### 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} (A^{ij}∂ _{i}u) = f in a smooth bounded domain Ω with Dirichlet boundary conditions. The coefficients A^{ij} 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 S_{μ} 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 u_{μ} ∈ 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 _{μ}^{m}), where h_{μ}^{m} is the typical size of the elements in S^{μ} and Ĥ^{1} is the Sobolev space of functions in H^{1} on each side of the interface. We give an explicit construction of GFEM spaces S_{μ} for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold, and present a numerical test.

Original language | English (US) |
---|---|

Pages (from-to) | 555-576 |

Number of pages | 22 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - Apr 17 2013 |

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### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Numerical Analysis*,

*51*(1), 555-576. https://doi.org/10.1137/100816031

}

*SIAM Journal on Numerical Analysis*, vol. 51, no. 1, pp. 555-576. https://doi.org/10.1137/100816031

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - A nonconforming generalized finite element method for transmission problems

AU - Mazzucato, Anna L.

AU - Nistor, Victor

AU - Qu, Qingqin

PY - 2013/4/17

Y1 - 2013/4/17

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=84876127976&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876127976&partnerID=8YFLogxK

U2 - 10.1137/100816031

DO - 10.1137/100816031

M3 - Article

AN - SCOPUS:84876127976

VL - 51

SP - 555

EP - 576

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 1

ER -