Numerical convergence and stability of mixed formulation with X-FEM cut-off

Saber, A. J.; Mansouri, Khalil; Renard, Yves; Arfaoui, Makrem; Moakher, Maher

doi:10.13052/17797179.2012.714724

Cited by 2 publications

(4 citation statements)

References 6 publications

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“…We cite in particular [8,20,17] following the CutFEM paradigm, and [1,14] following the X-FEM paradigm. The common feature of all these methods is that they discretize the variational formulation of the Stokes equation on the physical fluid domain Ω using the FE spaces defined on the background mesh occupying a domain Ω h , slightly larger than Ω.…”

Section: Introductionmentioning

confidence: 99%

ϕ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

Duprez¹,

Lleras²,

Lozinski³

2023

ESAIM: M2AN

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We present an immersed boundary method to simulate the creeping motion of a \AL{rigid particle} in a fluid described by the Stokes equations discretized thanks to a finite element strategy on unfitted meshes, called φ-FEM, that uses the description of the solid with a level-set function. One of the advantages of our method is the use of standard finite element spaces and classical integration tools, while maintaining the optimal convergence (theoretically in the H¹ norm for the velocity and L² for pressure; numerically also in the L² norm for the velocity).

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Section: Introductionmentioning

confidence: 99%

ϕ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

Duprez¹,

Lleras²,

Lozinski³

2023

ESAIM: M2AN

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“…Newer procedures may also be employed to improve the computed solution (see [5,13,14,32,33,35,37]). We point out, in particular, different versions of X-fem, PU-fem and G-Fem with or without (Heaviside) cut-off functions (see [2,9,10,36]). The reverse of the medal has to do with the condition number of the discrete problems to solve.…”

Section: Introductionmentioning

confidence: 99%

“…These problems suffer from bad and possibly very bad conditioning. We refer for instance to [2,10] for some convincing examples. Solving efficiently the discrete system, after carrying out a finite element approximation of any kind, requires some known effective techniques such as multi-grid, multi-scale or sub-structuring approaches.…”

Section: Introductionmentioning

confidence: 99%

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Computational Zooming in Near Unilateral Cracks by Schwarz Method with Total Overlap

Belgacem¹

2019

JMS

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We focus on the numerical solver of unilateral cracks by the Schwarz Method with Total Overlap. The aim is to isolate the treatment at the vicinity of the cracks from other regions of the computational domain. This avoids any direct interaction between specific approximations one may use around the singularities born at the tips of the cracks and more standard methods employed away from the cracks. We apply an iterative sub-structuring technique to capture the small structures by insulating the cracks into patches and making a zoom around each of them. The macro-problem is in turn set on the whole domain. As for the classical Schwarz method, the communication between the micro (local) and macro (global) levels is achieved iteratively through some suitable boundary conditions. The micro problem is fed by Dirichlet data along the (outer) boundary of the patches. The specificity of our approach is that the macro problem inherits transmission conditions. Although they are expressed across the cracks, the final algebraic system to invert is blind to the discontinuities of the solution. In fact, the stiffness matrix turns out to be the one related to a safe domain, as if cracks were closed or the unilateral singularities were switched off. Only the right hand side is affected by what happens at the vicinity of the cracks. This enables users to run one of many efficient algorithms found in the literature to solve the linear macro-problem. In the other hand side, in spite of the still bad conditioning and the non-linearity of the unilateral micro problems, they are reduced in size and may be inverted properly by convex optimization algorithms. A successful convergence analysis of this variant of the Schwarz Method is performed after adapting to the unilateral non-linearity the variational tools developed by P. L. Lions.

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Numerical convergence and stability of mixed formulation with X-FEM cut-off

Cited by 2 publications

References 6 publications

ϕ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

ϕ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

Computational Zooming in Near Unilateral Cracks by Schwarz Method with Total Overlap

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