Polynomial subspace operator preconditioning in 2-dimensional boundary element methods

A less-known advantage of boundary element methods when compared to volume discretization techniques is that single-layer and hyper-singular operators have opposite-order mapping properties, which allows their mutual preconditioning. State-of-the-art methods rely on discretizations over a barycentric refinement, which is especially in 3 spatial dimensions (3d) computationally expensive. In this paper we propose and analyze a novel construction in 2d that exploits partition of unity of continuous piecewise quadratic basis functions to discretize the hyper-singular operator on the original grid. By elementary calculus we prove that the resulting hyper-singular operator forms an optimal preconditioner to the single-layer operator discretized by piecewise constants. The efficiency of our approach is documented by numerical experiments performed on a GPU.