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arXiv · 1404.2744

Simultaneous quasi-optimal convergence in FEM-BEM coupling

Jens Markus Melenk·Dirk Praetorius·Barbara Wohlmuth
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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 C\'ea-type quasi-optimality and standard approximation results provide only $O(h^k)$ for the overall error in the natural product norm on $H^1(\Omega)\times H^{-1/2}(\Gamma)$.

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Jens Markus Melenk, Dirk Praetorius, Barbara Wohlmuth. 2014-04-10. Simultaneous quasi-optimal convergence in FEM-BEM coupling. https://doi.org/10.1002/mma.3374

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