Hierarchical Finite-Element Analysis of Multiscale Electromagnetic Problems via Sparse Operator-Adapted Wavelet Decomposition

TL;DR

This work tackles multiscale electromagnetic analysis with a finite-element formulation augmented by operator-adapted wavelet decompositions that decouple resolution levels, enabling independent scale solves and avoiding costly remeshing of coarser levels. The authors develop a hierarchical, sparse, matrix-based framework that constructs operator-adapted bases on unstructured meshes, computes refinement structures via local, linear-algebraic procedures, and solves level-wise systems with Krylov methods and SPAI-based preconditioning. They demonstrate near-linear computational performance ($\approx O(N)$) on 2D problems such as L- and U-shaped waveguides and a leaky MPSi waveguide, achieving errors comparable to conventional finest-level FE solutions while substantially reducing inter-scale coupling and memory usage. The approach offers a scalable path for high-fidelity multiscale EM simulations and can be extended to 3D and hp/adaptive polygonal hierarchies to further improve efficiency and applicability in complex geometries.

Abstract

In this paper, we present a finite element method (FEM) framework enhanced by an operator-adapted wavelet decomposition algorithm designed for the efficient analysis of multiscale electromagnetic problems. Usual adaptive FEM approaches, while capable of achieving the desired accuracy without requiring a complete re-meshing of the computational domain, inherently couple different resolution levels. This coupling requires recomputation of coarser-level solutions whenever finer details are added to improve accuracy, resulting in substantial computational overhead. Our proposed method addresses this issue by decoupling resolution levels. This feature enables independent computations at each scale that can be incorporated into the solutions to improve accuracy whenever needed, without requiring re-computation of coarser-level solutions. The main algorithm is hierarchical, constructing solutions from finest to coarser levels through a series of sparse matrix-vector multiplications. Due to its sparse nature, the overall computational complexity of the algorithm is nearly linear. Moreover, Krylov subspace iterative solvers are employed to solve the final linear equations, with ILU preconditioners that enhance solver convergence and maintain overall computational efficiency. The numerical experiments presented in this article verify the high precision and nearly linear computational complexity of the proposed algorithm.

Figures (9)

• Figure 1: Mesh hierarchy for different resolution levels. $\mathcal{M}^1$: coarsest-level mesh, $\mathcal{M}^2$: second-level mesh after one subdivision, and $\mathcal{M}^5$: fifth-level mesh after four subdivisions.
• Figure 2: L-shaped waveguide discontinuity numerical example analyzed with the sparse operator-adapted wavelet decomposition-based FEM. (a) Problem configuration. (b) Computed electric-field magnitude of the two-level solution, $\mathbbm{u}=\mathbbm{s}^{\text{coarse}}+\mathbbm{s}^{\text{detail,1}}$. (c) Second detail-level contribution, $\mathbbm{s}^{\text{detail,2}}$. (d) Third detail-level contribution, $\mathbbm{s}^{\text{detail,3}}$.
• Figure 3: U-shaped waveguide discontinuity numerical example analyzed with the sparse operator-adapted wavelet decomposition-based FEM. (a) Problem configuration. (b) Computed electric-field magnitude of the two-level solution, $\mathbbm{u}=\mathbbm{s}^{\text{coarse}}+\mathbbm{s}^{\text{detail,1}}$. (c) Second detail-level contribution, $\mathbbm{s}^{\text{detail,2}}$. (d) Third detail-level contribution, $\mathbbm{s}^{\text{detail,3}}$.
• Figure 4: Electric field distribution in the leaky MPSi waveguide, computed with a five-level multiscale method. The total field is reconstructed as $\mathbbm{E}=\mathbbm{E}^{\text{coarse}}+\sum_{k=1}^{4}\mathbbm{E}^{\text{detail},\,k}$. The two in-phase point sources are marked by black dots, and the white dashed rectangle highlights the region whose multilevel meshes are shown in Fig. \ref{['Fig-mesh']}. Axes are in units of the wavelength $\lambda$.
• Figure 5: Zoomed view of the meshes corresponding to the region indicated by the white dashed rectangle in Fig. 4, for the five-level mesh hierarchy: (a) coarsest-level mesh and (b) third-level mesh.