Sparse Operator-Adapted Wavelet Decomposition Using Polygonal Elements for Multiscale FEM Problems

TL;DR

The paper introduces a sparse operator-adapted wavelet finite element framework built on a coarsening-based polygonal mesh hierarchy to tackle multiscale EM problems. By achieving full scale decoupling, it allows independent, additive refinement that preserves near-linear computational complexity and reduces memory usage. The approach uses Whitney edge elements on convex polygons, precomputed operator-agnostic matrices, and efficient QR-based null-space computations to construct operator-adapted bases. Numerical experiments on wedge scattering and MPSi slabs demonstrate accurate near- and far-field results with substantial memory savings and scalable performance. This method offers a practical, adaptive alternative for multiscale FEM analyses in complex geometries.

Abstract

We develop a sparse multiscale operator-adapted wavelet decomposition-based finite element method (FEM) on unstructured polygonal mesh hierarchies obtained via a coarsening procedure. Our approach decouples different resolution levels, allowing each scale to be solved independently and added to the entire solution without the need to recompute coarser levels. At the finest level, the meshes consist of triangular elements which are geometrically coarsened at each step to form convex polygonal elements. Smooth field regions of the domain are solved with fewer, larger, polygonal elements, whereas high-gradient regions are represented by smaller elements, thereby improving memory efficiency through adaptivity. The proposed algorithm computes solutions via sequences of hierarchical sparse linear-algebra operations with nearly linear computational complexity.

Figures (11)

• Figure 1: Illustration of a three-level mesh hierarchy produced by the proposed coarsening algorithm. From left to right: $\mathcal{M}^{3}$ (input finest-level uniform mesh) with 38 triangular elements; $\mathcal{M}^{2}$ (obtained by coarsening the finest mesh once) with 18 quadrilaterals and 2 triangles; and $\mathcal{M}^{1}$ (coarsest-level mesh) with 4 hexagons, 1 pentagon, 9 quadrilaterals, and 1 triangle.
• Figure 2: Mesh coarsening procedure.
• Figure 3: Illustration of a mesh hierarchy produced by the proposed coarsening algorithm for the wedge scattering problem. From bottom to top: $\mathcal{M}^4$ (input finest-level uniform triangular mesh), $\mathcal{M}^3$ (obtained by coarsening the finest mesh once), and $\mathcal{M}^1$ (coarsest).
• Figure 4: Illustration of a mesh hierarchy produced by the proposed coarsening algorithm for the wedge scattering problem. From bottom to top: $\mathcal{M}^4$ (input finest-level adaptive triangular mesh), $\mathcal{M}^3$ (obtained by coarsening the finest mesh once), and $\mathcal{M}^1$ (coarsest).
• Figure 5: Geometry for the PEC wedge–scattering example. A perfectly conducting wedge with interior (opening) angle $2\alpha$ is illuminated by a plane wave incident at angle $\phi'$. The observation angle $\phi$ is measured counterclockwise from the wedge bisector ($0^\circ$); the wedge faces lie at $\phi=\pm\alpha$ degrees. Axes indicate the 2D $x$–$y$ plane.