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.publishedVersio

Robust block preconditioners for biot’s model

In this paper, we design robust and efficient block preconditioners for the two-field formulation of Biot’s consolidation model, where stabilized finite-element discretizations are used. The proposed block preconditioners are based on the well-posedness of the discrete linear systems. Block diagonal (norm-equivalent) and block triangular preconditioners are developed, and we prove that these methods are robust with respect to both physical and discretization parameters. Numerical results are presented to support the theoretical results

Preconditioning techniques for the coupled Stokes–Darcy problem: spectral and field-of-values analysis

We study the performance of some preconditioning techniques for a class of block three-by-three linear systems of equations arising from finite element discretizations of the coupled Stokes–Darcy flow problem. In particular, we investigate preconditioning techniques including block preconditioners, constraint preconditioners, and augmented Lagrangian-based ones. Spectral and field-of-value analyses are established for the exact versions of these preconditioners. The result of numerical experiments are reported to illustrate the performance of inexact variants of the various preconditioners used with flexible GMRES in the solution of a 3D test problem with large jumps in the permeability

Preconditioning of Hybridizable Discontinuous Galerkin Discretizations of the Navier-Stokes Equations

The incompressible Navier-Stokes equations are of major interest due to their importance in modelling fluid flow problems. However, solving the Navier-Stokes equations is a difficult task. To address this problem, in this thesis, we consider fast and efficient solvers. We are particularly interested in solving a new class of hybridizable discontinuous Galerkin (HDG) discretizations of the incompressible Navier-Stokes equations, as these discretizations result in exact mass conservation, are locally conservative, and have fewer degrees of freedom than discontinuous Galerkin methods (which is typically used for advection dominated flows). To achieve this goal, we have made various contributions to related problems, as I discuss next. Firstly, we consider the solution of matrices with 2x2 block structure. We are interested in this problem as many discretizations of the Navier-Stokes equations result in block linear systems of equations, especially discretizations based on mixed-finite element methods like HDG. These systems also arise in other areas of computational mathematics, such as constrained optimization problems, or the implicit or steady state treatment of any system of PDEs with multiple dependent variables. Often, these systems are solved iteratively using Krylov methods and some form of block preconditioner. Under the assumption that one diagonal block is inverted exactly, we prove a direct equivalence between convergence of 2x2 block preconditioned Krylov or fixed-point iterations to a given tolerance, with convergence of the underlying preconditioned Schur-complement problem. In particular, results indicate that an effective Schur-complement preconditioner is a necessary and sufficient condition for rapid convergence of 2x2 block-preconditioned GMRES, for arbitrary relative-residual stopping tolerances. A number of corollaries and related results give new insight into block preconditioning, such as the fact that approximate block-LDU or symmetric block-triangular preconditioners offer minimal reduction in iteration over block-triangular preconditioners, despite the additional computational cost. We verify the theoretical results numerically on an HDG discretization of the steady linearized Navier--Stokes equations. The findings also demonstrate that theory based on the assumption of an exact inverse of one diagonal block extends well to the more practical setting of inexact inverses. Secondly, as an initial step towards solving the time-dependent Navier-Stokes equations, we investigate the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time HDG discretization of the scalar advection-diffusion equation. The motivation for this study is two-fold. First, the HDG discretization of the velocity part of the momentum block of the linearized Navier-Stokes equations is the HDG discretization of the vector advection-diffusion equation. Hence, efficient and fast solution of the advection-diffusion problem is a prerequisite for developing fast solvers for the Navier-Stokes equations. The second reason to study this all-at-once space-time problem is that the time-dependent advection-diffusion equation can be seen as a ``steady'' advection-diffusion problem in (d+1)-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. We present numerical examples which demonstrate the effectiveness of AIR as a preconditioner for time-dependent advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems. As the final topic of this thesis, we extend two state-of-the-art preconditioners for the Navier-Stokes equations, namely, the pressure convection-diffusion and the grad-div/augmented Lagrangian preconditioners to HDG discretizations. Our preconditioners are simple to implement, and our numerical results show that these preconditioners are robust in h and only mildly dependent on the Reynolds numbers

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems