Searcharxiv⌕ Search

arXiv subjects

Filipe Cumaru

Publications and source records attributed to Filipe Cumaru.

2 recordsLinked to original sources
arXiv2025

Two-level additive Schwarz preconditioners for reduced integration methods

Incompressible fluid flow problems appear frequently in different applications. The discretization of such problems may result in large and ill-conditioned systems of linear equations. We consider the case of the Stokes equations discretized using a reduced integration method which approximates the incompressibility constraint by a penalty term thus allowing the problem to be solved only in terms of the velocity unknowns. We investigate the numerical scalability of a two-level overlapping additive Schwarz method with a reduced dimension generalized Dryja-Smith-Widlund (RGDSW) coarse space. In addition, we discuss the parallel implementation of the examples using the Fast and Robust Overlapping Schwarz (FROSch) package for additive Schwarz preconditioners and the NGSolve library, which implements multiple finite element space formulations.

math.NA↗
arXiv2025

Sharpened PCG Iteration Bound for High-Contrast Heterogeneous Scalar Elliptic PDEs

A new iteration bound for the preconditioned conjugate gradient (PCG) method is presented that more accurately captures convergence for systems with clustered eigenspectra, where the classical condition number-based bound is too pessimistic. By using the edge eigenvalues of each cluster in the spectral distribution, the bound is shown to be orders of magnitude sharper than the classical bound for certain examples. Its effectiveness is demonstrated on a high-contrast elliptic PDE preconditioned with a two-level overlapping Schwarz preconditioner, where the performance of different (algebraic) coarse spaces is successfully distinguished. A key contribution of this work is the observation that, for certain high-contrast problems, simpler coarse spaces can be made competitive in terms of PCG convergence. Conversely, more complex preconditioners are not always required. Finally, it is shown that the bound can be estimated effectively from Ritz values computed during early PCG iterations.

math.NA↗