The phase separation processes are typically modeled by Cahn-Hilliard equations. This equation was originally introduced to model phase separation in binary alloys, where phase stands for concentration of different components in alloy. When the binary alloy under preparation is subjected to a rapid reduction in temperature below a critical temperature, it has been experimentally observed that the concentration changes from a mixed state to a visibly distinct spatially separated two phase for binary alloy. This rapid reduction in the temperature, the so-called "deep quench limit", is modeled effectively by obstacle potential. The discretization of Cahn-Hilliard equation with obstacle potential leads to a block $2 \times 2$ {\em non-linear} system, where the $(1,1)$ block has a non-linear and non-smooth term. Recently a globally convergent Newton Schur method was proposed for the non-linear Schur complement corresponding to this non-linear system. The proposed method is similar to an inexact active set method in the sense that the active sets are first approximately identified by solving a quadratic obstacle problem corresponding to the $(1,1)$ block of the block $2 \times 2$ system, and later solving a reduced linear system by annihilating the rows and columns corresponding to identified active sets. For solving the quadratic obstacle problem, various optimal multigrid like methods have been proposed. In this paper, we study a non-standard norm that is equivalent to applying a block diagonal preconditioner to the reduced linear systems. Numerical experiments confirm the optimality of the solver and convergence independent of problem parameters on sufficiently fine mesh.
We consider the iterative solution of symmetric saddle-point matrices with a singular leading block. We develop a new ideal positive definite block diagonal preconditioner that yields a preconditioned operator with four distinct eigenvalues. We offer a few techniques for making the preconditioner practical, and illustrate the effectiveness of our approach with numerical experiments.
We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the MINRES algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distinct eigenvalues, 1 and -1, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely studied block diagonal preconditioner, specify the computational work associated with applying the new preconditioner inexactly, and survey a number of theoretical results for the block diagonal ca…
This paper deals with solving a class of three-by-three block saddle point problems. The systems are solved by preconditioning techniques. Based on an iterative method, we construct a block upper triangular preconditioner. The convergence of the presented method is studied in details. Finally, some numerical experiments are given to demonstrate the superiority of the proposed preconditioner over some existing ones.
In this paper, we execute the shift-splitting preconditioner for asymmetric saddle point problems with its (1,2) block's transposition unequal to its (2,1) block under the removed minus of its (2,1) block. The proposed preconditioner is stemmed from the shift splitting (SS) iteration method for solving asymmetric saddle point problems, which is convergent under suitable conditions. The relaxed version of the shift-splitting preconditioner is obtained as well. The spectral distributions of the related preconditioned matrices are given. Numerical experiments from the Stokes problem are offered t…
We study the performance of a new block preconditioner for a class of $3\times3$ block saddle point problems which arise from finite element methods for solving time-dependent Maxwell equations and some other practical problems. We also estimate the lower and upper bounds of eigenvalues of the preconditioned matrix. \cred{Finally, we examine our new preconditioner to accelerate the convergence speed of the GMRES method which shows the effectiveness of the preconditioner.
In this paper we consider multiple saddle point problems with block tridiagonal Hessian in a Hilbert space setting. Well-posedness and the related issue of preconditioning are discussed. We give a characterization of all block structured norms which ensure well-posedness of multiple saddle point problems as a helpful tool for constructing block diagonal preconditioners. We subsequently apply our findings to a general class of PDE-constrained optimal control problems containing a regularization parameter $α$ and derive $α$-robust preconditioners for the corresponding optimality systems. Finally…
We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total pressure and fluid pressure). The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properti…
In this work, we consider the modeling of inclusions in the material using an unfitted finite element method. In the unfitted methods, structured background meshes are used and only the underlying finite element space is modified to incorporate the discontinuities, such as inclusions. Hence, the unfitted methods provide a more flexible framework for modeling the materials with multiple inclusions. We employ the method of Lagrange multipliers for enforcing the interface conditions between the inclusions and matrix, this gives rise to the linear system of equations of saddle point type. We utili…
For nonsymmetric block three-by-three singular saddle point problems arising from the Picard iteration method for a class of mixed finite element scheme, recently Salkuyeh et al. in (D.K. Salkuyeh, H. Aslani, Z.Z. Liang, An alternating positive semi-definite splitting preconditioner for the three-by-three block saddle point problems, Math. Commun. 26 (2021) 177-195) established an alternating positive semi-definite splitting (APSS) method. In this work, we analyse the semi-convergence of the APSS method for solving a class of nonsymmetric block three-by-three singular saddle point problems. Th…
We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and-Cell (MAC) finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results s…
Recently, in (M. Masoudi, D.K. Salkuyeh, An extension of positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems, Computers \& Mathematics with Application, https://doi.org/10.1016/j.camwa.2019.10.030, 2019) an extension of the positive definite and skew-Hermitian splitting (EPSS) iteration method for nonsingular generalized saddle point problems has been presented. In this article, we study semi-convergence of the EPSS method for singular generalized saddle problems. Then a special case of EPSS (SEPSS) preconditioner is applied to the non…
We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied in details and its induced preconditioner is examined for accelerating the convergence speed of generalized minimal residual (GMRES) method. More precisely, we analyze the eigenvalue distribution of the preconditioned matrix. Numerical experiments are reported to demonstrate the effectiveness of the proposed preconditioner.
We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming HDG schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique \cite{Xu96}. A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matri…
A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time-space fractional diffusion equation
University of Electronic Science and Technology of China（电子科技大学）；Southwestern University of Finance and Economics（西南财经大学）；Lanzhou University of Technology（兰州理工大学）
In this paper, a robust and effective preconditioner for the fast Method of Moments(MoM) based Hierarchal Electric Field Integral Equation(EFIE) solver is proposed using symmetric near-field Schur's complement method. In this preconditioner, near-field blocks are scaled to a diagonal block matrix and these near-field blocks are replaced with the scaled diagonal block matrix which reduces the near-field storage memory and the overall matrix vector product time. Scaled diagonal block matrix is further used as a preconditioner and due to the block diagonal form of the final preconditioner, no add…
The preconditioned iterative solution of large-scale saddle-point systems is of great importance in numerous application areas, many of them involving partial differential equations. Robustness with respect to certain problem parameters is often a concern, and it can be addressed by identifying proper scalings of preconditioner building blocks. In this paper, we consider a new perspective to finding effective and robust preconditioners. Our approach is based on the consideration of the natural physical units underlying the respective saddle-point problem. This point of view, which we refer to …
An effective power based parallel preconditioner is proposed for general large sparse linear systems. The preconditioner combines a power series expansion method with some low-rank correction techniques, where the Sherman-Morrison-Woodbury formula is utilized. A matrix splitting of the Schur complement is proposed to expand the power series. The number of terms used in the power series expansion can control the approximation accuracy of the preconditioner to the inverse of the Schur complement. To construct the preconditioner, graph partitioning is invoked to reorder the original coefficient m…
We propose an unconditionally robust and highly effective preconditioner for general symmetric positive definite (SPD) matrices based on structured incomplete factorization (SIF), called enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing to the matrix and then compressing appropriate scaled off-diagonal blocks. Here, we use an enhanced scaling-and-compression strategy to design the new eSIF preconditioner. Some subtle modifications are made, such as the use of two-sided block triangular pr…
For 2x2 block matrices, it is well-known that block-triangular or block-LDU preconditioners with an exact Schur complement (inverse) converge in at most two iterations for fixed-point or minimal-residual methods. Similarly, for saddle-point matrices with a zero (2,2)-block, block-diagonal preconditioners converge in at most three iterations for minimal-residual methods, although they may diverge for fixed-point iterations. But, what happens for non-saddle-point matrices and block-diagonal preconditioners with an exact Schur complement? This note proves that minimal-residual methods applied to …
In recent years, solvers for finite-element discretizations of linear or linearized saddle-point problems, like the Stokes and Oseen equations, have become well established. There are two main classes of preconditioners for such systems: those based on block-factorization approach and those based on monolithic multigrid. Both classes of preconditioners have several critical choices to be made in their composition, such as the selection of a suitable relaxation scheme for monolithic multigrid. From existing studies, some insight can be gained as to what options are preferable in low-performance…
The paper discusses a reuse of matrix factorization as a building block in the Augmented Lagrangian (AL) and modified AL preconditioners for non-symmetric saddle point linear algebraic systems. The strategy is applied to solve two-dimensional incompressible fluid problems with efficiency rates independent of the Reynolds number. The solver is then tested to simulate motion of a surface fluid, an example of a 2D flow motivated by an interest in lateral fluidity of inextensible viscous membranes. Numerical examples include the Kelvin--Helmholtz instability problem posed on the sphere and on the …
