for a diffusion problem with rough and high contrast coefficients. The construction of the multiscale finite element space is based on the localized orthogonal decomposition methodology and it involves solutions of local finite element eigenvalue problems. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for the homogeneous Dirichlet boundary value problem for the Poisson equation on smooth or convex domains.} Simple explicit error estimates are established under conditions that can be verified from the outputs of the computation.

A Spectral LOD Method for Multiscale Problems with High Contrast

TL;DR

This work develops a spectral coarse-space localized-orthogonal-decomposition (LOD) finite element method for diffusion problems with rough, high-contrast coefficients. The approach builds local auxiliary spaces from eigenfunctions on coarse elements and uses a nonconforming kernel projection to form an ideal multiscale space, subsequently localized via a few conjugate gradient iterations to yield a fast online solve. The authors prove explicit energy- and -error estimates that depend on coarse mesh size , the number of CG steps , and the contrast through computable constants, while numerical experiments show near-FE performance and contrast-independence of errors. The method offers a practical bridge between coarse and fine scales, with offline basis construction and online inexpensive solves, applicable to general 2D/3D domains with simplicial meshes and elements.

Abstract

We present a multiscale finite element method for a diffusion problem with rough and high contrast coefficients. The construction of the multiscale finite element space is based on the localized orthogonal decomposition methodology and it involves solutions of local finite element eigenvalue problems. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for the homogeneous Dirichlet boundary value problem for the Poisson equation on smooth or convex domains.} Simple explicit error estimates are established under conditions that can be verified from the outputs of the computation.

Paper Structure

This paper contains 20 sections, 6 theorems, 94 equations, 9 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Auxiliary Space $V_{aux}$
  3. The operator $\Pi_{aux}$
  4. The ideal (or global) multiscale finite element method
  5. A Basis for $\mathrm{Ker}\,\Pi_{aux}$
  6. A Basis for ${V}_{aux}^{\hbox{$\rm {ms},h$}}$
  7. A localized multiscale finite element method
  8. The Modified Gram-Schmidt Algorithm
  9. Construction of the Special Basis Functions of $\mathrm{Ker}\,\Pi_{aux}$
  10. Basis Vectors/Functions Associated with the Interior of an Element of $\mathcal{T}_H$
  11. Basis Vectors/Functions Associated with the Interior of an Edge of $\mathcal{T}_H$
  12. Basis Vectors/Functions Associated with an Interior Vertex of $\mathcal{T}_H$
  13. A Matrix Form of the Correction Equation \ref{['eq:Corrector']}
  14. A Localized Multiscale Finite Element Method
  15. Analysis of the Localized Multiscale Finite Element Method
  16. ...and 5 more sections

Key Result

Lemma 3.1

We have and

Figures (9)

  • Figure 1.1: A coarse mesh $\mathcal{T}_H$ with 16 elements (left) and a fine mesh $\mathcal{T}_h$ (right).
  • Figure 4.1: The dual nodes $\hat{p}_1^{(i)}$, $\hat{p}_2^{(i)}$ and $\hat{p}_3^{(i)}$ in $K_i$ represented by the red dots, where $L_i=3$.
  • Figure 4.2: The three possibilities for $p\in \mathcal{V}_h\setminus\mathcal{N}_{aux}$
  • Figure 7.1: Four high contrast channels.
  • Figure 7.2: Relative energy errors (left) and relative $L^2$ errors (right) for Example \ref{['example:PS']}.
  • ...and 4 more figures

Theorems & Definitions (37)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • proof
  • ...and 27 more