{
  "id": "2002.01781",
  "version": "v1",
  "published": "2020-02-05T13:33:51.000Z",
  "updated": "2020-02-05T13:33:51.000Z",
  "title": "A probabilistic approach for exact solutions of determinist PDE's as well as their finite element approximations",
  "authors": [
    "Joel Chaskalovic"
  ],
  "comment": "17 pages, 4 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "A probabilistic approach is developed for the exact solution $u$ to a determinist partial differential equation as well as for its associated approximation $u^{(k)}_{h}$ performed by $P_k$ Lagrange finite element. Two limitations motivated our approach: on the one hand, the inability to determine the exact solution $u$ to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation $u^{(k)}_{h}$. We thus fill this knowledge gap by considering the exact solution $u$ together with its corresponding approximation $u^{(k)}_{h}$ as random variables. By way of consequence, any function where $u$ and $u_{h}^{(k)}$ are involved as well. In this paper, we focus our analysis to a variational formulation defined on $W^{m,p}$ Sobolev spaces and the corresponding a priori estimates of the exact solution $u$ and its approximation $u^{(k)}_{h}$ to consider their respective $W^{m,p}-$norm as a random variable, as well as the $W^{m,p}$ approximation error with regards to $P_k$ finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1 < k_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-05T13:33:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N15",
      "65N30",
      "65N75"
    ],
    "keywords": [
      "exact solution",
      "finite element approximations",
      "probabilistic approach",
      "determinist pdes",
      "lagrange finite element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}