Context. In type Ia supernova (SN Ia) envelopes a huge number of lines of different elements overlap within their thermal Doppler widths, and this problem is exacerbated by the circumstance that up to 20% of these lines can have a line optical depth greater than 1. The stagnation of the lambda iteration in such optically thick cases is one of the fundamental physical problems inherent in the iterative solution of the non-LTE problem, and the failure of a lambda iteration to converge is a point of crucial importance whose physical significance must be understood completely. Aims. We discuss a general problem related to radiative transfer under the physical conditions of supernova ejecta that involves a failure of the usual non-LTE iteration scheme to converge when multiple strong opacities belonging to different physical transitions come together, similar to the well-known situation where convergence is impaired even when only a single process attains large optical depths. The convergence problem is independent of the chosen frequency and depth grid spacing, independent of whether the radiative transfer is solved in the comoving or observer's frame, and independent of whether a common complete-linearization scheme or a conventional accelerated lambda iteration (ALI) is used. The problem appears when all millions of line transitions required for a realistic description of SN Ia envelopes are treated in the frame of a comprehensive non-LTE model. The only way out of this problem is a complete-linearization approach which considers all ions of all elements simultaneously, or an adequate generalization of the established ALI technique which accounts for the mutual interaction of the strong spectral lines of different elements and which thereby unfreezes the "stuck" state of the iteration. Methods. The physics of the atmospheres of SN Ia are strongly affected by the high-velocity expansion of the ejecta, dominating the formation of the spectra at all wavelength ranges. Thus, hydrodynamic explosion models and realistic model atmospheres that take into account the strong deviation from local thermodynamic equilibrium are necessary for the synthesis and analysis of the spectra. In this regard one of the biggest challenges we have found in the modeling of the radiative transfer in SN Ia is the fact that the radiative energy in the UV has to be transferred only via spectral lines into the optical regime in order to be able to leave the ejecta. However, convergence of the model toward a state where this is possible is impaired when using the standard procedures. We report on improvements in our approach of computing synthetic spectra for SN Ia with respect to (i) an improved and sophisticated treatment of many thousands of strong lines that interact intricately with the "pseudo-continuum" formed entirely by Doppler-shifted spectral lines, (ii) an improved and expanded atomic database, and (iii) the inclusion of energy deposition within the ejecta arising from the radioactive decay of mostly 56 Ni and 56 Co. Results....