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

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 Pp basis have a faster exponential convergence with respect to the number of degrees of freedom than their counterparts employing the tensor product Qp 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 Pp 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 Qp 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.