according to the uniformity in arclength. The tensor (\[M-hessian\]) is optimal [@HS03] for the $L^2$ norm of linear interpolation error. In our computation, we use a least squares fitting strategy for Hessian recovery (e.g., see [@Kamens09PhD; @KaHu2013]).

We now describe the formulation of the MMPDE-based mesh equation. A key component of the formulation is the $\bM$-uniform mesh concept with which any (nonuniform) adaptive mesh is viewed as a uniform one in some metric. It is known [@Hua06; @HR11] that such an $\bM$-uniform mesh $\cTh$ approximately satisfies the equidistribution and alignment conditions $$\begin{aligned}
& |K| \sqrt{\det(\bM_K)} = \frac{\sigma_h |K_c|}{|\Omega_c|} , \quad \forall K \in \cTh
\label{eq-1}
\\
&\frac{1}{d} {\text{trace}}\left( 
    \left( F'_K \right)^{-1}
      \bM_K^{-1} \left (F'_K \right )^{-T}
      \right)
      =
      \det \left( 
    \left( F'_K \right)^{-1}
      \bM_K^{-1} \left (F'_K \right )^{-T}
      \right)^{\frac{1}{d}} , \quad \forall K \in \cTh
\label{ali-1}\end{aligned}$$ where $|K|$ is the volume of $K$, $\bM_K$ is the average of $\bM$ over $K$, $\det(\cdot)$ and ${\text{trace}}(\cdot)$ denote the determinant and trace of a matrix, respectively, $|K_c|$ is the volume of the element $K_c \in \cTc$ corresponding to $K$, $F'_K$ is the Jacobian matrix of the affine mapping $F_K: K_c \to K$, and $$\sigma_h = \sum_{K \in \cTh} |K| \sqrt{\det(\bM_K)}, \quad 
|\Omega_c | 