دانلود رایگان مقاله پری کاندیشنر بلوک FETI-DP برای استوکس و سیستم خطی ارتجاعی مخلوط

Abstract

The aim of this work is to construct and analyze a FETI-DP type domain decomposition preconditioner for isogeometric discretizations of the Stokes and mixed linear elasticity systems. This method extends to the isogeometric analysis context the preconditioner previously proposed by Tu and Li (2015) for finite element discretizations of the Stokes system. The resulting isogeometric FETI-DP algorithm is proven to be scalable in the number of subdomains and has a quasi-optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes. Extensive two-dimensional numerical experiments validate the theory, investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity, and show its robustness with respect to domain deformation, material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces.

6. Conclusions

We have developed a FETI-DP type domain decomposition preconditioner for isogeometric discretizations of the Stokes and mixed linear elasticity systems. This work extends to isogeometric discretizations the method proposed by Li and Tu in their pioneering papers [13,14] for finite element discretizations of the Stokes system. Following the algebraic theory of [14] and applying some isogeometric technical tools developed in our previous works [5,10], we can prove that the proposed FETI-DP method is scalable in the number of subdomains and quasi-optimal in the ratio of subdomain and element sizes. Numerical experiments in the plane have confirmed the scalability and quasi-optimality of the proposed method. The numerical tests investigating the behavior of the preconditioner with respect to spline polynomial degree and regularity have shown that, in case of maximal subdomains interface regularity kΓ = p − 1, the FETI-DP preconditioner works fine for p = 2, 3, 4 and starts degenerating for p ≥ 5. The reason of such sub-optimal behavior for p ≥ 5 and maximal interface regularity could be attributed to the non-optimal choice of scaling functions, as observed in our previous isogeometric BDDC paper [5] for scalar elliptic problems. A possible remedy could be the use of deluxe scaling functions, see e.g. [10], but the extension of deluxe scaling to saddle point problems is still an open problem even for standard finite element discretizations. In case of reduced subdomains interface regularity, instead, the FETI-DP preconditioner presents a quasi-optimal behavior up to p = 8. Finally, further numerical tests have shown the robustness of the FETI-DP solver with respect to domain deformations, material incompressibility and the presence of discontinuities of the elastic coefficients across subdomains.