assumptions on the regularity of the PDE’s solution. We control the nonlinear terms by proving a uniform $L^{3+\delta}$ regularity for the discrete harmonic vector fields in curved polyhedra, establishing a discrete Sobolev compact embedding inequality ${\bf H}_h({\rm curl,div})\hookrightarrow
\hookrightarrow {\bf L}^{3+\delta}$ for the functions in the Nédélec element space, and introducing a $\ell^2(W^{1,3+\delta})$ estimate for fully discrete finite element solutions of parabolic equations, where $\delta>0$ is some constant which depends on the given domain.

Main results {#MainR}
============

A decoupled mixed FEM with bounded discrete energy
--------------------------------------------------

In this subsection, we introduce our assumptions on the domain and define the fully discrete finite element method to be considered in this paper. Then we introduce a discrete energy function (different from the free energy) and sketch a proof for a basic energy inequality satisfied by the finite element solution.

A [*curved polyhedron (or polygon)*]{} is a bounded Lipschitz domain $\Omega\subset\R^3$ (or $\Omega\subset\R^2$), possibly nonconvex and multi-connected, such that its boundary is locally $C^\infty$-isomorphic to the boundary of a polyhedron [@BS87], and there are $\frak M$ pieces of surfaces $\Sigma_1$, $\cdots$, $\Sigma_{\frak M}$ transversal to $\partial\Omega$ such that $\Sigma_i\cap\Sigma_j=\emptyset$ for $i\neq j$ and the domain $\Omega_0:=\Omega\backslash\Sigma$ is simply connected, where $\Sigma=\cup_{j=1}^{\frak M}\Sigma_j$ 