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In this paper, a general version of coupled-mode theory for frequency-domain scattering problems in integrated optics is proposed. As a prerequisite, a physically reasonable field template is required, that typically combines modes of the optical channels in the structure with coefficient functions of in principle arbitrary coordinates. Upon 1D discretizations of these amplitude functions into finite elements, a Galerkin procedure reduces the problem to a system of linear equations in the element coefficients, where given input amplitudes are included. Smooth approximate solutions are obtained by solving the system in a least squares sense. The versatility of the approach is illustrated by means of a series of 2D examples, including a perpendicular crossing of waveguides, and a grating-assisted rectangular resonator. As an Appendix, we show that, alternatively, a similar procedure can be derived by variational means, i.e., by restricting a suitable functional representation of the full 2D/3D vectorial scattering problem (with transparent influx boundary conditions for inhomogeneous exterior) to the respective field templates.