Higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1D

Deng, Quanling; Calo, Victor M.

doi:10.1016/j.cam.2019.112558

Cited by 7 publications

(17 citation statements)

References 44 publications

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“…The following lemma presents the existence of a local interpolation in V p h,E that is applicable to any element with an interface (i.e., any τ ∈ T h,Γ ). Proof of the lemma below has a slightly different flavor to the one given in [14].…”

Section: Fig 2 the Resulting Enriched Basis Functions As Applied Tomentioning

confidence: 99%

“…The next lemma is established in [14] for a problem with an interface. We extend it to multiple number of interfaces over Ω, whose proof closely follows that of [14].…”

Section: Fig 2 the Resulting Enriched Basis Functions As Applied Tomentioning

confidence: 99%

“…Recently, SGFEM has been implemented in two-dimensional parabolic (time-dependent) interface problem [36]. The higher order SGFEM has also been developed for the elliptic eigenvalue and source interface problem in [14]. Having said that, the application of SGFEM for nonlinear interface problems is very few to be found in literatures.…”

Section: Introductionmentioning

confidence: 99%

“…As reported in [30], SGFEM is applied to industrially relevant two phase/free-surface flow problems governed by the Navier-Stokes equation, which is nonlinear due to convective acceleration. The convergence analysis of the linear SGFEM was established in [5,6,25] and later was generalized to arbitrary order in 1D setting in [14].…”

Section: Introductionmentioning

confidence: 99%

“…This effort can be considered as a first attempt toward the ultimate goal of applying SGFEM to the unsaturated Richards equation for heterogeneous and layered soil system. As in [14], the approximate solution is represented in terms of the usual finite element basis and the enrichment functions aimed at capturing the effects of discontinuity. The resulting nonlinear algebraic system is solved by utilizing Newton's method of iteration.…”

Section: Introductionmentioning

confidence: 99%

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On the Application of Stable Generalized Finite Element Method for Quasilinear Elliptic Two-Point BVP

2021

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In this paper, we discuss an application of the generalized finite element method (GFEM) to approximate the solutions of quasilinear elliptic equations with multiple interfaces in one dimensional space. The problem is characterized by spatial discontinuity of the elliptic coefficient that depends on the unknown solution. It is known that unless the partition of the domain matches the discontinuity configuration, accuracy of standard finite element techniques significantly deteriorates and standard refinement of the partition may not suffice. The GFEM is a viable alternative to overcome this predicament. It is based on the construction of certain enrichment functions supplied to the standard finite element space that capture effects of the discontinuity. This approach is called stable (SGFEM) if it maintains an optimal rate of convergence and the conditioning of GFEM is not worse than that of the standard FEM. A convergence analysis is derived and performance of the method is illustrated by several numerical examples. Furthermore, it is known that typical global formulations such as FEMs do not enjoy the numerical local conservation property that is crucial in many conservation law-based applications. To remedy this issue, a Lagrange multiplier technique is adopted to enforce the local conservation. A numerical example is given to demonstrate the performance of proposed technique.

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Section: Fig 2 the Resulting Enriched Basis Functions As Applied Tomentioning

confidence: 99%

“…The next lemma is established in [14] for a problem with an interface. We extend it to multiple number of interfaces over Ω, whose proof closely follows that of [14].…”

Section: Fig 2 the Resulting Enriched Basis Functions As Applied Tomentioning

confidence: 99%