Partial expansion of a Lipschitz domain and some applications

Gopalakrishnan, Jay; Qiu, Weifeng

doi:10.1007/s11464-012-0189-2

Cited by 20 publications

(12 citation statements)

References 27 publications

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“…The decomposition (4.61) is a well-known regular-singular decomposition in the literature, whatever µ ∈ (W 1,∞ (Ω)) 3×3 or µ as assumed in section 2, with ||w 1 || 1 + ||p 1 || 1 ≤ C||curlw|| 0 ; see [13,14,34,42]. The second in (4.62) just follows from Assumption 1, with w ∈ J j=1 (H r (Ω j )) 3 .…”

Section: By the Inverse Estimates On Finite Dimensional Spacementioning

confidence: 98%

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

Duan¹,

Lin²,

Tan³

2016

SIAM J. Numer. Anal.

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Abstract. In this paper we propose and study a finite element method for a curlcurl-graddiv eigenvalue interface problem. Its solution may be of piecewise non-H 1 . We would like to approximate such a solution in an H 1 -conforming finite element space. With the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces, the proposed method is essentially a standard H 1 -conforming element method, up to element bubbles which can be statically eliminated at element levels. We first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds. We then analyze the underlying eigenvalue interface problem, and we obtain the error bounds O(h 2r 0 ) for eigenvalues which correspond to eigenfunctions in J j=1 (H r (Ω j )) 3 → (H r 0 (Ω)) 3 space, where the piecewise regularity r and the global regularity r 0 may belong to the most interesting interval [0,1].

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Section: By the Inverse Estimates On Finite Dimensional Spacementioning

confidence: 98%

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

Duan¹,

Lin²,

Tan³

2016

SIAM J. Numer. Anal.

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“…For a strongly Lipschitz domain, it is well-known that a Lipschitz collar can be defined using transversal vector fields near the boundary [28,8,18].…”

Section: Geometric Settingmentioning

confidence: 99%

Smoothed projections over weakly Lipschitz domains

Licht

2018

Math. Comp.

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Abstract. We develop finite element exterior calculus over weakly Lipschitz domains. Specifically, we construct commuting projections from L p de Rham complexes over weakly Lipschitz domains onto finite element de Rham complexes. These projections satisfy uniform bounds for finite element spaces with bounded polynomial degree over shape-regular families of triangulations. Thus we extend the theory of finite element differential forms to polyhedral domains that are weakly Lipschitz but not strongly Lipschitz. As new mathematical tools, we use the collar theorem in the Lipschitz category, and we show that the degrees of freedom in finite element exterior calculus are flat chains in the sense of geometric measure theory.

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“…Hence, first simultaneously solving two symmetric, positive definite problems (second-order elliptic interface problems) in parallel, and then solving a δ-regularization problem, we can obtain the desired solution. In addition, we could generalize the developed theory to those problems with mixed boundary conditions (i.e., u × n| Γ1 = 0, εu • n| Γ2 = 0, with Γ = Γ 1 ∪ Γ 2 ), since the two fundamental tools in our analysis are now available for mixed boundary conditions, the regular-singular decomposition [34] and the L 2 orthogonal decomposition [30].…”

Section: Conclusion and Extensionmentioning

confidence: 99%

A Delta-Regularization Finite Element Method for a Double Curl Problem with Divergence-Free Constraint

Duan¹,

Li²,

Tan³

et al. 2012

SIAM J. Numer. Anal.

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To deal with the divergence-free constraint in a double curl problem: curl µ −1 curl u = f and div εu = 0 in Ω, where µ and ε represent the physical properties of the materials occupying Ω, we develop a δ-regularization method: curl µ −1 curl u δ + δεu δ = f to completely ignore the divergence-free constraint div εu = 0. It is shown that u δ converges to u in H(curl ; Ω) norm as δ → 0. The edge finite element method is then analyzed for solving u δ. With the finite element solution u δ,h , quasi-optimal error bound in H(curl ; Ω) norm is obtained between u and u δ,h , including a uniform (with respect to δ) stability of u δ,h in H(curl ; Ω) norm. All the theoretical analysis is done in a general setting, where µ and ε may be discontinuous, anisotropic and inhomogeneous, and the solution may have a very low piecewise regularity on each material subdomain Ω j with u, curl u ∈ (H r (Ω j)) 3 for some 0 < r < 1, where r may be not greater than 1/2. To establish the uniform stability and the error bound for r ≤ 1/2, we have respectively developed a new theory for the K h ellipticity (related to mixed methods) and a new theory for the Fortin interpolation operator. Numerical results presented confirm the theory.

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Partial expansion of a Lipschitz domain and some applications

Cited by 20 publications

References 27 publications

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

Smoothed projections over weakly Lipschitz domains

A Delta-Regularization Finite Element Method for a Double Curl Problem with Divergence-Free Constraint

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