{
  "id": "1805.10744",
  "version": "v1",
  "published": "2018-05-28T02:42:52.000Z",
  "updated": "2018-05-28T02:42:52.000Z",
  "title": "Error estimates for Galerkin finite element methods for the Camassa-Holm equation",
  "authors": [
    "D. C. Antonopoulos",
    "V. A. Dougalis",
    "D. E. Mitsotakis"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We also consider an initial-boundary-value problem on a finite interval for the system form of CH and analyze the convergence of its standard Galerkin semidiscretization. Using the fourth-order accurate, explicit, \"classical\" Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the `peakon' type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-28T02:42:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M60",
      "35Q53"
    ],
    "keywords": [
      "error estimates",
      "camassa-holm equation",
      "standard galerkin finite element method",
      "system form",
      "models one-way propagation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}