= Galerkin orthogonality
{c}
{title2=$a(u-u_h,v_h)=0$}
Let $V_h\subset V$ be a <conforming finite element space>. Suppose the exact solution satisfies $a(u,v)=\ell(v)$ for every $v\in V$ and its <Galerkin method> approximation satisfies $a(u_h,v_h)=\ell(v_h)$ for every $v_h\in V_h$, using the same <bilinear form> and <linear functional>. Subtracting the two identities gives <Galerkin orthogonality>:
$$
a(u-u_h,v_h)=0\qquad(v_h\in V_h).
$$
This identity itself does not require symmetry. Its interpretation as an <orthogonal projection> in an <energy norm> does require a bounded symmetric <coercive bilinear form>: then $a$ is an <inner product>, $e=u-u_h$ is perpendicular to every available discrete correction, and
$$
\|u-v_h\|_a^2=\|u-u_h\|_a^2+\|u_h-v_h\|_a^2\qquad(v_h\in V_h).
$$
Consequently $u_h$ is the unique closest point in $V_h$ in the <energy norm>. This <Pythagorean theorem> in a function space explains the beauty of the <Galerkin method>: integrated local equations compute a global best approximation. The <Céa lemma> supplies the corresponding estimate in a chosen reference norm.
For the <finite element solution of the one-dimensional Poisson equation>, $a(w,v)=\int_0^1w'v'\,dx$ and $\|w\|_a=\|w'\|_{L^2(0,1)}$. Testing the error against the $i$th <piecewise-linear hat function> on a uniform <mesh> gives
$$
\frac1h\int_{x_{i-1}}^{x_i}e'\,dx
-\frac1h\int_{x_i}^{x_{i+1}}e'\,dx=0.
$$
In this model $e$ vanishes at every node, so each of these <integrals> is individually zero, although $e$ is positive inside every element. <Galerkin orthogonality> controls the derivative-weighted balance; it does not say that the pointwise error vanishes.
The exact identity assumes conformity and exact evaluation of the continuous forms, including any <numerical integration> that is used. If assembly instead uses $a_h$ and $\ell_h$, subtraction gives the consistency defect
$$
a(u-u_h,v_h)=(\ell-\ell_h)(v_h)+(a_h-a)(u_h,v_h).
$$
Thus inexact <numerical integration> need not preserve the projection. In the <Petrov-Galerkin method> the error is tested against a different test space, and for a nonsymmetric <bilinear form> the displayed identity need not describe perpendicularity in any <energy norm>.
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