2019/06/20 by Laurent Bourgeois, Bourgeois, Laurent, Lucas Chesnel +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1906.08700
openalex publication_date 2019/06/20 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We are interested in the classical ill-posed Cauchy problem for the Laplace\nequation. One method to approximate the solution associated with compatible\ndata consists in considering a family of regularized well-posed problems\ndepending on a small parameter \ε>0. In this context, in order to\nprove convergence of finite elements methods, it is necessary to get regularity\nresults of the solutions to these regularized problems which hold uniformly in\n\ε. In the present work, we obtain these results in smooth domains\nand in 2D polygonal geometries. In presence of corners, due the particular\nstructure of the regularized problems, classical techniques `a la Grisvard do\nnot work and instead, we apply the Kondratiev approach. We describe the\nprocedure in detail to keep track of the dependence in \ε in all the\nestimates. The main originality of this study lies in the fact that the limit\nproblem is ill-posed in any framework.\n