The inverse Robin problem covers the determination of the Robin parameter in an elliptic partial differential equation posed on a domain $\Omega$. Given the solution of the Robin problem on a subdomain $\omega \subset \Omega$ together with the elliptic problem's right hand sides, the aim is to solve this inverse Robin problem numerically. In this work, a computational method for the reconstruction of the Robin parameter inspired by a unique continuation method is established. The proposed scheme relies solely on first-order Lagrange finite elements ensuring a straightforward implementation. Under the main assumption that the Robin parameter is in a finite dimensional space of continuously differentiable functions it is shown that the numerical method is second order convergent in the finite element's mesh size. For noisy data this convergence rate is shown to hold true until the noise term dominates the error estimate. Numerical experiments are presented that highlight the feasibility of the Robin parameter reconstruction and that confirm the theoretical convergence results numerically.

翻译：逆Robin问题涉及确定定义在区域$\Omega$上的椭圆型偏微分方程中的Robin参数。给定Robin问题在子区域$\omega \subset \Omega$上的解以及椭圆型问题的右端项，目标是对该逆Robin问题进行数值求解。本文建立了一种受唯一延拓方法启发的Robin参数重构计算方法。所提方案仅依赖于一阶拉格朗日有限元，确保了实现的简便性。在Robin参数属于连续可微函数的有限维空间这一主要假设下，证明了该数值方法在有限元网格尺寸上具有二阶收敛性。对于含噪声数据，该收敛速率在噪声项主导误差估计前保持成立。数值实验展示了Robin参数重构的可行性，并在数值上验证了理论收敛结果。