{
  "id": "2206.08493",
  "version": "v1",
  "published": "2022-06-17T00:28:13.000Z",
  "updated": "2022-06-17T00:28:13.000Z",
  "title": "Nonconforming finite elements for the Brinkman and $-\\text{curl}Δ \\text{curl}$ problems on cubical meshes",
  "authors": [
    "Qian Zhang",
    "Min Zhang",
    "Zhimin Zhang"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We propose two families of nonconforming elements on cubical meshes: one for the $-\\text{curl}\\Delta\\text{curl}$ problem and the other for the Brinkman problem. The element for the $-\\text{curl}\\Delta\\text{curl}$ problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter $\\nu$. The lowest-order elements for the $-\\text{curl}\\Delta\\text{curl}$ and the Brinkman problems have 48 and 30 degrees of freedom, respectively. The two families of elements are subspaces of $H(\\text{curl};\\Omega)$ and $H(\\text{div};\\Omega)$, and they, as nonconforming approximation to $H(\\text{gradcurl};\\Omega)$ and $[H^1(\\Omega)]^3$, can form a discrete Stokes complex together with the Lagrange element and the $L^2$ element.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-17T00:28:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonconforming finite elements",
      "cubical meshes",
      "brinkman problem",
      "uniformly stable finite element method",
      "discrete stokes complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}