×

A nonconforming generalized finite element method for transmission problems. (English) Zbl 1370.65069

Summary: 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 \[ Pu := -\sum \partial_j(A^{ij}\partial_i u )= f \] in a smooth bounded domain \(\Omega\) with Dirichlet boundary conditions. The coefficients \(A^{ij}\) are piecewise smooth, possibly with jump discontinuities along a smooth, closed surface \(\Gamma\), called the interface, which does not intersect the boundary of the domain. We consider a sequence of approximation spaces \(S_\mu\) satisfying two conditions – (1) nearly zero boundary and interface matching and (2) approximability – which are similar to those in [I. Babuška et al. [J. Comput. Appl. Math. 218, No. 1, 175–183 (2008; Zbl 1153.65106)]. Then, if \(u_\mu\in S_\mu\), \(\mu \geq 1\), is a sequence of Galerkin approximations of the solution \(u\) to the interface problem, the approximation error \(\|u-u_\mu\|_{\hat H^1(\Omega)}\) is of order \(O(h_\mu^m)\), where \(h_\mu^m\) is the typical size of the elements in \(S^\mu\) and \(\hat H^1\) is the Sobolev space of functions in \(H^1\) on each side of the interface. We give an explicit construction of GFEM spaces \(S_\mu\) for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold, and present a numerical test.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Citations:

Zbl 1153.65106
PDFBibTeX XMLCite
Full Text: DOI Link