New a posteriori error estimates for full-space transmission problems

Abstract

In the present work, we derive functional upper bounds for the potential error arising from finite-element boundary-element coupling formulations for a nonlinear Poisson-type transmission problem. The proposed a posteriori error estimates are independent of the precise discretization scheme and provide guaranteed upper bounds for the potential error. The computation of these upper bounds is based on the solutions of local auxiliary finite element problems on patches in the interior domain and in a strip domain along the coupling boundary. Numerical experiments illustrate the performance of the proposed error estimation strategy for a related adaptive mesh-refinement strategy.

Figures (16)

• Figure 1: Meshes generated by Algorithm \ref{['algo:adap']} in Example \ref{['subsec:squareex']} for $\theta = 0.4$ and $k = 2$. Only triangles not colored in gray contribute to the computation of the error indicators \ref{['eq:indicators']}. The triangles of the patches $\widetilde{\omega}_\ell^{z,k}$ are depicted in blue, whereas the triangles in $\Omega$ are indicated in yellow and the remaining triangles in $\Omega^{\mathrm{ext}}$ in gray. The inner boundary $\Gamma$ is shown in red and the outer boundary of the union of the patches in green.
• Figure 2: Meshes generated by Algorithm \ref{['algo:adapint']} in Example \ref{['subsec:squareex']} for $\theta = 0.4$ and $k = 2$. The triangles of the patches $\omega_\ell^{z,k}$ are depicted in blue, while the remaining triangles are indicated in yellow. The outer boundary $\Gamma$ is shown in red and the inner boundary of the union of the patches in green.
• Figure 3: Convergence rates of the full error estimator $\widetilde{\eta}_\ell$ from \ref{['eq:fullest']} in Algorithm \ref{['algo:adap']} for $k = 2$ in Example \ref{['subsec:squareex']} for different marking parameters $0 < \theta \leq 1$ (left). Comparison of the different contributions $\eta_\ell^{\mathrm{int}}$, $\widetilde{\eta}_\ell^{\mathrm{ext}}$, $\widetilde{\mathrm{osc}}_\ell^{\mathrm{D}}$, and $\widetilde{\mathrm{osc}}_\ell^{\mathrm{N}}$ of $\widetilde{\eta}_\ell$ for $\theta = 0.4$ (right).
• Figure 4: Convergence rates of the full error estimator $\eta_\ell$ from \ref{['eq:fullestalt']} in Algorithm \ref{['algo:adapint']} for $k = 2$ in Example \ref{['subsec:squareex']} for different marking parameters $0 < \theta \leq 1$ (left). Comparison of the different contributions $\eta_\ell^{\mathrm{int}}$, $\eta_\ell^{\mathrm{ext}}$, $\mathrm{osc}_\ell^{\mathrm{D}}$, and $\mathrm{osc}_\ell^{\mathrm{N}}$ of $\eta_\ell$ for $\theta = 0.4$ (right).
• Figure 5: Comparison between the estimators $\eta_\ell, \eta_\ell^{\mathrm{ext}}$ from \ref{['eq:intindicators']}--\ref{['eq:fullestalt']} in Algorithm \ref{['algo:adapint']} and $\widetilde{\eta}_\ell$, $\widetilde{\eta}_\ell^{\mathrm{ext}}$ from \ref{['eq:indicators']}--\ref{['eq:fullest']} in Algorithm \ref{['algo:adap']} for Example \ref{['subsec:squareex']} with $k = 2$ and $\theta = 0.4$.

Theorems & Definitions (18)

• Remark 1
• Lemma 2
• proof
• Theorem 3: Functional upper bound
• proof
• Remark 4
• Corollary 5
• proof
• Corollary 6
• proof