On the exponent of exponential convergence of p-version FEM spaces

Abstract

We study the exponent of the exponential rate of convergence in terms of the number of degrees of freedom for various non-standard p-version finite element spaces employing reduced cardinality basis. More specifically, we show that serendipity finite element methods and discontinuous Galerkin finite element methods with total degree mathcal {P}_{p} basis have a faster exponential convergence with respect to the number of degrees of freedom than their counterparts employing the tensor product mathcal {Q}_{p} basis for quadrilateral/hexahedral elements, for piecewise analytic problems under p-refinement. The above results are proven by using a new p-optimal error bound for the L2-orthogonal projection onto the total degree mathcal {P}_{p} basis, and for the H1-projection onto the serendipity finite element space over tensor product elements with dimension d ≥ 2. These new p-optimal error bounds lead to a larger exponent of the exponential rate of convergence with respect to the number of degrees of freedom. Moreover, these results show that part of the basis functions in mathcal {Q}_{p} basis plays no roles in achieving the hp-optimal error bound in the Sobolev space. The sharpness of theoretical results is also verified by a series of numerical examples.