Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods

Large linear systems with sparse, non-symmetric matrices arise in the modeling of Markov chains or in the discretization of convection-diffusion problems. Due to their potential to solve sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non-symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M-matrix. In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non-symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M-matrix structure, an approach based on linear optimization yields optimally sparse M-matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an AMLI type method. While in the considered test problems, the M-matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure

A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential / capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes the resulting systems of nonlinear algebraic equations are solved with Newton's method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust and efficient. In particular no post-processing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1000 processors

Block strategies to compute the lambda modes associated with the neutron diffusion equation

[EN] Given a configuration of a nuclear reactor core, the neutronic distribution of the power can beapproximated by means of the multigroup neutron diffusion equation. This is an approximationof the neutron transport equation that assumes that the neutron current is proportional to thegradient of the scalar neutron ux with a diffusion coeffcient [1]. This approximation is known asthe Fick's first law. To define the steady-state problem, the criticality of the system must be forced.In this work, the -modes problem is used. That yields a generalized eigenvalue problem whoseeigenvector associated with the dominant eigenvalue represents the distribution of the neutron uxin steady-state.The spatial discretization of the equation is made by a continuous Galerkin high order finite elementmethod is applied [2] to obtain an algebraic eigenvalue problem. Usually, the matrices obtainedfrom the discretization are huge and sparse. Moreover, they have a block structure given by the different number of energy groups. In this work, block strategies are developed to optimize thecomputation of the associated eigenvalue problems.First, different block eigenvalue solvers are studied. On the other hand, the convergence of theseiterative methods mainly depends on the initial guess and the preconditioner used. In this sense,different multilevel techniques to accelerate the rate of convergence are proposed. Finally, the sizeof the problems can be suffciently large to be unfeasible to be solved in personal computers. Thus,a matrix-free methodology that avoids the allocation of the matrices in memory is applied [3].Three-dimensional benchmarks are used to show the effciency of the methodology proposed.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.Carreño, A.; Vidal Ferrándiz, A.; Ginestar Peiró, D.; Verdú, G. (2022). Block strategies to compute the lambda modes associated with the neutron diffusion equation. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 423-430. https://doi.org/10.4995/YIC2021.2021.13470OCS42343

Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements

We consider the e�cient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an e�ective Schur complement approximation. Numerical results illustrate the competitiveness of this approach

Multilevel Solvers for Unstructured Surface Meshes

Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner