On the exponent of exponential convergence of the $p$-version FEM spaces
Provides theoretical justification for using reduced-cardinality basis in p-version FEM to improve computational efficiency for analysts solving elliptic problems.
This paper proves that serendipity and discontinuous Galerkin finite element methods with total degree basis achieve faster exponential convergence per degree of freedom than tensor product basis for piecewise analytic problems under p-refinement, with theoretical sharpness confirmed by numerical examples.
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 $L^2$-orthogonal projection onto the total degree $\mathcal{P}_p$ basis, and for the $H^1$-projection onto the serendipity finite element space over tensor product elements with dimension $d\geq2$. 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.