A fast direct solver for elliptic problems on general meshes in 2D

An O(N) direct solver for integral equations on the plane

“…These solvers do not have optimal complexity (they typically require O(N 3/2 ) for the factorization stage in 2D, and O(N 2 ) in 3D), but are nevertheless popular (especially in 2D) due in part to their robustness, and in part to the unrivaled speed that can be attained for problems involving multiple right hand sides. Very recently, it has been demonstrated that by exploiting structured matrix algebra (such as, e.g., H-matrices, or HSS matrices), to manipulate the dense matrices that arise due to fill-in, direct solvers of linear or close to linear complexity can be constructed [17,28,30,35,40]. The direct solver described in this paper is conceptually similar to these algorithms in that they all rely on hierarchical domain decompositions, and efficient representations of operators that live on the interfaces between sub-domains.…”

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

“…Closely related to the concept of H-matrices and its arithmetic are "hierarchically semiseparable matrices", [Xia13,XCGL09,LGWX12] and the idea of "recursive skeletonization", [HG12,GGMR09,HY13a]; for discretizations of PDEs, we mention [HY13a,GM13,SY12,Mar09], and particular applications to boundary integral equations are [MR05,CMZ13,HY13b]. These factorization algorithms aim to exploit that some off-diagonal blocks of certain Schur complements are low rank.…”

A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method

“…Exploiting the internal structure in these matrices, the complexity of the build stage can be reduced from OðN 1:5 Þ to OðNÞ. This acceleration is analogous to what was done for classical nested dissection for finite-element and finite-difference matrices in [9,21,30,35]. Note that while the acceleration of the solve phase is trivial, it takes some work to exploit the more complicated structure in V s and W s .…”