Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

Montardini, Monica; Sangalli, Giancarlo; Tani, Mattia

doi:10.1016/j.cma.2018.04.017

Cited by 16 publications

(17 citation statements)

References 52 publications

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“…The extension of the overall solver beyond the Poisson model problem considered in this paper also requires an efficient preconditioner. We remark that the FD preconditioner has been generalized in [26], that considers the Stokes system and improves the preconditioner robustness with respect to the geometry parametrization. Furthermore the use of the FD preconditioner for conforming multi-patch domains is discussed in [29].…”

Section: Discussionmentioning

confidence: 99%

“…This situation is typical of modern high-order methods: the final performance is mainly related to the quality of the preconditioner. We have already obtained promising results for the Stokes system [26], and will devote further studies to the topic.…”

Section: Introductionmentioning

confidence: 91%

“…This approach, originally proposed in [24], was first used in the context of isogeometric analysis in [29]. The FD preconditioner has been further developed in [26] in order to improve its performance on complex geometries. However, for the sake of simplicity here we use its simplest version from [29].…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Matrix-free weighted quadrature for a computationally efficient isogeometric k-method

Sangalli

Tani

2018

Computer Methods in Applied Mechanics and Engineering

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The k-method is the isogeometric method based on splines (or NURBS, etc.) with maximum regularity. When implemented following the paradigms of classical finite element methods, the computational resources required by the k−method are prohibitive even for moderate degree. In order to address this issue, we propose a matrix-free strategy combined with weighted quadrature, which is an ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more important, greatly reduces memory consumption. Our strategy also requires an efficient preconditioner for the linear system iterative solver. In this work we deal with an elliptic model problem, and adopt a preconditioner based on the Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our numerical tests show that the isogeometric solver based on MF-WQ is faster than standard approaches (where the main cost is the matrix formation by standard Gaussian quadrature) even for low degree. But the main achievement is that, with MF-WQ, the k-method gets orders of magnitude faster by increasing the degree, given a target accuracy. Therefore, we are able to show the superiority, in terms of computational efficiency, of the high-degree k-method with respect to low-degree isogeometric discretizations. What we present here is applicable to more complex and realistic differential problems, but its effectiveness will depend on the preconditioner stage, which is as always problem-dependent. This situation is typical of modern high-order methods: the overall performance is mainly related to the quality of the preconditioner.Matrix-free isogeometric analysis 2 of degree p goes up to C p−1 . We denote k-method the isogeometric method where splines of highest possible regularity are employed. In this setting, higher accuracy is achieved by k-refinement, that is, raising the degree p and refining the mesh, with C p−1 regularity at the new inserted knots, see [18].The k-method leads to several advantages, such as higher accuracy per degree-of-freedom and improved spectral accuracy, see for example [18,12,5,6,19,14]. At the same time, the k-method brings significant challenges at the computational level. Indeed, using standard finite element routines, its computational cost grows too fast with respect to the degree p, making degree raising unfavourable from the viewpoint of computational efficiency. As a consequence, in complex isogeometric simulations quadratic C 1 NURBS are preferred, see [4,27].The computational cost of the k-method is well understood in literature. The standard formation of the Poisson (Galerkin) stiffness matrix, by Gauss quadrature and element assembly, takes O(N p 9 ) floating point operations (flops) in 3D, where N is the number of degrees of freedom (see, e.g., [8] for the details). Computational limitations appear also in the solution phase of the linear system: the large support overlap associated with B-splines makes the associated Galerkin matrices ...

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 91%

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Matrix-free weighted quadrature for a computationally efficient isogeometric k-method

Sangalli

Tani

2018

Computer Methods in Applied Mechanics and Engineering

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“…In this section, we present a strategy to partially incorporate G in the preconditioner, without increasing its computational cost. The same idea has been used in [32] for the Stokes problem. We begin by splitting the bilinear form A(·, ·) as…”

Section: Inclusion Of the Geometry Information In The Preconditionermentioning

confidence: 99%

Space–time least–squares isogeometric method and efficient solver for parabolic problems

et al. 2019

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In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensorproduct construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degreesof-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization.A knot vector in [0, 1] is a sequence of non-decreasing points Ξ := {0 = ξ 1 ≤ · · · ≤ ξ m+p+1 = 1}, where m and p are positive integers. We use open knot vectors, that is ξ 1 = · · · = ξ p+1 = 0 and ξ m = · · · = ξ m+p+1 = 1. Then, according to Cox-De Boor recursion formulas (see [14]), the univariate B-splines are piecewise polynomials b i,p : (0, 1) → R defined as for p = 0:

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“…Then, we address the ill-conditioning issues stemming from our choice of super-penalty parameters. We adapt the block preconditioner based on an inexact Schur complement reduction (SCR) introduced in [27,28] and we combine it with a preconditioner tailored to the isogeometric discretization of the Kirchhoff plate, where we exploit the tensor product structure of B-splines and an efficient algorithm for the solution of the arising Sylvester-like system; for a detailed derivation we refer to [30,31,37]. Finally, we show through several numerical benchmarks the optimal convergence properties of the presented methodology, where our approach does not suffer from boundary locking, especially on very coarse meshes.…”

Section: Introductionmentioning

confidence: 99%

A projected super-penalty method for the $$C^1$$-coupling of multi-patch isogeometric Kirchhoff plates

2021

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This work focuses on the development of a super-penalty strategy based on the $$L^2$$ L 2 -projection of suitable coupling terms to achieve $$C^1$$ C 1 -continuity between non-conforming multi-patch isogeometric Kirchhoff plates. In particular, the choice of penalty parameters is driven by the underlying perturbed saddle point problem from which the Lagrange multipliers are eliminated and is performed to guarantee the optimal accuracy of the method. Moreover, by construction, the method does not suffer from boundary locking, especially on very coarse meshes. We demonstrate the applicability of the proposed coupling algorithm to Kirchhoff plates by studying several benchmark examples discretized by non-conforming meshes. In all cases, we recover the optimal rates of convergence achievable by B-splines where we achieve a substantial gain in accuracy per degree-of-freedom compared to other choices of the penalty parameters.

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Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

Cited by 16 publications

References 52 publications

Matrix-free weighted quadrature for a computationally efficient isogeometric k-method

Matrix-free weighted quadrature for a computationally efficient isogeometric k-method

Space–time least–squares isogeometric method and efficient solver for parabolic problems

A projected super-penalty method for the $$C^1$$-coupling of multi-patch isogeometric Kirchhoff plates

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