= Solution
Consider a bounded <Lipschitz domain> $\Omega$ and homogeneous <Dirichlet boundary conditions> for
$$
-\nabla\cdot(A(x)\nabla u)+c(x)u=f.
$$
Assume $A$ is a bounded real <symmetric matrix> field with $\xi^TA(x)\xi\geq a_0|\xi|^2$ uniformly, $c\geq0$ is bounded, and $f$ defines a bounded linear functional on $V=H_0^1(\Omega)$, the <zero-boundary Sobolev space>. The <weak formulation> is
$$
a(u,v)=\ell(v)\quad(v\in V),\qquad
a(u,v)=\int_\Omega\nabla v^TA\nabla u+cuv,\quad\ell(v)=\langle f,v\rangle.
$$
The <Cauchy-Schwarz inequality> gives continuity $|a(u,v)|\leq M\|u\|_V\|v\|_V$. Uniform ellipticity and the <Poincaré inequality> give a <coercive bilinear form>, $a(v,v)\geq\alpha\|v\|_V^2$. The <Lax-Milgram theorem> therefore supplies a unique <weak solution>, with $\|u\|_V\leq\|\ell\|_{V'}/\alpha$. Its proof represents $a$ by a <bounded linear operator> $T$ using the <Riesz representation theorem>; coercivity gives an injective operator with closed image, and a zero <orthogonal complement> makes that image all of $V$. This yields existence, uniqueness and the displayed estimate.
For a <conforming finite element space> $V_h\subseteq V$, the <Galerkin method> finds $u_h\in V_h$ such that $a(u_h,v_h)=\ell(v_h)$ for every $v_h\in V_h$. With a basis $\phi_1,\ldots,\phi_N$, this is the linear system $KU=F$, where the <stiffness matrix> is $K_{ij}=a(\phi_j,\phi_i)$ and $F_i=\ell(\phi_i)$. Its <positive-definite matrix> property proves unique discrete solvability. Local support gives a <sparse matrix> through elementwise assembly.
For the symmetric problem, the <Ritz method> minimizes $J(v)=a(v,v)/2-\ell(v)$ over $V_h$. Differentiating in every direction $v_h$ gives precisely the <Galerkin method> equations. Conversely, if $u_h$ solves them, $J(u_h+w_h)-J(u_h)=a(w_h,w_h)/2$, proving it is the unique minimum. The same argument over $V$ gives $J(v)-J(u)=\|v-u\|_a^2/2$, where $\|v\|_a=\sqrt{a(v,v)}$ is the <energy norm>. Thus the <Ritz method> chooses the best trial function in that norm.
Subtracting the continuous and discrete equations gives <Galerkin orthogonality>, $a(u-u_h,v_h)=0$. For arbitrary $v_h\in V_h$ and $e=u-u_h$, it follows that $a(e,e)=a(e,u-v_h)$. Coercivity and continuity prove the <Céa lemma>:
$$
\alpha\|e\|_V^2\leq M\|e\|_V\|u-v_h\|_V,
\qquad
\|u-u_h\|_V\leq\frac M\alpha\inf_{v_h\in V_h}\|u-v_h\|_V.
$$
In the symmetric <energy norm>, the <Cauchy-Schwarz inequality> improves the constant to one. Equivalently, <Galerkin orthogonality> gives the <Pythagorean identity> $\|u-v_h\|_a^2=\|u-u_h\|_a^2+\|u_h-v_h\|_a^2$. These estimates establish stability and show that approximation properties of the trial space determine convergence.
For continuous piecewise-linear elements on a <shape-regular mesh> in dimensions at most three, a <finite element interpolation estimate> gives $\|u-I_hu\|_{H^1}\leq Ch\|u\|_{H^2}$ when $u\in H^2(\Omega)$. Combining it with the <Céa lemma> yields $O(h)$ <energy norm> error. More generally, degree-$p$ elements give $O(h^p)$ error in $H^1$ under the corresponding regularity and approximation assumptions. Mere mesh refinement cannot supply an order whose required solution regularity is absent.
An additional order in the <L2 norm> follows under a dual <elliptic regularity> assumption. Solve $a(v,z)=(e,v)_{L^2}$ and assume $\|z\|_{H^2}\leq C\|e\|_{L^2}$. The <Aubin–Nitsche duality argument> uses <Galerkin orthogonality> to obtain
$$
\|e\|_{L^2}^2=a(e,z-I_hz)\leq Ch\|e\|_{H^1}\|z\|_{H^2}
\leq Ch\|e\|_{H^1}\|e\|_{L^2}.
$$
Therefore $\|e\|_{L^2}\leq Ch\|e\|_{H^1}$, giving $O(h^2)$ for the piecewise-linear case with $u\in H^2$. This improvement explicitly requires regularity of the dual problem.
As a concrete example, solve $-u''=1$ on $(0,1)$ with zero endpoint values. On a uniform mesh of spacing $h$, the <piecewise-linear hat functions> give $K_{jj}=2/h$, $K_{j,j\pm1}=-1/h$ and $F_j=h$. The discrete equations are $2U_j-U_{j-1}-U_{j+1}=h^2$, whose solution is $U_j=x_j(1-x_j)/2$. The exact solution is $u(x)=x(1-x)/2$, so $u_h$ is its piecewise-linear nodal interpolant. On each interval $[x_j,x_{j+1}]$,
$$
u-u_h=\frac12(x-x_j)(x_{j+1}-x).
$$
Integrating the squared error and squared derivative error gives the explicit rates
$$
\boxed{\|u-u_h\|_{H_0^1}=\frac h{\sqrt{12}},\qquad\|u-u_h\|_{L^2}=\frac{h^2}{\sqrt{120}},}
$$
where the $H_0^1$ norm here is $\|v'\|_{L^2}$.
For nonsymmetric coercive problems, the <Galerkin method>, <Galerkin orthogonality> and the <Céa lemma> still apply, but the symmetric quadratic minimization interpretation of the <Ritz method> generally does not. Nonconforming trial spaces, inexact integration and noncoercive equations need additional arguments beyond the conforming coercive theory proved here.
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