A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation

Abstract

This work concerns the use of the method of quasi-reversibility to solve the Cauchy problem for Laplace's equation. We describe a mixed formulation of that method and its relationship with a classical formulation. A discretized formulation using finite elements of class C 0 is derived from the mixed formulation, and convergence of the solution of that discretized problem with noisy data to the exact solution is analyzed. Finally, a simple numerical example is implemented in order to show the feasibility of the method.

Iterative Method to Solve a Data Completion Problem for Biharmonic Equation for Rectangular Domain

“…Among the deterministic method, we find the most ancient of them is the one based on optimization tools introduced by Kohn and Vogelius [5], we mention the method of Quasi-reversibility introduced by Latès since 1960 [6], and recently used by [7], [8], Thikhonv method [9] and the iterative method [10].…”

A globally convergent numerical method for a coefficient inverse problem with backscattering data

“…So, the resulting over-determined boundary value problem is solved via the Quasi-Reversibility Method (QRM). The QRM was first proposed in [19] and was developed further in [6,7,9,13,14,15]. It has proven to be effective for solving boundary value problems with over-determined boundary conditions.…”

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

“…In order to solve Cauchy problems for elliptic equations, many regularization methods have been introduced. 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.…”

“…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].…”