{
  "id": "2106.03191",
  "version": "v1",
  "published": "2021-06-06T17:48:15.000Z",
  "updated": "2021-06-06T17:48:15.000Z",
  "title": "An $L^p$-weak Galerkin method for second order elliptic equations in non-divergence form",
  "authors": [
    "Waixiang Cao",
    "Junping Wang",
    "Yuesheng Xu"
  ],
  "comment": "36 pages",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This article presents a new primal-dual weak Galerkin method for second order elliptic equations in non-divergence form. The new method is devised as a constrained $L^p$-optimization problem with constraints that mimic the second order elliptic equation by using the discrete weak Hessian locally on each element. An equivalent min-max characterization is derived to show the existence and uniqueness of the numerical solution. Optimal order error estimates are established for the numerical solution under the discrete $W^{2,p}$ norm, as well as the standard $W^{1,p}$ and $L^p$ norms. An equivalent characterization of the optimization problem in term of a system of fixed-point equations via the proximity operator is presented. An iterative algorithm is designed based on the fixed-point equations to solve the optimization problems. Implementation of the iterative algorithm is studied and convergence of the iterative algorithm is established. Numerical experiments for both smooth and non-smooth coefficients problems are presented to verify the theoretical findings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-06T17:48:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "65N12",
      "65N15",
      "35J15",
      "35B45"
    ],
    "keywords": [
      "second order elliptic equation",
      "non-divergence form",
      "optimization problem",
      "iterative algorithm",
      "primal-dual weak galerkin method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}