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

Abstract:

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 $\|u-u_h\|_1$ 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 $\|\boldsymbol {\sigma }-\boldsymbol {\sigma }_h\|_{H(div)}$ 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.