Benchmark configurations for quantitative validation and comparison of incompressible interfacial flow codes, which model two-dimensional bubbles rising in liquid columns, are proposed. The benchmark quantities: circularity, center of mass, and mean rise velocity are defined and measured to monitor convergence toward a reference solution. Comprehensive studies are undertaken by three independent research groups, two representing Eulerian level set finite-element codes and one representing an arbitrary Lagrangian-Eulerian moving grid approach.\ud The first benchmark test case considers a bubble with small density and viscosity ratios, which undergoes moderate shape deformation. The results from all codes agree very well allowing for target reference values to be established. For the second test case, a bubble with a very low density compared to that of the surrounding fluid, the results for all groups are in good agreement up to the point of break up, after which all three codes predict different bubble shapes. This highlights the need for the research community to invest more effort in obtaining reference solutions to problems involving break up and coalescence.\ud Other research groups are encouraged to participate in these benchmarks by contacting the authors and submitting their own data. The reference data for the computed benchmark quantities can also be supplied for validation purposes

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Comparative benchmark results for different solution methods for fluidstructure interaction problems are given which have been developed as collaborative project in the DFG Research Unit 493. The configuration consists of a laminar incompressible channel flow around an elastic object. Based on this benchmark configuration the numerical behavior of different approaches is analyzed exemplarily. The methods considered range from decoupled approaches which combine Lattice Boltzmann methods with hp-FEM techniques, up to strongly coupled and even fully monolithic approaches which treat the fluid and structure simultaneously.

A simple nonconforming quadrilateral Stokes element based on "rotated" multi-linear shape functions is analyzed. On strongly nonuniform meshes the usualparametric version of this element suffers from a lack of consistency, while its nonparametric counterpart turns out to be convergent with optimal orders. This theoretical result is confirmed by numerical tests.

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