Existence of $\mathscr{H}$-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator

We consider discretizations of the hyper-singular integral operator on closed (connected) surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block rank. We cover in particular the data-sparse format of $\mathscr{H}$-matrices. We show this approximability result for two types of discretizations. The first one is a saddle-point formulation, which incorporates the constraint of vanishing mean of the solution. The second discretization is based on a stabilized hyper-singular operator, which leads to symmetric positive-definite matrices. In this latter setting we also show that the hierarchical Cholesky factorization can be approximated at an exponential rate in the block rank.