Aubin–Nitsche duality argument

For a symmetric elliptic variational problem with Galerkin orthogonality, let $e=u-u_h$ and solve the dual problem $a(v,z)=(e,v{)}_{L^2}$ for every $v\in H_0^1(\Omega )$. If elliptic regularity gives $\Vert z{\Vert }_{H^2}\le C\Vert e{\Vert }_{L^2}$, then a finite element interpolation estimate givesThus $\Vert e{\Vert }_{L^2}\le Ch\Vert e{\Vert }_{H^1}$. Combining this with the Céa lemma and an $O(h)$ energy norm error gives an $O(h^2)$ L2 norm error. The additional order depends on the dual regularity assumption, which can fail on unsuitable domains.

For $-(p{u}^{\prime }{)}^{\prime }+qu=f$ with one homogeneous Neumann condition and one fixed Dirichlet value, the dual problem has error forcing and homogeneous corresponding boundary data. Galerkin orthogonality with its nodal interpolant yields an element-residual identity with no jump terms, because the interpolation error vanishes at every node. With $p_{\ast }=\min p>0$, $q_{\ast }=\min q>0$, $q^{\ast }=\max q$, one reliable constant isThe proof combines a dual $H^2$ estimate with two interval Poincaré inequalities. Constants are independent of the nonuniform mesh.