Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary Observations

• Published in: Applied mathematics & optimization Vol. 84; no. 2; pp. 2289 - 2325
• Main Authors: Hào, Dinh Nho, Quyen, Tran Nhan Tam, Son, Nguyen Thanh
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer Nature B.V
• Subjects: Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Control; Convergence; Galerkin method; Inverse problems; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Noise levels; Noise measurement; Numerical and Computational Physics; Optimization; Regularization; Simulation; Systems Theory; Theoretical; Inverse source problem; Parabolic problem; 35R30; Convergence rates; Ill-posedness; Crank–Nicolson Galerkin method; 65J22; 35R25; 65J20; Tikhonov regularization; Source condition
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-020-09710-2

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 a quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and a corresponding convergence rate are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.