Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE

摘要

Different numerical methods have been proposed in the literature for solving the Cauchy problem for linear elliptic equations modelling different physical phenomena (Laplace equation for the stationary heat equation, Lamé operator in elasticity, etc.). The aim of this paper is to situate these methods, fixed point methods and domain decomposition based techniques in a general variational framework, and show the equivalence between them. A generalization of the energy gap method proposed by the authors for Fourier-like boundary conditions is studied. Then a comparison of these methods by using an analytical example and two numerical problems with complicated geometry or boundary conditions is performed in order to estimate their numerical performances. It appeared that no method stood out from the others due to better or worse performance. According to the situations, the classification of the performances in terms of conditioning, an essential factor because the Cauchy problem is ill-posed, or in terms of the capacity to deal with strongly spatially variable data, depends on the problem dealt with. The issue of regularization is not addressed here because it is method-dependent and distorts the appreciation of the basic performance of the different approaches.