High-order Galerkin methods exhibit superior convergence properties and computational efficiency. To achieve optimal performance, they require highly regular meshes, including curved elements. High-order meshing capabilities, however, are often not available in commercial mesh generators. And if they are, they often lack robustness for complex geometries, leading to time-consuming manual user intervention. In addition, many industrial applications involve moving parts, e.g., pistons, valves or rotors, which necessitates frequent remeshing. 
High-order embedded methods are an emerging technology that has the potential to naturally overcome many of these issues. Embedded methods, also known as fictitious domain, immersed or cut-cell methods, enable automatic mesh generation by intersecting a simple background mesh with the actual geometry, where the geometry itself can be described by different geometric models – e.g. by spline surfaces or level-sets. They are also very promising with respect to moving geometries, since they are able to accommodate topology changes without additional user input or remeshing.
Obviously, high-order embedded domain methods do not come for free and require additional techniques beyond established Galerkin technology. In this context, this mini-symposium aims at bringing together researchers from across the computational engineering and mathematics communities to discuss new methodological developments and results that support high-order embedded analysis. Questions of interest include, but are not limited to, accurate quadrature variants in cut cells, handling of small cut cells and their negative effects, interaction with different geometric models, accurate and stable imposition of constraints along embedded boundaries and interfaces, the use of specially tailored basis functions, and new fields of applications in solid and fluid mechanics, multi-physics, and optimization. Emphasis is also placed on the treatment of evolving/moving domains in a high-order context, where the changing shape and topology, the addition and deletion of cut cells, time integration, and maintaining high-order accuracy represent particular challenges.