Residual-based stabilized formulation for the solution of inverse elliptic partial differential equations

• Published in: Computers & mathematics with applications (1987) 2020-09, Vol.80 (5), p.822-836
• Main Authors: Tyagi, Mohit, Barbone, Paul E., Oberai, Assad A.
• Format: Article
• Language: English
• Subjects: Conduction heating; Convergence; Elastic moduli; Elliptic functions; Finite element method; Finite-element and Galerkin methods; Forward problem; Grid refinement (mathematics); Inverse problems; Lagrange multiplier; Material properties; Optimization; Partial differential equations; Plane stress; Saddle points; Stability
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2020.04.016

We consider an inverse problem where the forward problem is that of linear plane stress elasticity, or equivalently, that of linear heat/hydraulic conduction. We demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem. In particular, it is stable for the response variable and the Lagrange multiplier, but not for the material property field. This lack of stability implies that we are unable to prove optimal convergence with mesh refinement for the overall problem. We overcome this difficulty by adding to the saddle point problem a residual-based term that provides sufficient stability, and prove optimal convergence in an energy-like norm. We verify these estimates through simple numerical examples. We note that while we have considered a specific model for an inverse elliptic problem in this manuscript, similar ideas could be developed for a broad class of inverse elliptic problems.