Past exam of the finite element methods for partial differential equations course of the University of Oxford / 2023 / 3 / c

The finite-dimensional vector subspace $V_h\subset V$ is closed, so it is a Hilbert space with the inherited Sobolev norm. The bounded bilinear form, coercive bilinear form, and bounded linear functional estimates from part (b) restrict to $V_h$. Applying the Lax-Milgram theorem proves

For any $v_h\in V_h$, put $d_h=v_h-u_h\in V_h$. Expansion using symmetry of the bilinear form givesThus $u_h$ minimizes $J$ on $V_h$, uniquely by the coercive bilinear form estimate. Similarly, the continuous weak formulation gives the exact identityCombining the two identities provesMore precisely, $J(v_h)-J(u_h)=\frac{1}{2}\Vert v_h-u_h{\Vert }_a^2$ and $J(u_h)-J(u)=\frac{1}{2}\Vert u_h-u{\Vert }_a^2$, with the energy norm justified next. These identities connect the energy excess directly to the squared error.