Complex fluid–fluid interfaces determine for a large part the macroscopic material properties of foams and emulsions that appear in applications such as food, materials processing and consumer care products. As a step towards predicting these properties, a 2D axisymmetric and a 3D finite element model have been developed for simulating the dynamics of a single Newtonian drop in a Newtonian matrix fluid with a rheologically complex sharp interface in between. Interfaces with constant interfacial tension and with viscous, elastic and viscoelastic extra interfacial stresses are considered. The model has been validated by means of the method of manufactured solutions and by comparison with results from other studies using different discretisation methods. Higher-order convergence in space and time is obtained, demonstrating correct implementation of our numerical methods. Benchmark solutions for a drop with a Kelvin–Voigt interface under simple shear flow have been provided. Compared to a viscous interface, the drop deformation becomes smaller and the drop becomes less oriented in the direction of flow if interfacial elasticity is added.