On the Coupling of Boundary Integral and Finite Element Methods

Let Ω ᶜ be the complementary of a bounded regular domain in R² of boundary Γ. We consider the problem (1) \[ \{ { {array}{*{20}{c}} {Δ u = f;} & {{in}\;{Ω ^c},} \\ {u{|_Γ } = {u0,}} & {} \\ {array} } .\] where f has its support in a bounded subdomain Ω ₁ of Ω ᶜ. Let Γ ₂ be the common boundary of Ω ₁ and Ω ₂ = Ω ᶜ - Ω ₁. We solve the problem (1) by using an equivalent system of equations involving an integral equation on Γ ² coupled with the equation: (2) \[ \{ { {array}{*{20}{c}} {Δ u = f} & {{in}\;{Ω _1},} \\ {u{|_Γ } = {u_0},} & {} \\ {u{|_{{Γ _2}}} = λ .} & {} \\ {array} } .\] We introduce a finite element approximation of Eq. (2) and of the integral equation and we prove optimal error estimates.