An alternative extended linear system for boundary value problems on locally perturbed geometries

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

“…In the context of integral equations, some of them include the Hierarchically Block Separable (HBS) solver [9], the Hierarchically Semi-Separable (HSS) [10,11], the Hierarchical Interpolative Factorization (HIF) [12] and the H or H 2 -matrix methods [13]. The techniques developed in [7,8] for the extended linear system (ELS), designed for problems with locally perturbed geometries, can be coupled with any of the above direct solver approaches. In this work, we employ a HBS solver for the ELS.…”

“…Contributions. Motivated by the applications mentioned above, we apply the solution technique given in [7,8] to Stokes flow problems defined on complex geometries, some of which are adapted from real application geometry data. The linear system associated with discretization of an integral equation for Stokes flow has a physical null space corresponding to the pressure being unique up to a constant which can be corrected via an analytic technique.…”

“…This is even more cumbersome when local refinement is added to the original discretization. To accommodate this challenge, we modify the solution technique, particularly the part that constructs a low-rank approximation matrix to capture the changes in the discretization, referred to as the update matrix in [7,8], to maintain the stability of the solver. However, even with this modification, for some problems the direct solver in [7,8] may fail to reach the desired tolerance.…”

“…To accommodate this challenge, we modify the solution technique, particularly the part that constructs a low-rank approximation matrix to capture the changes in the discretization, referred to as the update matrix in [7,8], to maintain the stability of the solver. However, even with this modification, for some problems the direct solver in [7,8] may fail to reach the desired tolerance. This is because that fast direct solvers such as the HBS solver may not reach the desired tolerance for ill-conditioned problems and that the accuracy of the local perturbation solver is restricted by the accuracy of the fast direct solver precomputed for the original discretization.…”

“…We present a fast algorithm that avoids re-building the inverse operator from scratch by precomputing an inverse operator corresponding to a reference mesh and rapidly updating it whenever the boundary discretization is locally refined (or coarsened). This work is as an extension of Zhang-Gillman [7,8], where Laplace BIEs on locally-perturbed geometries were considered. The central idea is that the discretized BIE on the walls can be written as an extended version of the linear system for the original geometry and a fast direct solver on the original geometry can be reused to reduce the computational burden of solving the problems on the refined discretization.…”