arXiv subjects
Publications and source records attributed to Jose Pablo Lucero Lorca.
We introduce a new direct multilevel method for solving arbitrary complex square linear systems that uses a regular smoother and an arbitrary but equal number of pre- and post-smoothings. Through careful analysis of the error propagation operator, we cluster the spectrum of this operator. This allows us to write a direct K-cycle version of the method.
Multi-level preconditioners for Discontinuous Galerkin (DG) discretizations are widely used to solve elliptic equations, and a main ingredient of such solvers is the interpolation operator to transfer information from the coarse to the fine grid. Classical interpolation operators give continuous interpolated values, but since DG solutions are naturally discontinuous, one might wonder if one should not use discontinuous interpolation operators for DG discretizations. We consider a discontinuous interpolation operator with a parameter that controls the discontinuity, and determine the optimal choice for the discontinuity, leading to the fastest solver for a specific 1D symmetric interior penalty DG discretization model problem. We show in addition that our optimization delivers a perfectly clustered spectrum with a high geometric multiplicity, which is very advantageous for a Krylov solver using the method as its preconditioner. Finally, we show the applicability of the optimal choice to higher dimensions.
We present robust and highly parallel multilevel non-overlapping Schwarz preconditioners, to solve an interior penalty discontinuous Galerkin finite element discretization of a system of steady state, singularly perturbed reaction-diffusion equations with a singular reaction operator, using a GMRES solver. We provide proofs of convergence for the two-level setting and the multigrid V-cycle as well as numerical results for a wide range of regimes.