On the coupling of boundary integral and finite element methods

Abstract:

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