Block Preconditioners Research Articles

A comparison of block preconditioners for isogeometric analysis discretizations of the incompressible Navier–Stokes equations

AbstractWe deal with numerical solution of the incompressible Navier–Stokes equations discretized using the isogeometric analysis (IgA) approach. Similarly to finite elements, the discretization leads to sparse nonsymmetric saddle‐point linear systems. The IgA discretization basis has several specific properties different from standard FEM basis, most importantly a higher interelement continuity leading to denser matrices. We are interested in iterative solution of the resulting linear systems using a Krylov subspace method (GMRES) preconditioned with several state‐of‐the‐art block preconditioners. We compare the efficiency of the ideal versions of these preconditioners for three model problems (for both steady and unsteady flow in two and three dimensions) and investigate their properties with focus on the IgA specifics, that is, various degree and continuity of the discretization basis. Our experiments show that the block preconditioners can be successfully applied to the systems arising from high continuity IgA, moreover, that the high continuity can bring some benefits in this context. For example, some of the preconditioners, whose convergence is h‐dependent in the steady case, seem to be less sensitive to the mesh refinement for higher continuity discretizations. In the unsteady case, we generally get faster convergence for higher continuity than for C0 continuous discretizations of the same degree for most of the preconditioners.

A high performance level-block approximate LU factorization preconditioner algorithm

Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as single-level approaches or within a multilevel framework. We will present a block incomplete factorization that is based on skillfully blocking the system initially and throughout the factorization. Our objective is to develop algebraic block preconditioners for the efficient solution of such systems by Krylov subspace methods. We will demonstrate how our level-block approximate algorithm outperforms a single level-block scalar method often by orders of magnitude on modern architectures, thus paving the way for its prospective use inside various multilevel incomplete factorization approaches or other applications where the core part relies on an efficient incomplete factorization algorithms.

A new block preconditioner and improved finite element solver of Poisson-Nernst-Planck equation

The Poisson-Nernst-Planck (PNP) equation is one important continuum model for studying charge transport in ion channels, which is a common phenomenon and plays a key role in molecular biosciences. In this paper, to improve the current PNP solvers, according to the equation structure, a new block preconditioner was proposed and proved that the preconditioned linear system has bounded eigenvalues independent of mesh sizes under some conditions, thus guaranteeing that the convergence rate of the preconditioned linear solver is independent of mesh sizes. Meanwhile, the commonly-used solution decomposition schemes in classic continuum models for isolating the singularities were presented and then one was chosen for solving PNP when zero initial guess was used for the Newton method. Furthermore, an efficient and improved finite element PNP solver was proposed by further combining the two-grid method as an acceleration technique. Then the new program package was fulfilled based on the state-of-the-art finite element library FEniCS and the efficient scientific library PETSc. Finally, numerical simulations on a test model with analytical solution as well as some tests on protein cases were carried out to validate the new program package and our theoretical results.

Using partial spectral information for block diagonal preconditioning of saddle-point systems

Considering saddle-point systems of the Karush–Kuhn–Tucker (KKT) form, we propose approximations of the “ideal” block diagonal preconditioner based on the exact Schur complement proposed by Murphy et al. (SIAM J Sci Comput 21(6):1969–1972, 2000). We focus on the case where the (1,1) block is symmetric and positive definite, but with a few very small eigenvalues that possibly affect the convergence of Krylov subspace methods like Minres. Assuming that these eigenvalues and their associated eigenvectors are available, we first propose a Schur complement preconditioner based on this knowledge and establish lower and upper bounds on the preconditioned Schur complement. We next analyse theoretically the spectral properties of the preconditioned KKT systems using this Schur complement approximation in two spectral preconditioners of block diagonal forms. In addition, we derive a condensed “two in one” formulation of the proposed preconditioners in combination with a preliminary level of preconditioning on the KKT system. Finally, we illustrate on a PDE test case how, in the context of a geometric multigrid framework, it is possible to construct practical block preconditioners that help to improve on the convergence of Minres.

Algebraic Multigrid Block Preconditioning for Multi-Group Radiation Diffusion Equations

The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume discretization of multi-group radiation diffusion equations, whose coefficient matrices can be rearranged into the $(G+2)\times(G+2)$ block form, where $G$ is the number of energy groups. The preconditioning techniques are based on the monolithic classical algebraic multigrid method, physical-variable based coarsening two-level algorithm and two types of block Schur complement preconditioners. The classical algebraic multigrid is applied to solve the subsystems that arise in the last three block preconditioners. The coupling strength and diagonal dominance are further explored to improve performance. We use representative one-group and twenty-group linear systems from capsule implosion simulations to test the robustness, efficiency, strong and weak parallel scaling properties of the proposed methods. Numerical results demonstrate that block preconditioners lead to mesh- and problem-independent convergence, and scale well both algorithmically and in parallel.

A New Block Preconditioner for Implicit Runge--Kutta Methods for Parabolic PDE Problems

A new preconditioner based on a block factorization into lower triangular, diagonal, and upper triangular factors (an $LDU$ factoriaztion) with algebraic multigrid subsolves for scalability is intr...

A constrained transport divergence-free finite element method for incompressible MHD equations

In this paper we study finite element method for three-dimensional incompressible resistive magnetohydrodynamic equations, in which the velocity, the current density, and the magnetic induction are divergence-free. It is desirable that the discrete solutions should also satisfy divergence-free conditions exactly especially for the momentum equations. Inspired by constrained transport method, we devise a new stable mixed finite element method that can achieve the goal. We also prove the well-posedness of the discrete solutions. To solve the resulting linear algebraic equations, we propose a GMRES solver with an augmented Lagrangian block preconditioner. By numerical experiments, we verify the theoretical results and demonstrate the quasi-optimality of the discrete solver with respect to the number of degrees of freedom. A comparison with other discretization using lid driven cavity is also given.

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

AbstractIn this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle‐point system. The Schur complement for the generalized saddle‐point system is very difficult or even impossible to approximate, which prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle‐point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simplified wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare the performance of the MSSS preconditioner with the performance of the state‐of‐the‐art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for the control of the Navier–Stokes equation.

Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems

Several physical phenomena are described by systems of partial differential equations (PDEs) that, after space discretization, yield the solution of saddle point algebraic linear systems. In realistic three-dimensional numerical simulations, these linear systems are large scale and ill-conditioned, thus they require the development of effective solvers. The aim of this work is the construction and numerical validation of parallel block preconditioners for a set of three-dimensional saddle point problems discretized by the low order virtual element method (VEM). VEM is a recent numerical technology for the approximation of PDEs on polygonal and polyhedral meshes. We focus on the following systems of PDEs: stationary Maxwell equations in the mixed Kikuchi formulation; elliptic equations in mixed form; Stokes system; linear elasticity in the mixed Hellinger–Reissner formulation. We provide two parallel block preconditioners: one based on the approximate Schur complement and the other on a regularization technique. Several numerical experiments are run in parallel on a Linux cluster. We analyze the performance of the iterative solvers in terms of GMRES iterations and computational time. We verify the robustness of the solvers with respect to different polyhedral meshes and the scalability of both the assembling and solution time by varying the number of processors. The performance of the two iterative solvers is also compared with state-of-the-art parallel direct linear solvers.

A fast boundary element based solver for localized inelastic deformations

SummaryWe present a numerical method for the solution of nonlinear geomechanical problems involving localized deformation along shear bands and fractures. We leverage the boundary element method to solve for the quasi‐static elastic deformation of the medium while rigid‐plastic constitutive relations govern the behavior of displacement discontinuity (DD) segments capturing localized deformations. A fully implicit scheme is developed using a hierarchical approximation of the boundary element matrix. Combined with an adequate block preconditioner, this allows to tackle large problems via the use of an iterative solver for the solution of the tangent system. Several two‐dimensional examples of the initiation and growth of shear‐bands and tensile fractures illustrate the capabilities and accuracy of this technique. The method does not exhibit any mesh dependency associated with localization provided that (i) the softening length‐scale is resolved and (ii) the plane of localized deformations is discretized a priori using DD segments.

Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number

We review a number of preconditioners for the advection-diffusion operator and for the Schur complement matrix, which, in turn, constitute the building blocks for Constraint and Triangular Preconditioners to accelerate the iterative solution of the discretized and linearized Navier-Stokes equations. An intensive numerical testing is performed onto the driven cavity problem with low values of the viscosity coefficient. We devise an efficient multigrid preconditioner for the advection-diffusion matrix, which, combined with the commuted BFBt Schur complement approximation, and inserted in a 2×2 block preconditioner, provides convergence of the Generalized Minimal Residual (GMRES) method in a number of iteration independent of the meshsize for the lowest values of the viscosity parameter. The low-rank acceleration of such preconditioner is also investigated, showing its great potential.

An observation on the uniform preconditioners for the mixed Darcy problem

AbstractWhen solving a multiphysics problem one often decomposes a monolithic system into simpler, frequently single‐physics, subproblems. A comprehensive solution strategy may commonly be attempted, then, by means of combining strategies devised for the constituent subproblems. When decomposing the monolithic problem, however, it may be that requiring a particular scaling for one subproblem enforces an undesired scaling on another. In this manuscript we consider the H(div)‐based mixed formulation of the Darcy problem as a single‐physics subproblem; the hydraulic conductivity, K, is considered intrinsic and not subject to any rescaling. Preconditioners for such porous media flow problems in mixed form are frequently based on H(div) preconditioners rather than the pressure Schur complement. We show that when the hydraulic conductivity, K, is small the pressure Schur complement can also be utilized for H(div)‐based preconditioners. The proposed approach employs an operator preconditioning framework to establish a robust, K‐uniform block preconditioner. The mapping property of the continuous operator is a key component in applying the theoretical framework point of view. As such, a main challenge addressed here is establishing a K‐uniform inf‐sup condition with respect to appropriately weighted Hilbert intersection‐ and sum spaces.

Block preconditioners for mixed-dimensional discretization of flow in fractured porous media

In this paper, we are interested in an efficient numerical method for the mixed-dimensional approach to modeling single-phase flow in fractured porous media. The model introduces fractures and their intersections as lower-dimensional structures, and the mortar variable is used for flow coupling between the matrix and fractures. We consider a stable mixed finite element discretization of the problem, which results in a parameter-dependent linear system. For this, we develop block preconditioners based on the well-posedness of the discretization choice. The preconditioned iterative method demonstrates robustness with regard to discretization and physical parameters. The analytical results are verified on several examples of fracture network configurations, and notable results in reduction of number of iterations and computational time are obtained.

A stabilized hybrid mixed finite element method for poroelasticity

In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We introduce a perturbation of the bilinear form of the displacement which allows for the elimination of the bubble functions. Together with eliminating Darcy's velocity by hybridization, we obtain an eliminated system whose size is the same as the classical P1-RT0-P0 discretization. Based on the well-posedness of the eliminated system, we design block preconditioners that are parameter-robust. Numerical experiments are presented to confirm the theoretical results of the stabilized scheme as well as the block preconditioners.

Robust block preconditioners for poroelasticity

In this paper we develop and analyze several preconditioners for the linear systems arising from discretized poroelasticity problems. The preconditioners include one block preconditioner for the two-field Biot model and several preconditioners for the classical three-field Biot model. We manage to analyze these different preconditioners under a same theoretical framework and show that all of them are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.

Approximate inverse-based block preconditioners in poroelasticity

We focus on the fully implicit solution of the linear systems arising from a three-field mixed finite element approximation of Biot’s poroleasticity equations. The objective is to develop algebraic block preconditioners for the efficient solution of such systems by Krylov subspace methods. In this work, we investigate the use of approximate inverse-based techniques to decouple the native system of equations and obtain explicit sparse approximations of the Schur complements related to the physics-based partitioning of the unknowns by field type. The proposed methods are tested in various numerical experiments including real-world applications dealing with petroleum and geotechnical engineering.

The nested block preconditioning technique for the incompressible Navier–Stokes equations with emphasis on hemodynamic simulations

We develop a novel iterative solution method for the incompressible Navier–Stokes equations with boundary conditions coupled with reduced models. The iterative algorithm is designed based on the variational multiscale formulation and the generalized-α scheme. The spatiotemporal discretization leads to a block structure of the resulting consistent tangent matrix in the Newton–Raphson procedure. As a generalization of the conventional block preconditioners, a three-level nested block preconditioner is introduced to attain a better representation of the Schur complement, which plays a key role in the overall algorithm robustness and efficiency. This approach provides a flexible, algorithmic way to handle the Schur complement for problems involving multiscale and multiphysics coupling. The solution method is implemented and benchmarked against experimental data from the nozzle challenge problem issued by the US Food and Drug Administration. The robustness, efficiency, and parallel scalability of the proposed technique are then examined in several settings, including moderately high Reynolds number flows and physiological flows with strong resistance effect due to coupled downstream vasculature models. Two patient-specific hemodynamic simulations, covering systemic and pulmonary flows, are performed to further corroborate the efficacy of the proposed methodology.

Topology optimization of repetitive near-regular shell structures using preconditioned conjugate gradients method

During the topology optimization process, the stiffness matrix of the structure changes continuously. Suppose we have a structure whose stiffness matrix is in a special form such as block tridiagonal matrix at the beginning of topology optimization process. The question now arises: How this special property can be used in the topology optimization process in an efficient manner? To find the answer, we examined the use of an appropriate form of the stiffness matrix at the beginning of the optimization process by constructing preconditioners for conjugate gradient method, where this approximate method is utilized to solve the nested analysis equations in topology optimization. As we already know, using a suitable nodal ordering of the repetitive near-regular shell structure, the stiffness matrix will have a suitable block form. In this study, this block form is utilized to construct some of the well-known block preconditioners. Constructing preconditioners only once at the beginning and employing in the entire design process of optimization is previously proposed as an effective approach for the examples they studied. Here, we examine the effectiveness of this method on shell structures.

A hybrid interface preconditioner for monolithic fluid\u2013structure interaction solvers

We propose a hybrid interface preconditioner for the monolithic solution of surface-coupled problems. Powerful preconditioning techniques are crucial when it comes to solving large monolithic systems of linear equations efficiently, especially when arising from coupled multi-physics problems like in fluid–structure interaction. Existing physics-based block preconditioners have proven to be efficient, but their error assessment reveals an accumulation of the error at the coupling surface. We address this issue by combining them with an additional additive Schwarz preconditioner, whose subdomains span across the interface on purpose. By performing cheap but accurate subdomain solves that do not depend on the separation of physical fields, this error accumulation can be reduced effectively. Numerical experiments compare the performance of the hybrid preconditioner to existing approaches, demonstrate the increased efficiency, and study its parallel performance.

The inexact Euler-extrapolated block preconditioners for a class of complex linear systems

By utilizing Euler-extrapolated technique, we construct several inexact block preconditioners for solving a broad class of complex linear systems. The spectral properties of the preconditioned matrices are studied. Moreover, we determine the quasi-optimal parameter of these preconditioners. Numerical results are given to verify the effectiveness of the proposed inexact preconditioners.