Multigrid Finite Element Methods for Electromagnetic Field Modeling

Abstract

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Scattered Field Computation With an Extended Feti-Dpem2 Method

“…A lot of efforts have been devoted to the development of iterative methods such as multi-grid methods [13], but the problem is still open (for a review see for example [14]). Among the different schemes proposed in order to solve large scale models and preserve the versatility of the method, one can cite the Domain Decomposition Method (DDM) and its different evolutions [15][16][17][18].…”

Faster 3D finite element time domain – Floquet absorbing boundary condition modelling using recursive convolution and vector fitting

“…more accurate results are expected if the mesh is refined and higher order elements are employed [21,22]. In [23] (page 392) a similar shift is obtained between two sets of results when modeling a similar structure. The first set of results was obtained using the 6 d.o.f.…”

Maximally Orthogonal High-Order Basis Functions Have a Well-Conditioned Gram Matrix

“…However, the crux is that many techniques exist for removing the ill-conditioning originating from the equation. Examples are multigrid, multiresolution or operator preconditioning [8], [9], [11]. Though these techniques are very effective, they cannot remove the illconditioning that comes from the linear dependence between finite elements if the Gram matrix bears no resemblance to any physical operator relevant to the problem being solved.…”

“…This in turn leads to ill-conditioned matrices resulting from the discretization of partial differential equations or integral equations. While the ill-conditioning due to the pseudo-differential nature of the equation (be it partial differential or integral) can usually be mitigated using multigrid, multiresolution or Calderón preconditioners [8]- [11], this is usually not as easily done for the ill-conditioning due to linear dependence between the different basis functions. It is therefore of great practical importance to find high-order subdomain finite element spaces that show sufficient linear independence between the various finite elements.…”