Orientation embedded high order shape functions for the exact sequence elements of all shapes

Abstract

A unified construction of high order shape functions is given for all four classical energy spaces (H 1 , H(curl), H(div) and L 2 ) and for elements of "all" shapes (segment, quadrilateral, triangle, hexahedron, tetrahedron, triangular prism and pyramid). The discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element. The shape functions are conforming, hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes. Expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape. The polynomial order is allowed to differ for each separate topological entity (vertex, edge, face or interior) in the mesh, so the shape functions can be used to implement local p adaptive finite element methods. Each topological entity may have its own orientation, and the shape functions can have that orientation embedded by a simple permutation of arguments.i

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

“…Since the pioneer work by Raviart and Thomas in 1977, this formulation received considerable attention, and different kinds of approximation spaces were constructed in the 1980s . In recent years, mixed formulations have received renovated interest due to their accuracy and local conservation property …”

Discrete least-squares finite element methods

“…The assembly of the normal equation for the DPG method can easily be incorporated into any finite element code supporting exact sequence conforming shape functions [3,36]. Recall that the DPG Gram matrix G is block diagonal.…”

“…should be locally derived from an exact sequence of spaces of polynomial order p, such as the ones found in [36] for triangles and quadrilaterals. Indeed, for a master square, (0, 1)…”

“…For regular enough elements, this is often possible when V = L 2 -indeed, this was verified in the FOSLS experiments in Section 5.1. The interested reader may consult [7,36] and references therein, to explore such orthogonal or sparsity-optimized shape functions.…”

“…FOSLS) and DLS methods (with ∆p ≥ 1). Moreover, because of the uniformity of the mesh and from using L 2 -orthogonal shape functions at the master element level [36], we produced a diagonal Gram matrix,…”