Simultaneous quasi-optimal convergence in FEM-BEM coupling

Abstract

We consider the symmetric FEM-BEM coupling that connects two linear elliptic second order partial differential equations posed in a bounded domain $\Omega$ and its complement, where the exterior problem is restated by an integral equation on the coupling boundary $\Gamma =\partial \Omega$. We assume that the corresponding transmission problem admits a shift theorem for data in $H^{-1+s}$, $s\in [-1,-1+s_0]$, $s_0>1/2$. 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 $H^{-1/2}(\Gamma )$-norm is obtained for the flux variable, while 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 $H^1(\Omega )\times H^{-1/2}(\Gamma )$.

References in corpus (2)

• On optimal $L^2$- and surface flux convergence in FEM (extended version)
• $L^2$-error estimates for finite element approximations of boundary fluxes

Cited by in corpus (5)

• FEM-BEM mortar coupling for the Helmholtz problem in three dimensions
• A time-dependent FEM-BEM coupling method for fluid-structure interaction in 3d
• On optimal $L^2$- and surface flux convergence in FEM (extended version)
• Mortar coupling of $hp$-discontinuous Galerkin and boundary element methods for the Helmholtz equation
• On superconvergence of Runge-Kutta convolution quadrature for the wave equation