We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a
bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound
the distance between an arbitrary parameter E>0 and the spectrum {Ej}
in terms of the boundary L2-norm of a normalized trial solution u of the
Helmholtz equation (Δ+E)u=0. We also bound the L2-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
improve 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.Comment: 18 pages, 3 figure