On the exponent of exponential convergence of the $p$-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 $L^2$-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 $d\ge 2$. 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.

References in corpus (1)

• A priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems