{
  "id": "1910.07768",
  "version": "v1",
  "published": "2019-10-17T08:37:07.000Z",
  "updated": "2019-10-17T08:37:07.000Z",
  "title": "Convergence analysis of a numerical scheme for a tumour growth model",
  "authors": [
    "Jerome Droniou",
    "Neela Nataraj",
    "Gopikrishnan Chirappurathu Remesan"
  ],
  "comment": "40 pages, 18 figures, appendix of some classical results",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.AP"
  ],
  "abstract": "We consider a one-spatial dimensional tumour growth model that consists of three variables; volume fraction and velocity of tumour cells, and nutrient concentration. The model variables form a coupled system of semi-linear advection equation (hyperbolic), simplified stationary Stokes equation (elliptic), and linear diffusion equation (parabolic) with appropriate conditions on the time-dependent boundary which is governed by an ordinary differential equation. An equivalent formulation in a fixed domain is used to overcome the difficulty associated with the time-dependent boundary in the original model. Though this reduces complexity of the model, the tight coupling between the component equations and their non-linear nature offer challenges in proving suitable \\emph{a priori} estimates. A numerical scheme that employs a finite volume method for the hyperbolic equation, a finite element method for the elliptic equation and a mass-lumped finite element method for the parabolic equation is developed. We establish the existence of a time interval $(0,T_{\\ast})$ over which the convergence of the scheme using compactness techniques is proved, thus proving the existence of a solution. Numerical tests and justifications that confirm the theoretical findings conclude the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-17T08:37:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M08",
      "65M12",
      "65M60",
      "35R37",
      "35J15",
      "35K10",
      "35L02",
      "46B50"
    ],
    "keywords": [
      "numerical scheme",
      "convergence analysis",
      "finite element method",
      "one-spatial dimensional tumour growth model",
      "non-linear nature offer challenges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}