5 research outputs found
Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic equations with boundary observations
This work is devoted to an inverse problem of identifying a source term
depending on both spatial and time variables in a parabolic equation from
single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method
is applied to the least squares functional with an quadratic stabilizing
penalty term. The convergence of finite dimensional regularized approximations
to the sought source as measurement noise levels and mesh sizes approach to
zero with an appropriate regularization parameter is proved. Moreover, under a
suitable source condition, an error bound and corresponding convergence rates
are proved. Finally, several numerical experiments are presented to illustrate
the theoretical findings.Comment: Inverse source problem, Tikhonov regularization, Crank-Nicolson
Galerkin method, Source condition, Convergence rates, Ill-posedness,
Parabolic proble
A convergent adaptive finite element method for elliptic Dirichlet boundary control problems
This paper concerns the adaptive finite element method for elliptic Dirichlet boundary control problems in the energy space. The contribution of this paper is twofold. First, we rigorously derive efficient and reliable a posteriori error estimates for finite element approximations of Dirichlet boundary control problems. As a by-product, a priori error estimates are derived in a simple way by introducing appropriate auxiliary problems and establishing certain norm equivalence. Secondly, for the coupled elliptic partial differential system that resulted from the first-order optimality system, we prove that the sequence of adaptively generated discrete solutions including the control, the state and the adjoint state, guided by our newly derived a posteriori error indicators, converges to the true solution along with the convergence of the error estimators. We give some numerical results to confirm our theoretical findings
A convergent adaptive finite element method for electrical impedance tomography
In this work, we develop and analyse an adaptive finite element method for efficiently solving electrical impedance tomography—a severely ill-posed nonlinear inverse problem of recovering the conductivity from boundary voltage measurements. The reconstruction technique is based on Tikhonov regularization with a Sobolev smoothness penalty and discretizing the forward model using continuous piecewise linear finite elements. We derive an adaptive finite element algorithm with an a posteriori error estimator involving the concerned state and adjoint variables and the recovered conductivity. The convergence of the algorithm is established, in the sense that the sequence of discrete solutions contains a convergent subsequence to a solution of the optimality system for the continuous formulation. Numerical results are presented to verify the convergence and efficiency of the algorithm