 Square modified Helmholtz Dirichlet-to-Neumann coefficients

ID: square-modified-helmholtz-dirichlet-to-neumann-coefficients

Let

R_{jn} = \int_{\partial Ω} f \partial_{n} v_{jn} d s

ρ_{jn} = e^{- ω_{n}} R_{jn}

δ_{n} = e^{- 2 ω_{n}}

. The opposite-side normal-trace coefficients obey

c_{B n} = \frac{ρ _{B n} - δ _{n} ρ _{T n}}{1 - δ _{n}^{2}}, c_{T n} = \frac{ρ _{T n} - δ _{n} ρ _{B n}}{1 - δ _{n}^{2}},

with the same formulas for

L, R

. When the known Dirichlet boundary condition is integrated exactly, these are exact coefficients, since orthogonality eliminates all other unknown modes.

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