In the setting of 3D local mesh refinement, we present the theoretical construction and the implementation aspects of the Bramble-Pasciak-Xu (BPX) preconditioner. The refinement under consideration is the 3D local redgreen refinement procedure introduced by Bornemann-Erdmann-Kornhuber (BEK). We outline how to construct the theoretical optimality of the BPX preconditioner in the setting of ellip...

The original TPABLO algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries. This block diagonal can then be used as a preconditioner. We propose and analyze three extensions of this approach: we incorporate a nonsymmetric permutation to obtain a la...

This paper introduces the sparsifying preconditioner for the pseudospectral approximation of highly indefinite systems on periodic structures, which include the frequency-domain response problems of the Helmholtz equation and the Schrödinger equation as examples. This approach transforms the dense system of the pseudospectral discretization approximately into a sparse system via an equivalent i...

When one approximates elliptic equations by the spectral collocation method on the Chebyshev-Gauss-Lobatto (CGL) grid, the resulting coefficient matrix is dense and illconditioned. It is known that a good preconditioner, in the sense that the preconditioned system becomes well conditioned, can be constructed with finite difference on the CGL grid. However, there is a lack of an efficient solver...

This paper presents a preconditioner for non-overlapping Schwarz methods applied to the Helmholtz problem. Starting from a simple analytic example, we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain. The preconditioner works by propagating information globally by concurrently sweeping in both directio...

Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higher-fidelity simulations remains. Present day CFD simulations are limited by the lack of an effi...

We develop a preconditioner for systems arising from space-time finite element discretizations of parabolic equations. The preconditioner is based on a transformation of the coupled system into block diagonal form and an efficient solution strategy for the arising 2 × 2 blocks. The suggested strategy makes use of an inexact factorization of the Schur complement of these blocks, for which unifor...

We present a domain decomposition method suitable for the drift-diiusion equations. The new scheme is applied to the linear systems resulting from Gummel's method and corresponds to a preconditioned conjugate gradient technique for the Schur complement of the linear systems. In designing the preconditioner two problems are addressed : anisotropic phenomena and large scale variations in the magn...

In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. The proposed preconditioner is extracted form a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some proper...

For solving large dense complex linear systems that arise in electromagnetic calculations, we perform experiments using a general purpose spectral low rank update preconditioner in the context of the GMRES method preconditioned by an approximate inverse preconditioner. The goal of the spectral preconditioner is to improve the convergence properties by shifting by one the smallest eigenvalues of...