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

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.