BUSCADOR RECOLECTA

Isogeometric Analysis (IGA) is a computational technique for the numerical approximation of partial differential equations (PDEs). This technique is based on the use of spline-type basis functions, that are able to hold a global smoothness and allow to exactly capture a wide set of common geometries. The current rise of this approach has encouraged the search of fast solvers for isogeometric discretizations and nowadays this topic is receiving a lot of attention. In this framework, a desired property of the solvers is the robustness with respect to both the polynomial degree p and the mesh size h. For this task, in this paper we propose a two-level method such that a discretization of order p is considered in the first level whereas the second level consists of a linear or quadratic discretization. On the first level, we suggest to apply one single iteration of a multiplicative Schwarz method. The choice of the block-size of such an iteration depends on the spline degree p, and is supported by a local Fourier analysis (LFA). At the second level one is free to apply any given strategy to solve the problem exactly. However, it is also possible to get an approximation of the solution at this level by using an h-multigrid method. The resulting solver is efficient and robust with respect to the spline degree p. Finally, some numerical experiments are given in order to demonstrate the good performance of the proposed solver.

Two‐phase flow problems in porous media can be found in several areas, such as Geomechanics, Hydrogeology, Engineering and Biomedicine, for example. Typically, these processes are mathematically modeled by a highly nonlinear system of coupled partial differential equations. The nonlinearity of the system makes the design and implementation of robust numerical solvers a challenging task. In this work we consider the flow of two immiscible and incompressible fluids within a non‐deformable porous medium. A mixed pressure‐saturation formulation is adopted, allowing the transition from the unsaturated to saturated zones and maintaining numerical mass conservation. A cell‐centered finite volume method and an implicit Euler scheme are considered for the spatial and time discretization of the problem. In this work, we propose a solution method for two‐phase flow problems which is based on the combination of the modified Picard linearization method and a very simple cell‐centered multigrid algorithm that performs efficiently even for heterogeneous random media. This is shown in the numerical experiments, where two test problems are presented to demonstrate the robustness of the proposed solver.

This work presents an overview of the most relevant results obtained by the authors regarding the numerical solution of the Biot’s consolidation problem by preconditioning techniques. The emphasis here is on the design of parameter-robust preconditioners for the efficient solution of the algebraic system of equations resulting after proper discretization of such poroelastic problems. The classical two- and three-field formulations of the problem are considered, and block preconditioners are presented for some of the discretization schemes that have been proposed by the authors for these formulations. These discretizations have been proved to be well-posed with respect to the physical and discretization parameters, what provides a framework to develop preconditioners that are robust with respect to such parameters as well. In particular, we construct both norm-equivalent (block diagonal) and field-of-value-equivalent (block triangular) preconditioners, which are proved to be parameter-robust. The theoretical results on this parameter-robustness are demonstrated by considering typical benchmark problems in the literature for Biot’s model.

In recent years, the gradual saturation of parallelization in space has been a strongmotivation for the design and analysis of new parallel-in-time algorithms. Amongthese methods, the parareal algorithm, first introduced by Lions, Maday and Turinici[9], has received significant attention., The work of Andrés Arrarás and Carmen Rodrigo was supported by the Spanish State Research Agency under project PGC2018-099536-A-I00 (MCIU/AEI/FEDER, UE). The work of Francisco J. Gaspar and Laura Portero was supported by the Spanish State Research Agency under project PID2019-105574GB-I00 (AEI/10.13039/501100011033).