Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust Multiscale Methods for Helmholtz equations in high contrast heterogeneous media

Xingguang Jin, Changqing Ye, Eric T. Chung

TL;DR

The paper addresses solving the Helmholtz equation in highly heterogeneous media with large wavenumber $k$, where standard FEM struggles due to pollution and high-contrast coefficients. It develops the Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) to construct multiscale trial and test spaces via local spectral problems and oversampling-based energy minimization, enabling a stable, low-dimensional Petrov–Galerkin formulation. The authors prove inf-sup stability and an a priori error estimate under a resolution condition, demonstrate exponential decay of basis functions, and validate the method through three numerical experiments showing reduced pollution and accurate coarse-mesh solutions even for high-contrast coefficients. The work advances scalable multiscale modeling for wave propagation in complex media and suggests practical pathways for robust simulations at higher frequencies and in perforated domains.

Abstract

In this paper, we provide the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to solve Helmholtz equations in heterogeneous medium. This novel multiscale method is specifically designed to overcome problems related to pollution effect, high-contrast coefficients, and the loss of hermiticity of operators. We establish the inf-sup stability and give an a priori error estimate for this method under a number of established assumptions and resolution conditions. The theoretical results are validated by a set of numerical tests, which further show that the multiscale technique can effectively capture pertinent physical phenomena.

Robust Multiscale Methods for Helmholtz equations in high contrast heterogeneous media

TL;DR

The paper addresses solving the Helmholtz equation in highly heterogeneous media with large wavenumber , where standard FEM struggles due to pollution and high-contrast coefficients. It develops the Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) to construct multiscale trial and test spaces via local spectral problems and oversampling-based energy minimization, enabling a stable, low-dimensional Petrov–Galerkin formulation. The authors prove inf-sup stability and an a priori error estimate under a resolution condition, demonstrate exponential decay of basis functions, and validate the method through three numerical experiments showing reduced pollution and accurate coarse-mesh solutions even for high-contrast coefficients. The work advances scalable multiscale modeling for wave propagation in complex media and suggests practical pathways for robust simulations at higher frequencies and in perforated domains.

Abstract

In this paper, we provide the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to solve Helmholtz equations in heterogeneous medium. This novel multiscale method is specifically designed to overcome problems related to pollution effect, high-contrast coefficients, and the loss of hermiticity of operators. We establish the inf-sup stability and give an a priori error estimate for this method under a number of established assumptions and resolution conditions. The theoretical results are validated by a set of numerical tests, which further show that the multiscale technique can effectively capture pertinent physical phenomena.
Paper Structure (11 sections, 12 theorems, 144 equations, 8 figures, 4 tables)

This paper contains 11 sections, 12 theorems, 144 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The construction of the CEM-GMsFEM basis function
  4. Multiscale basis functions
  5. Analysis
  6. Numerical experiments
  7. Model problem 1
  8. Model problem 2
  9. Model problem 3
  10. Conclusion
  11. Acknowledgment

Key Result

Lemma 3.1

In each $K_j\in\mathcal{T}_H$, for all $v\in H^1(K_j)$, where $\Lambda=\max_{1\leq j\leq N}\lambda_j^{l_j+1}$, and

Figures (8)

  • Figure 1: An illustration of the two-scale mesh, a fine element $h$, a coarse element $K_j$ and its oversampling coarse element $K_j^m$ with the oversmapling layer $m=1$.
  • Figure 2: Numerical results for the Helmholtz equations with homogeneous coefficients where the relative errors of the proposed method with different numbers of oversampling layers $m$ and the $Q1$ FEM are calculated with respect to the coarse mesh size $H$. Subplots (a) and (b) which the relative errors are measured in the $L_2$ norm and energy norm, respectively.
  • Figure 3: Model Problem 2
  • Figure 4: Subplots (a) and (b) show the relative errors of the proposed method for the pointwise isotropic coefficients with different numbers of oversampling layers $m$ with respect to the coarse mesh size $H$, but measured in different norms.
  • Figure 5: Solutions by using $H=1/40$ and oversampling layer equals to 3. Subplot (a), (b),(c) represent reference solution , CEM-GMsFEM solution and Difference of these two solutions, respectively .
  • ...and 3 more figures

Theorems & Definitions (25)

  • Lemma 3.1
  • Remark
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 15 more