A nonconforming generalized finite element method for transmission problems

Anna L. Mazzucato, Victor Nistor, Qingqin Qu

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

We obtain quasi-optimal rates of convergence for transmission (interface) problems on domains with smooth, curved boundaries using a nonconforming generalized finite element method (GFEM). More precisely, we study the strongly elliptic problem Pu := -∑∂j (Aijiu) = f in a smooth bounded domain Ω with Dirichlet boundary conditions. The coefficients Aij are piecewise smooth, possibly with jump discontinuities along a smooth, closed surface Γ, called the interface, which does not intersect the boundary of the domain. We consider a sequence of approximation spaces Sμ satisfying two conditions-(1) nearly zero boundary and interface matching and (2) approximability-which are similar to those in Babuška, Nistor, and Tarfulea, [J. Comput. Appl. Math., 218 (2008), pp. 175-183]. Then, if uμ ∈ S μ, μ ≥ 1, is a sequence of Galerkin approximations of the solution u to the interface problem, the approximation error ∥u-u μĤ1(Ω) is of order O(h μm), where hμm is the typical size of the elements in Sμ and Ĥ1 is the Sobolev space of functions in H1 on each side of the interface. We give an explicit construction of GFEM spaces Sμ for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold, and present a numerical test.

Original languageEnglish (US)
Pages (from-to)555-576
Number of pages22
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number1
DOIs
StatePublished - Apr 17 2013

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Generalized Finite Element Method
Nonconforming Finite Element Method
Transmission Problem
Finite element method
Sobolev spaces
Interface Problems
Optimal Rate of Convergence
Interfaces (computer)
Boundary conditions
Curved Boundary
Approximation Space
Approximability
Galerkin Approximation
Approximation Error
Intersect
Elliptic Problems
Sobolev Spaces
Dirichlet Boundary Conditions
Bounded Domain
Discontinuity

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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A nonconforming generalized finite element method for transmission problems. / Mazzucato, Anna L.; Nistor, Victor; Qu, Qingqin.

In: SIAM Journal on Numerical Analysis, Vol. 51, No. 1, 17.04.2013, p. 555-576.

Research output: Contribution to journalArticle

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AB - We obtain quasi-optimal rates of convergence for transmission (interface) problems on domains with smooth, curved boundaries using a nonconforming generalized finite element method (GFEM). More precisely, we study the strongly elliptic problem Pu := -∑∂j (Aij∂ iu) = f in a smooth bounded domain Ω with Dirichlet boundary conditions. The coefficients Aij are piecewise smooth, possibly with jump discontinuities along a smooth, closed surface Γ, called the interface, which does not intersect the boundary of the domain. We consider a sequence of approximation spaces Sμ satisfying two conditions-(1) nearly zero boundary and interface matching and (2) approximability-which are similar to those in Babuška, Nistor, and Tarfulea, [J. Comput. Appl. Math., 218 (2008), pp. 175-183]. Then, if uμ ∈ S μ, μ ≥ 1, is a sequence of Galerkin approximations of the solution u to the interface problem, the approximation error ∥u-u μ∥ Ĥ1(Ω) is of order O(h μm), where hμm is the typical size of the elements in Sμ and Ĥ1 is the Sobolev space of functions in H1 on each side of the interface. We give an explicit construction of GFEM spaces Sμ for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold, and present a numerical test.

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