{
  "id": "1608.01035",
  "version": "v1",
  "published": "2016-08-03T00:23:37.000Z",
  "updated": "2016-08-03T00:23:37.000Z",
  "title": "Wavenumber-explicit analysis for the Helmholtz $h$-BEM: error estimates and iteration counts for the Dirichlet problem",
  "authors": [
    "Jeffrey Galkowski",
    "Eike H. Müller",
    "Euan A. Spence"
  ],
  "categories": [
    "math.NA",
    "math.AP"
  ],
  "abstract": "We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method using the standard second-kind combined-field integral equations. We sharpen previously-existing results on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions, and we prove new bounds on how the number of GMRES iterations must grow with $k$ in order to have the error in the iterative solution bounded independently of $k$. Despite the fact that all the integral-operator bounds used in these arguments are sharp in their $k$-dependence, numerical experiments demonstrate that although the bounds on $h$ and the number of iterations are sufficient, they are not necessary for many geometries. We prove these results by proving new, sharp bounds on norms of the Helmholtz single- and double-layer boundary integral operators as mappings from $L^2(\\Gamma)\\rightarrow H^1(\\Gamma)$ (where $\\Gamma$ is the boundary of the obstacle), and then using these in conjunction with existing results. The new $L^2(\\Gamma)\\rightarrow H^1(\\Gamma)$ bounds are obtained using estimates on the restriction to the boundary of eigenfunctions of the Laplacian, building on recent work by the first author and collaborators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-03T00:23:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J05",
      "65F10",
      "65N22",
      "65N38",
      "65R20",
      "78A45"
    ],
    "keywords": [
      "dirichlet problem",
      "error estimates",
      "iteration counts",
      "wavenumber-explicit analysis",
      "standard second-kind combined-field integral equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}