Implementation of continuous hp-adaptive finite element spaces without limitations on hanging sides and distribution of approximation orders

Adaptive techniques using hp refinement are known to be one of the most efficient methodologies to accelerate the convergence of finite element algorithms. However, the implementation of computational tools for the development of hp-adaptive algorithms is intricate and depends strongly on the data structure. There exist few computational environments available to the scientific finite element community capable to implement hp-adaptive approximation spaces for the complete family of finite element topologies, and which implement hanging sides. This article describes a methodology for the development of continuous hp-adaptive finite element approximation spaces, without constraints on the refinement strategy concerning the difference of levels and approximation orders between neighboring elements. The shape functions are hierarchical, and the coefficient constraints associated with hanging sides that can occur in non-conformal geometric meshes are defined using L2-projections. The topological and functional aspects of the construction are described in one, two and three dimensions, for a variety of geometric entities (line, triangle, quadrilateral, tetrahedron, pyramid, prism, and hexahedron). The implementation is demonstrated in the object-oriented scientific computational environment NeoPZ (http://github.com/labmec/neopz). NeoPZ is a general finite element approximation software, which incorporates a variety of variational formulations. Validation of the refinement methodology is demonstrated by two and three dimensional numerical experiments.