being more than one cell is sometimes forbidden and rarely desired; the goal of some mesh improvement techniques (e.g. pillowing) is to remove these configurations. In some contexts, a distinction is made between a topological mesh and a geometric mesh whose embedding satisfies certain quality criteria. Important mesh variants that are not CW complexes include non-conformal meshes where cells do not meet strictly face-to-face, but the cells nonetheless partition the domain. An example of this is an octree, where an element face may be partitioned by the faces of adjacent elements. Such meshes are useful for flux-based simulations. In overset grids, there are multiple conformal meshes that overlap geometrically and do not partition the domain; see e.g., Overflow, the OVERset grid FLOW solver. So-called meshless or meshfree methods often make use of some mesh-like discretization of the domain, and have basis functions with overlapping support. Sometimes a local mesh is created near each simulation degree-of-freedom point, and these meshes may overlap and be non-conformal to one another. Implicit triangulations are based on a delta complex: for each triangle the lengths of its edges, and a gluing map between face edges. (please expand) High-order elements Many meshes use linear elements, where the mapping from the abstract to realized element is linear, and mesh edges are straight segments. Higher order polynomial mappings are common, especially quadratic. A primary goal for higher-order elements is to more accurately represent the domain boundary, although they have accuracy benefits in the interior of the mesh as well. One of the motivations for cubical meshes is that linear cubical elements have some of the same numerical advantages as quadratic simplicial elements. In the isogeometric analysis simulation technique, the mesh cells containing the domain boundary use the CAD representation directly instead of a linear or polynomial approximation. Mesh improvement Improving a mesh involves changing its discrete connectivity, the continuous geometric position of its cells, or both. For discrete changes, for simplicial elements one swaps edges and inserts/removes nodes. The same kinds of operations are done for cubical (quad/hex) meshes, although there are fewer possible operations and local changes have global consequences. E.g., for a hexahedral mesh, merging two nodes creates cells that are not hexes, but if diagonally-opposite nodes on a quadrilateral are merged and this is propagated into collapsing an entire face-connected column of hexes, then all remaining cells will still be hexes. In adaptive mesh refinement,