{
  "id": "1906.08700",
  "version": "v1",
  "published": "2019-06-20T15:42:47.000Z",
  "updated": "2019-06-20T15:42:47.000Z",
  "title": "On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates",
  "authors": [
    "Laurent Bourgeois",
    "Lucas Chesnel"
  ],
  "categories": [
    "math.AP",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter $\\varepsilon>0$. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in $\\varepsilon$. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due the particular structure of the regularized problems, classical techniques \\`a la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in $\\varepsilon$ in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-20T15:42:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplace equation",
      "error estimates",
      "quasi-reversibility solutions",
      "finite elements methods",
      "regularized problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}