Simultaneous quasi-optimal convergence rates in FEM-BEM coupling

S. Nicaise We consider the symmetric finite element–boundary element coupling that connects two linear elliptic second-order partial differential equations posed in a bounded domain Ω and its complement, where the exterior problem is restated as an integral equation on the coupling boundary Γ = ∂Ω. Under the assumption that the corresponding transmission problem admits a shift theorem for data in H^{−1 + s},s∈[0,s₀],s₀>1/2, we analyze the discretization by piecewise polynomials of degree k for the domain variable and piecewise polynomials of degree k − 1 for the flux variable on the coupling boundary. Given sufficient regularity, we show that (up to logarithmic factors) the optimal convergence O(h^{k + 1/2}) in the H^−1/2(Γ)-norm is obtained for the flux variable, whereas classical arguments by Céa-type quasi-optimality and standard approximation results provide only O(h^k) for the overall error in the natural product norm on H¹(Ω) × H^−1/2(Γ).