Parallel Fast Direct Error-Controlled Scattering Solutions via an $\mathcal{H}$-Matrix-Accelerated Locally Corrected Nyström Method for the Combined Field Integral Equation

A parallel, fast, direct, high-order solution of the Locally Corrected Nyström (LCN) method discretization of the combined field integral equation (CFIE) is presented for solving scattering problems involving perfect electric conductors (PECs) of arbitrary shape. The discrete LCN operator is represented using the hierarchical matrix (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{H}$</tex>-matrix) framework to accelerate the filling and solving processes, while consuming a fraction of the memory conventionally required for the dense system. The solver is validated for an exact parametrization of the surface of a sphere with quadrilateral patches (i.e., a mapped sphere). The accuracy is also studied for high-order solutions of arbitrary shapes, demonstrating a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{O}(h^{p})$</tex> convergence. Results from this direct solver are indicative that the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{H}$</tex>-matrix-accelerated LCN method will provide a flexible error-controllable preconditioner for general scattering problems.