On the coupling of boundary integral and finite element methods

Abstract

Let Ω c {\Omega ^c} be the complementary of a bounded regular domain in R 2 {{\mathbf {R}}^2} of boundary Γ \Gamma . We consider the problem (1) \[ { Δ u = f ; a m p ; in Ω c , u | Γ = u 0 , a m p ; \left \{ {\begin {array}{*{20}{c}} {\Delta u = f;} & {{\text {in}}\;{\Omega ^c},} \\ {u{|_\Gamma } = {u_{0,}}} & {} \\ \end {array} } \right . \] where f has its support in a bounded subdomain Ω 1 {\Omega _1} of Ω c {\Omega ^c} . Let Γ 2 {\Gamma _2} be the common boundary of Ω 1 {\Omega _1} and Ω 2 = Ω c − Ω 1 {\Omega _2} = {\Omega ^c} - {\Omega _1} . We solve the problem (1) by using an equivalent system of equations involving an integral equation on Γ 2 {\Gamma ^2} coupled with the equation: (2) \[ { Δ u = f a m p ; in Ω 1 , u | Γ = u 0 , a m p ; u | Γ 2 = λ . a m p ; \left \{ {\begin {array}{*{20}{c}} {\Delta u = f} \hfill & {{\text {in}}\;{\Omega _1},} \hfill \\ {u{|_\Gamma } = {u_0},} \hfill & {} \hfill \\ {u{|_{{\Gamma _2}}} = \lambda .} \hfill & {} \hfill \\ \end {array} } \right . \] We introduce a finite element approximation of Eq. (2) and of the integral equation and we prove optimal error estimates.