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

## TL;DR

This paper demonstrates that certain non-standard finite element spaces, like serendipity and total degree basis methods, achieve faster exponential convergence rates than traditional tensor product bases for analytic problems, supported by new error bounds and numerical tests.

## Contribution

It introduces new $p$-optimal error bounds for specific finite element bases, showing improved exponential convergence rates and revealing that some basis functions are unnecessary for optimal error bounds.

## Key findings

- Serendipity and total degree basis methods have faster exponential convergence than tensor product bases.
- New $p$-optimal error bounds are established for projections onto these bases.
- Numerical examples confirm the theoretical convergence rates.

## 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 $L^2$-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.