Weak coupling of solutions of first-order least-squares method

A theoretical analysis of a first-order least-squares finite element method for second-order self-adjoint elliptic problems is presented. We investigate the coupling effect of the approximate solutions $u_h$ for the primary function $u$ and $\boldsymbol{\sigma}_h$ for the flux $\boldsymbol{\sigma}=-\mathcal A\nabla u$. We prove that the accuracy of the approximate solution $u_h$ for the primary function $u$ is weakly affected by the flux $\boldsymbol{\sigma}=-\mathcal A\nabla u$. That is, the bound for $\Vert u-u_h\Vert _1$ is dependent on $\boldsymbol{\sigma}$, but only through the best approximation for $\boldsymbol{\sigma}$ multiplied by a factor of meshsize $h$. Similarly, we provide that the bound for $\Vert\boldsymbol{\sigma}-\boldsymbol{\sigma}_h\Vert _{H(div)}$ is dependent on $u$, but only through the best approximation for $u$ multiplied by a factor of the meshsize $h$. This weak coupling is not true for the non-selfadjoint case. We provide the numerical experiment supporting the theorems in this paper.