Well-Conditioned Frames for High Order Finite Element Methods (Journal of Computational Mathematics, 2021)
The purpose of this paper is to discuss representations of high order $C^0$ finite elementspaces on simplicial meshes in any dimension. When computing with high order piecewisepolynomials the conditioning of the basis is likely to be important. The main result of thispaper is a construction of representations by frames such that the associated $L^2$ conditionnumber is bounded independently of the polynomial degree. To our knowledge, such a representation has not been presented earlier. The main tools we will use for the constructionis the bubble transform, introduced previously in [1], and properties of Jacobi polynomialson simplexes in higher dimensions. We also include a brief discussion of preconditionediterative methods for the finite element systems in the setting of representations by frames.