From the literature, it is known that the choice of basis functions in hp-FEM heavily influences the computational cost in order to obtain an approximate solution. Depending on the choice of the reference element, suitable tensor product like basis functions of Jacobi polynomials with different weights lead to optimal properties due to condition number and sparsity. This paper presents biorthogonal basis functions to the primal basis functions mentioned above. The authors investigate hypercubes and simplices as reference elements, as well as the cases of $H^1$ and H(Curl). The functions can be expressed sums of tensor products of Jacobi polynomials with maximal two summands.

翻译：文献表明，在hp-FEM中，基函数的选择对求解近似解的计算成本具有显著影响。根据参考单元的选择，加权雅可比多项式的张量积型基函数可因条件数和稀疏性而实现最优特性。本文提出与上述原始基函数对偶的双正交基函数。作者研究了超立方体和单纯形作为参考单元的情形，并覆盖了$H^1$和H(Curl)空间的应用场景。所构造函数可表示为至多两项雅可比多项式张量积之和的形式。