Adaptive finite element methods for the identification of distributed parameters in elliptic equation
The authors consider the problem of determining the unknown functional coefficient \(q\) in the elliptic equation \[ -\nabla(q(x) \nabla u)=f, \quad x \in \Omega; \quad u=0, \quad x \in \partial \Omega \] by the observation data \(u=z \in L_2(\Omega)\). The domain \(\Omega\) is a bounded open subset of \(\mathbb R^n\) \((n \geq 3)\) with \(\partial \Omega \in C^3\) or \(\Omega\) is a parallelepiped. The original inverse problem is reduced to the variational problem \[ \min_{q \in K} \{ \| u(q)-z\|^2_{L_2(\Omega)}+ \beta \| \nabla q \|^2_{L_2(\Omega)} \}, \] where \(K=\{ q\in W^{1,t}: 0<\alpha \leq q(x) \leq \nu, q|_{\partial \Omega}=q^* \}\), \(t>n\).
scientific article; zbMATH DE number 5314283
scientific article