An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity

Energy methods for Cauchy problems of evolutions equations

“…This method, already presented for stationary problems in [1][2][3][4], is based on the idea of exploiting simultaneously and symmetrically the overspecified data measured on the accessible boundary. Next, two well-posed problems are defined.…”

“…A two-fields based energy error method has been designed for general symmetric elliptic operators, leading to efficient numerical algorithms that have been implemented in various situations, including 3D heterogeneous ones with non linear boundary conditions, see Andrieux and Baranger [1][2][3]. The method is based on the definition of an appropriate energy error functional, as a function of the fields on the boundary that are out of reach.…”

An iterative method for the Cauchy problem in linear elasticity with fading regularization effect

“…The references [1,5,8,9,11,12,14,20,21,23,25,26] propose different methods of solving the Cauchy problem for the Laplace equation. References [2,3,13,22,23,27,28,29,30,31,32,33,34,35,36,37,38,40,41] deal with the Cauchy problem in linear elasticity. These methods can be classified as Tikhonov type methods [15,22,27,29,33,34,35,38,39,40,41], quasi-reversibility type methods [5,21,25], iterative methods [1,2,3,8,9,…”

“…References [2,3,13,22,23,27,28,29,30,31,32,33,34,35,36,37,38,40,41] deal with the Cauchy problem in linear elasticity. These methods can be classified as Tikhonov type methods [15,22,27,29,33,34,35,38,39,40,41], quasi-reversibility type methods [5,21,25], iterative methods [1,2,3,8,9,11,12,13,14,15,20,23,26,28,32,36,37],... Quasi reversibility methods and Tikhonov regularization methods present the advantage of leading to well posed problems after mod...…”

“…Some iterative methods are based on the use of a sequence of well-posed problems and others on the minimization of an energy-like functional. Numerical algorithms are implemented using different numerical methods, such as the finite element method (FEM) [1,2,3,5,8,9,11,13,27,38], the boundary element method (BEM) [12,14,20,22,26,23,28,29,30,31,32,33,36,37,40,41], the finite difference method [21] or meshless methods [34,35]. Some papers present comparisons between different numerical methods [10,31,37].…”