Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues [article]

A. H. Barnett, Andrew Hassell
2010 arXiv   pre-print
We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain Ω⊂^n with piecewise smooth boundary. We bound the distance between an arbitrary parameter E > 0 and the spectrum {E_j } in terms of the boundary L^2-norm of a normalized trial solution u of the Helmholtz equation (Δ + E)u = 0. We also bound the L^2-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all E greater than a small constant, and
more » ... mprove upon the best-known bounds of Moler--Payne by a factor of the wavenumber √(E). One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.
arXiv:1006.3592v1 fatcat:sykwejspvbgedjypyef2yejfhi