Wideband Directional $\mathcal{H}^2$-Matrix Compression for the Electric Field Integral Equation with Geometry-Adaptive Cluster Trees

RESUMO

We present an efficient wideband construction of directional $\mathcal{H}^2$-matrices for the electrical field integral equation that, in contrast to existing constructions, supports not only box trees but also geometry-adaptive cluster trees. To accommodate geometry-adaptive cluster trees, we determine the number of directions from the electrical size of each cluster (instead of the level of a box tree), construct the directions using Spherical-Fibonacci points, and establish a hierarchy between the direction sets of a cluster and its children through an angular nearest-neighbor mapping. We construct the nested directional representation using the incomplete adaptive cross approximation, for which we introduce a robustified tree-mimicry pivoting strategy that prevents premature convergence for block-structured matrices arising from certain geometries and meshes. Numerical results demonstrate that the proposed approach achieves the desired accuracy, requires no more storage than the octree-based construction and substantially less when the geometry or discretization is poorly matched to octree clustering, and exhibits the expected $\mathcal{O}(N\log N)$ scaling for high-frequency problems.