Adaptive finite element methods for the identification of distributed parameters in elliptic equation. (English) Zbl 1148.65086
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 \(K=\{ q\in W^{1,t}: 0<\alpha \leq q(x) \leq \nu, q|_{\partial \Omega}=q^* \}\), \(t>n\).
\[ -\nabla(q(x) \nabla u)=f, \quad x \in \Omega; \quad u=0, \quad x \in \partial \Omega \]
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 \(K=\{ q\in W^{1,t}: 0<\alpha \leq q(x) \leq \nu, q|_{\partial \Omega}=q^* \}\), \(t>n\).
Reviewer: Mikhail Yu. Kokurin (Yoshkar-Ola)
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35R30 | Inverse problems for PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
parameter identification; finite element; least-squares; Gauss-Newton method; elliptic equation; inverse problemReferences:
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