Practical unstructured splines: Algorithms, multi-patch spline spaces, and some applications to numerical analysis

•Efficient algorithms for unstructured simplex spline spaces.•Multi-patch spline domains including non-convex, non-simply-connected patches.•Fully unstructured multi-patch discontinuous Galerkin-isogeometric analysis.•Usual discontinuous Galerkin basis is obtained as a special case.•Block-diagonal mass matrix, useful for time-explicit hyperbolic problems.