A domain decomposition method for Isogeometric multi-patch problems with inexact local solvers

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Abstract

In Isogeometric Analysis, the computational domain is often described as multi-patch, where each patch is given by a tensor product spline/NURBS parametrization. In this work we propose a FETI-like solver where local inexact solvers exploit the tensor product structure at the patch level. To this purpose, we extend to the isogeometric framework the so-called All-Floating variant of FETI, that allows us to use the Fast Diagonalization method at the patch level. We construct then a preconditioner for the whole system and prove its quasi-robustness with respect to the local mesh-size h and patch-size H: precisely the condition number of the preconditioned system is bounded by the square of the logarithm of Hh. Our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree p.

Introduction

Isogeometric Analysis (IgA) was introduced in the seminal paper [1] as an extension of finite element analysis. The key idea is to use the same basis functions that describe the computational domain, typically B-splines, NURBS or extensions, also to represent the unknown solution of the partial differential equations.

In this work, we are concerned with the numerical solution of large isogeometric compressible linear elasticity problems in multi-patch domains, that is, domains defined as the union of several patches, each described by a different tensor product spline/NURBS parametrization (see [2]). We restrict ourselves to C0 global continuity. Indeed, imposing higher continuity among patches, while interesting especially in the context of isogeometric analysis, is a difficult task and a topic of current active research (see for example [3]), then beyond the scope of this paper. However, within each patch, we want to consider high-degree and high-continuity spline approximation, whose advantages are evidenced in literature, see for example [4], [5], [6], [7], [8], [9]. It is also known that the development of linear solvers for high-degree and high-continuity spline-based isogeometric analysis is a challenging task, see [10], [11].

Our starting point is [12], where it has been shown the potential of the Fast Diagonalization (FD) method to construct fast solvers for elliptic isogeometric problems. The FD method is a direct solver introduced in [13], that can be applied to problems with a Sylvester-like structure. In general, elliptic isogeometric problems do not possess the required Sylvester-like structure, even on a single patch, unless the patch parametrization is trivial. However, [12] constructs efficient preconditioners (that is, inexact solvers) with the required structure on a single patch. Similarly, here we use FD as an inexact and fast solver for problems at the patch level.

Our approach is based on the Finite Element Tearing and Interconnecting (FETI) idea, that, after its appearance in  [14], has been widely developed and adopted in finite element solvers, see [15]. IETI, the isogeometric version of FETI, has been introduced in [16]. In particular, we develop in this paper an All-Floating IETI (in short AF-IETI) method, the isogeometric version of the All-Floating FETI introduced in [17], which is in turn similar to the so-called total-FETI of [18]. With this variant of FETI, both the global continuity of the solution and the Dirichlet boundary conditions are weakly imposed by Lagrange multipliers. The choice of the AF-IETI formulation is crucial for us since it yields the Sylvester-like structure that we need to use FD as inexact local solver. To allow inexact solvers, a saddle point formulation as in [19] is also required.

The abstract framework and mathematical techniques of [19] can be used here and give us a bound of the condition number of the preconditioned system which is explicit with respect to the patch-size H and the local mesh-size h, and only depends on the logarithm of their ratio. Our numerical tests also indicate that the performance of the preconditioner does not deteriorate as the degree p is increased. Furthermore, to show the potential of the proposed inexact AF-IETI, we compare numerically its performance to AF-IETI with the exact local solvers: on these tests our results indicate that the inexact approach, while requiring more iterations as expected, is orders of magnitude faster than the exact one.

Domain decomposition methods represent an active research area in isogeometric analysis. We recall the overlapping Schwarz methods studied in [20], [21], the BDDC methods with related preconditioners studied from [22], the dual–primal approach introduced in [16] and further studied in [23], [24]. Domain decomposition approaches with inexact local solvers have been studied in [25], [26]. The case of trimmed domains has been recently addressed in [27].

The paper is organized as follows. In Section 2 we present the basics of multi-patch based IgA and in Section 3 we introduce the model problem as well as its discrete formulation. The AF-IETI method is described in Section 4, while exact and inexact local solvers are introduced and analysed in Section 5. Numerical results are reported in Section 6 and, finally, Section 7 contains some conclusions and future directions of research.

Section snippets

B-splines

Given two integers m,p>0, we introduce a knot vector Ξ{0=ξ1ξm+p+1=1} in the interval [0,1], where m and p are, respectively, the number of basis functions that will be built from the knot vector and their polynomial degree. We consider open knot vectors, i.e. we set ξ1==ξp+1=0 and ξm+1==ξm+p+1=1. Following Cox–de Boor recursion formulas [28], univariate B-splines are piecewise polynomials defined for i=1,,m as follows:

for p=0 b̂i,0(η)=1if ξiη<ξi+1,0otherwise,

for p1 b̂i,p(η)=ηξiξi+pξ

Model problem and its discretization

Let ΩRd be a computational domain described by a multi-patch spline parametrization, as in Section 2.2, and let Ω denote its boundary. Suppose that Ω=ΩDΩN with ΩDΩN=, where ΩD has positive measure. Let f[L2(Ω)]d, g[L2(ΩN)]d and HD1(Ω){vH1(Ω)s.t.v=0 on ΩD}. Then, the variational formulation of the compressible linear elasticity problem we consider reads: Findu[HD1(Ω)]ds.t.forallv[HD1(Ω)]d a(u,v)=F,v,where we define a(u,v)2μΩε(u):ε(v)dx+λΩuvdx,F,vΩfvdx+ΩNgvds

All-Floating IETI method

In this section, we present the All-Floating IETI (AF-IETI) method, an extension to IgA of the AF-FETI method, introduced in [29], (see also [30]). In this formulation, the FETI interface includes the whole boundary Ω without distinction between Dirichlet and Neumann boundary.

Let Ω be the computational domain described as the union of Npatch isogeometric patches Ω(k) for k=1,,Npatch, as detailed in Section 2.2. We work with the computational spaces Vh(k)Vh(k)dandVhΠk=1NpatchVh(k),where Vh(k)

Solving the local problems

This section deals with the definition of PA and PS. These operators are selected block-diagonal, where each block corresponds to a patch. In particular, both the application of PA1 and the application of PS correspond to the solution of patch-wise elliptic problems. We discuss two possible choices: the first involves exact solvers for A+DHM and BΓSBΓT, while the second represents an inexact version that makes the application of the whole preconditioner B1 more efficient.

Numerical results

In this section we assess the performance of the preconditioning strategies. We compare the exact and inexact local solvers introduced in Sections 5.1 Exact local solvers, 5.2 Inexact local solvers, respectively, in three dimensional domains. We use the version of the inexact local solvers that incorporates some information on the geometry parametrization, that is detailed in Section 5.2.3. We show the results only for the choices of PS (5.35), (5.36), for exact and inexact local solvers,

Conclusions

In this paper, we studied a combination of the FD solver with a domain decomposition method of FETI type in a general multi-patch isogeometric setting, where the subdomains of the method coincide with the isogeometric patches. We focused on the All-Floating (domain) version of FETI, where both the continuity and Dirichlet boundary conditions are weakly imposed by Lagrange multipliers. We built a preconditioner for the resulting saddle-point linear system in which the FD method is used in the

CRediT authorship contribution statement

Michał Bosy: Formal analysis, Methodology, Software, Writing - original draft. Monica Montardini: Formal analysis, Methodology, Software, Validation, Writing - original draft. Giancarlo Sangalli: Conceptualization, Supervision, Formal analysis, Methodology, Writing - review & editing. Mattia Tani: Conceptualization, Formal analysis, Methodology, writing - review & editing.

Acknowledgements

The authors were partially supported by the European Research Council through the FP7 Ideas Consolidator Grant HIGEOM n.616563. The second, third and fourth authors are members of the Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INDAM) and the second author was partially supported by INDAM-GNCS “Finanziamento Giovani Ricercatori 2019–20” for the project “Efficiente risoluzione dell’equazione di Navier–Stokes in ambito isogeometrico”. These supports are

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