We consider a mixed-boundary-value/interface problem for the elliptic operator P=-Σij∂i(aij∂ju)=f on a polygonal domain ΩâŠR2 with straight sides. We endowed the boundary of Ω partially with Dirichlet boundary conditions u=0 on ∂DΩ, and partially with Neumann boundary conditions Σijνiaij∂ju=0 on ∂NΩ. The coefficients aij are piecewise smooth with jump discontinuities across the interface Γ, which is allowed to have singularities and cross the boundary of Ω. In particular, we consider "triple-junctions" and even "multiple junctions". Our main result is to construct a sequence of Generalized Finite Element spaces sn that yield "hm-quasi-optimal rates of convergence", m≥1, for the Galerkin approximations unε sn of the solution u. More precisely, we prove that ||-u-un||-≤Cdim(s n)-m/2||-f||Hm-1(Ω), where C depends on the data for the problem, but not on f, u, or n and dim(sn)→∞. Our construction is quite general and depends on a choice of a good sequence of approximation spaces Sn′ on a certain subdomain W that is at some distance to the vertices. In case the spaces Sn′ are Generalized Finite Element spaces, then the resulting spaces sn are also Generalized Finite Element spaces.
All Science Journal Classification (ASJC) codes
- Computational Mathematics
- Applied Mathematics