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 uh for the primary function u and σh for the flux σ = −A∇u. We prove that the accuracy of the approximate solution uh for the primary function u is weakly affected by the flux σ = −A∇u. That is, the bound for ‖u − uh‖1 is dependent on σ, but only through the best approximation for σ multiplied by a factor of meshsize h. Similarly, we provide that the bound for ‖σ− σh‖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.