Application of the Sparse Cardinal Sine Decomposition to 3D Stokes Flows

Abstract

In boundary element method (BEM), one encounters linear system with a dense and non-symmetric square matrix which might be so large that inverting the linear system is too prohibitive in terms of cpu time and/or memory. Each usual powerful treatment (Fast Multipole Method, H-matrices) developed to deal with this issue is optimized to efficiently perform matrix vector products. This work presents a new technique to adequately and quickly handle such products: the Sparse Cardinal Sine Decomposition. This approach, recently pioneered for the Laplace and Helmholtz equations, rests on the decomposition of each encountered kernel as series of radial Cardinal Sine functions. Here, we achieve this decomposition for the Stokes problem and implement it in MyBEM, a new fast solver for multi-physical BEM. The reported computational examples permit us to compare the advocated method against a usual BEM in terms of both accuracy and convergence.

FEM-BEM Coupling for Electromagnetism with the Sparse Cardinal Sine Decomposition,

“…The equations for each of the electric and magnetic fields are obtained by taking the curl of both equations in (3) and substitute back with the other one. One obtains…”

“…The heart of the BEM lies in the integral representation of an electromagnetic field (E, H) that solves (3) in Ω + and inside Ω − = R 3 \ Ω + , and which satisfies furthermore the Silver-Muller condition (4). One has, calling Γ = ∂Ω + the surface of the object (see e.g.…”

“…using the first equation in (3), and where we have denoted by J tr = H tr × n the electric interior current. Note that the boundary integral on the right-hand side of the equation is taken only on Γ d .…”

“…More recently a new compression technique, the Sparse Cardinal Sine Decomposition (SCSD), was developed for the same goal [1,4], and applied to point to point gravitational interactions or acoustic scattering [2]. A more recent application to the Stokes problem must also be pointed out [3].…”

FEM and BEM simulations with the Gypsilab framework

“…the scatterer), which lowers the dimension of the object that needs to be discretized. Nevertheless, due to computer limitations, those methods may require a special compression technique such as the Fast Multipole Method (FMM) [18,15], the H-matrix storage [20] or the more recent Sparse Cardinal Sine Decomposition [2,3,4], in order to be applied to relevant problems. In terms of software packages available for the simulation with this kind of methods, and probably due to the technicality sketched above, the list is much shorter than before.…”