Hybridized CutFEM for Elliptic Interface Problems

Burman, Erik; Elfverson, Daniel; Hansbo, Peter; Larson, Mats G.; Larsson, Karl

doi:10.1137/18m1223836

Cited by 19 publications

(17 citation statements)

References 33 publications

(34 reference statements)

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“…In this section we establish the convergence of the iterative procedure ( 8)- (12). This Theorem is one of the novel contributions of this paper.…”

Section: Convergence Theoremmentioning

confidence: 96%

“…The use of classical methods involves remeshing at each time step, which requires difficult-to-implement and expensive refinement-and-derefinement algorithms [47,13,35,38,4]. Expensive meshing/remeshing can be avoided by resorting to unfitted methods, such as GFEM/XFEM [32,6] (and, more generally, partition of unit methods, ( [43]), Immersed Finite Element [49,41], CutFEM ( [12]), or hierarchical methods such as the finite cell method with local enrichment [31] or hp-d methods [21,45,52,55,61]. Alternatively, one can resort to methods that facilitate the remeshing procedure by allowing polygonal elements, with possibly curved edges/faces, such as the virtual element method [20,5,10,14,1].…”

Section: Introductionmentioning

confidence: 99%

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A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

Viguerie¹,

Bertoluzza²,

Veneziani³

et al. 2021

Preprint

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Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and preconditioning. These difficulties are increased if the region of localized dynamics changes in time. Overlapping domain decomposition methods, which split the problem at the continuous level, show promise due to their ease of implementation and computational efficiency. Accordingly, the present work aims to further develop the mathematical theory of such methods at both the continuous and discrete levels. For the continuous formulation of the problem, we provide a full convergence analysis. For the discrete problem, we show how the described method may be interpreted as a Gauss-Seidel scheme or as a Neumann series approximation, establishing a convergence criterion in terms of the spectral radius of the system. We then provide a spectral scaling argument and provide numerical evidence for its justification.

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“…In this section we establish the convergence of the iterative procedure ( 8)- (12). This Theorem is one of the novel contributions of this paper.…”

Section: Convergence Theoremmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

Viguerie¹,

Bertoluzza²,

Veneziani³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We now suppose that u g is defined on Ω Γ h , cf. (8), rather than on the whole of Ω h . We keep the primal unknown u in (9) and we want to impose…”

Section: Dirichlet Conditionsmentioning

confidence: 98%

“…The simplest meaningful example in the realm of linear elasticity is given by structures consisting of multiple materials having different elasticity parameters. This situation has already been treated in XFEM [12,2,31,30], CutFEM [8,20,19,24], and SBM [25] paradigms. We are now going to demonstrate the applicability of φ-FEM in this context.…”

Section: Test Casementioning

confidence: 98%

$ϕ$-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer

Cotin¹,

Duprez²,

Lleras³

et al. 2021

Preprint

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One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods using the simple computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely developed. In the present work, we investigate the performances of an unfitted method called φ-FEM that converges optimally and uses classical finite element spaces so that it can be easily implemented using general FEM libraries. The main idea is to take into account the geometry thanks to a level set function describing the boundary or the interface. Up to now, the φ-FEM approach has been proposed, tested and substantiated mathematically only in some simplest settings: Poisson equation with Dirichlet/Neumann/Robin boundary conditions. Our goal here is to demonstrate its applicability to some more sophisticated governing equations arising in the computational mechanics. We consider the linear elasticity equations accompanied by either pure Dirichlet boundary conditions or by the mixed ones (Dirichlet and Neumann boundary conditios co-existing on parts of the boundary), an interface problem (linear elasticity with material coefficients abruptly changing over an internal interface), a model of elastic structures with cracks, and finally the heat equation. In all these settings, we derive an appropriate variant of φ-FEM and then illustrate it by numerical tests on manufactured solutions. We also compare the accuracy and efficiency of φ-FEM with those of the standard fitted FEM on the meshes of similar size, revealing the substantial gains that can be achieved by φ-FEM in both the accuracy and the computational time.

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“…For instance, we may consider bifurcating cracks where a so called Kirchhoff condition holds along the intersection, cracks that meet the boundary, and cracks which are piecewise smooth. We refer to [5,8,11,24], for details on how to construct CutFEM for bifurcating cracks and how to handle the Kirchhoff condition weakly in a systematic manner. The regularity of the exact solution may be locally lower due to nonconvex corners and edges and therefore there may be a need for adaptive mesh refinement.…”

Section: Remarkmentioning

confidence: 99%

A cut finite element method for a model of pressure in fractured media

2020

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We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.

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Hybridized CutFEM for Elliptic Interface Problems

Cited by 19 publications

References 33 publications

A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

$ϕ$-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer

A cut finite element method for a model of pressure in fractured media

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