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In this thesis, the novel Tensor Diffusion approach for the numerical simulation of viscoelastic fluid flows is introduced. Therefore, it is assumed that the extra-stress tensor can be decomposed into a product of the strain-rate tensor and a (nonsymmetric) tensor-valued viscosity function. As a main potential advantage, which can be demonstrated for fully developed channel flows, the underlying complex material behaviour can be explicitly described by means of the so-called Diffusion Tensor. Consequently, this approach offers the possibility to reduce the complete nonlinear viscoelastic three-field model to a generalised Stokes-like problem regarding the velocity and pressure fields, only. This is enabled by inserting the Diffusion Tensor into the momentum equation of the flow model, while the extra-stress tensor or constitutive equation can be neglected. As a result, flow simulations of viscoelastic fluids could be performed by applying techniques particularly designed for solving the (Navier-)Stokes equations, which leads to a way more robust and efficient numerical approach. But, a conceptually improved behaviour of the numerical scheme concerning viscoelastic fluid flow simulations may be exploited with respect to discretisation and solution techniques of typical three- or four-field formulations as well. In detail, an (artificial) diffusive operator, which is closely related to the nature of the underlying material behaviour, is inserted into the (discrete) problem by means of the Diffusion Tensor. In this way, certain issues particularly regarding the flow simulation of viscoelastic fluids without a Newtonian viscosity contribution, possibly including realistic material and model parameters, can be resolved. In a first step, the potential benefits of the Tensor Diffusion approach are illustrated in the framework of channel flow configurations, where several linear and nonlinear material models are considered for characterising the viscoelastic material behaviour. In doing so, typical viscoelastic flow ...