Estimates on Neumann Eigenfunctions at the Boundary, and the “method of Particular Solutions” for Computing Them

نویسندگان

  • ANDREW HASSELL
  • ALEX BARNETT
چکیده

We consider the method of particular solutions for numerically computing eigenvalues and eigenfunctions of the Laplacian on a smooth, bounded domain Ω in Rn with either Dirichlet or Neumann boundary conditions. This method constructs approximate eigenvalues E, and approximate eigenfunctions u that satisfy ∆u = Eu in Ω, but not the exact boundary condition. An inclusion bound is then an estimate on the distance of E from the actual spectrum of the Laplacian, in terms of (boundary data of) u. We prove operator norm estimates on certain operators on L(∂Ω) constructed from the boundary values of the true eigenfunctions, and show that these estimates lead to sharp inclusion bounds in the sense that their scaling with E is optimal. This is advantageous for the accurate computation of large eigenvalues. The Dirichlet case can be treated using elementary arguments and will appear in [5], while the Neumann case seems to require much more sophisticated technology. We include preliminary numerical examples for the Neumann case.

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تاریخ انتشار 2011