BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations (original) (raw)
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Multi space reduced basis preconditioners for parametrized Stokes equations
Computers & mathematics with applications, 2019
In this work we introduce a two-level preconditioner for the ecient solution of large scale saddle point linear systems arising from the nite element (FE) discretization of parametrized Stokes equations. The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning method proposed in [12], and relies on the combination of an approximated block (ne grid) preconditioner with a reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructed either with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter of fact, each RB coarse component is tailored to perform a single iteration of the iterative method at hand. The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time. Numerical test cases dealing with Stokes ows in three dimensional parameterdependent geometries are considered to assess the numerical performance of the proposed technique in dierent large scale computational settings. A detailed comparison with both the current state of the art of i) standard RB methods and ii) preconditioning techniques for Stokes equations highlights the better eciency of the proposed methodology.
2000
Balancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing NeumannNeumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomain...
2018
In this work we introduce a new two-level preconditioner for the ecient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" ne grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built upon a new Multi Space Reduced Basis (MSRB) method that we introduce for the rst time in this paper, where a reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the exible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are wellsuited to perform a single iteration, by addressing the error components which have not been treated yet. The Krylov iterations employed to solve the resulting preconditioned system targets small tolerances with a very small iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in dierent large scale computational settings related to parametrized advection diusion equations and compared with the current state of the art algebraic multigrid preconditioners.
Journal of Computational Science, 2015
The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf-sup stable and pointwise divergence-free. When applied to discretized Stokes equations, these spaces generate a symmetric and indefinite saddle-point linear system. Krylov subspace methods are usually the most efficient procedures to solve such systems. One of such methods, for symmetric systems, is the Minimum Residual Method (MINRES). However, the efficiency and robustness of Krylov subspace methods is closely tied to appropriate preconditioning strategies. For the discrete Stokes system, in particular, block-diagonal strategies provide efficient preconditioners. In this paper, we compare the performance of block-diagonal preconditioners for several block choices. We verify how the eigenvalue clustering promoted by the preconditioning strategies affects MINRES convergence. We also compare the number of iterations and wall-clock timings. We conclude that among the building blocks we tested, the strategy with relaxed inner conjugate gradients preconditioned with incomplete Cholesky provided the best results.
2017
Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this paper we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical eviden...
2013
The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf-sup stable and divergence-free. When applied to discretize Stokes equations, it generates a symmetric and indefinite linear system of saddle-point type. Krylov subspace methods are usually the most efficient procedures to solve such systems. One of such methods, for symmetric systems, is the Minimum Residual Method (MINRES). However, the efficiency and robustness of Krylov subspace methods is closely tied to appropriate preconditioning strategies. For the discrete Stokes system, in particular, block-diagonal strategies provide efficient preconditioners. In this paper, we compare the performance of block-diagonal preconditioners for several block choices. We verify how eigenvalue clustering promoted by the preconditioning strategies affects MINRES convergence. We also compare the number of iterations and wall-clock timings. We conclude that an incomplete Cholesky block-diagonal preconditioning strategy with relaxed inner conjugate gradients iterations provides the best computational strategy when compared to other block-diagonal and global solution strategies.
Electronic Transactions on Numerical Analysis, 2018
Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions. The onelevel domain decomposition preconditioners are based on the solution of local problems. This has the undesired consequence that the results are not scalable, it means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why in this work we introduce, and test numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider the nearly incompressible elasticity problems and Stokes equations, and discretise them by using two finite element methods, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations.
SIAM Journal on Scientific Computing, 2010
This paper presents an efficient numerical solver for the finite element approximation of the incompressible Navier-Stokes equations within a moving three dimensional domain. The moving domain is modelled using the Arbitrary Lagrangian-Eulerian (ALE) formulation. Applying a finite element approximation leads to the solution of a large sparse system of equations. In this work we look at the application of the F p preconditioner within GMRES for efficiently solving such systems. Numerical results are presented for tetrahedral and hexahedral elements, and both structured and unstructured meshes. In all cases GMRES convergence rates are seen to be independent of mesh size. Finally, we show an application of this problem for modelling fluid flow within the heart.
ℋ︁-Matrix Preconditioners for Symmetric Saddle-Point Systems from Meshfree Discretization
Numerical Linear Algebra with Applications, 2008
Meshfree methods are suitable for solving problems on irregular domains, avoiding the use of a mesh. To deal with the boundary conditions, we can use Lagrange multipliers and obtain a sparse, symmetric and indefinite system of saddle-point type. Many methods have been developed to solve the indefinite system. Previously, we presented an algebraic method to construct an LU-based preconditioner for the saddle-point system obtained by meshfree methods, which combines the multilevel clustering method with the H-matrix arithmetic. The corresponding preconditioner has both H-matrix and sparse matrix subblocks. In this paper we refine the above method and propose a way to construct a pure H-matrix preconditioner. We compare the new method with the old method, JOR and smoothed algebraic multigrid methods. The numerical results show that the new preconditioner outperforms the preconditioners based on the other methods.
The Mixed Finite Element Multigrid Preconditioned MINRES Method for Stokes Equations
Journal of Applied Fluid Mechanics
The study considers the saddle point problem arising from the mixed finite element discretization of the steady state Stokes equations. The saddle point problem is an indefinite system of linear equations, a feature that degrades the performance of any iterative solver. The heart of the study is the construction of fast, robust and effective iterative solution methods for such systems. Specific attention is given to the preconditioned MINRES solver PMINRES which is carefully treated for the solution of the Stokes equations. The study concentrates on the block preconditioner applied to the MINRES to effectively solve the whole coupled system. We combine iterative techniques with the MINRES as preconditioner approximations to produce an efficient solver for indefinite system of equations. We consider different preconditioner approximations of the building blocks of the preconditioner and compare their effects in accelerating the MINRES iterative scheme. We give a detailed overview of the algorithmic aspects and the theoretical convergence analysis of our solver. We study the MINRES method with the following preconditioner approximations: diagonal, multigrid v-cycle, preconditioned conjugate gradient and Chebyshev semi iteration methods. A comparative analysis of the preconditioner approximations show that the multigrid method is a suitable accelerator for the MINRES method. The application of the preconditioner becomes mandatory as evidenced by poor performance of the MINRES as compared to PMINRES. We study the problem in a two dimensional setting using the Hood-Taylor Q 2 − Q 1 stable pair of finite elements. The incompressible flow iterative solution software(IFISS) matlab toolbox is used to assemble the matrices. We present the numerical results to illustrate the efficiency and robustness of the MINRES scheme with the multigrid preconditioner.