= Solution
The starting point of a conforming <finite element method> is a <weak formulation> in a <Hilbert space> $X$. For example, a homogeneous <Dirichlet boundary condition> on a bounded <Lipschitz domain> gives $X=H_0^1(\Omega)$. One seeks
$$
a(u,v)=\ell(v)\qquad(v\in X),
$$
where $a$ is a <bounded bilinear form>, $|a(w,v)|\leq L\|w\|_X\|v\|_X$, and $\ell\in X^*$ is a bounded <linear functional>. A <conforming finite element space> $X_h\subset X$ is finite dimensional and is typically built from functions that are polynomial on each element of a <finite element mesh>. With a basis $\varphi_j$, the <Galerkin method> imposes the same identity only for $v_h\in X_h$, giving
$$
\boxed{Kc=F,\qquad K_{ij}=a(\varphi_j,\varphi_i),\quad F_i=\ell(\varphi_i).}
$$
The <stiffness matrix> is sparse when basis supports overlap only locally. Local element integrals assemble its entries; the <mass matrix> represents an $L^2$ term. This separates the choice of trial functions from the variational principle determining their coefficients.
The fundamental well-posedness theorem is the <Lax-Milgram theorem>: if $a(v,v)\geq\gamma\|v\|_X^2$ for some $\gamma>0$, there is a unique solution for each $\ell\in X^*$, with $\|u\|_X\leq\|\ell\|_{X^*}/\gamma$. The same theorem on $X_h$ gives a unique discrete solution. The constants must be independent of the mesh for uniform <stability of a numerical method>. Boundary conditions are built into the trial space when they are essential constraints; flux conditions arise through boundary terms in <integration by parts> and are natural constraints in the <weak formulation>.
When $a$ is symmetric and a <coercive bilinear form>, the <Ritz method> minimizes
$$
\mathcal E(v)=\tfrac12a(v,v)-\ell(v)
$$
over $X_h$. Its first variation is exactly the <Galerkin method>, so \b[Ritz minimization and Galerkin testing coincide for symmetric coercive problems]. The <energy norm> $\|v\|_a=\sqrt{a(v,v)}$ turns the approximation into an <orthogonal projection>. Indeed, subtracting the exact and discrete weak equations gives <Galerkin orthogonality>, $a(u-u_h,v_h)=0$. For every $v_h\in X_h$ the <Pythagorean identity> becomes
$$
\|u-v_h\|_a^2=\|u-u_h\|_a^2+\|u_h-v_h\|_a^2,
$$
so
$$
\boxed{\|u-u_h\|_a=\inf_{v_h\in X_h}\|u-v_h\|_a.}
$$
This exact best-approximation property is the central advantage of the <Ritz method>.
Symmetry is unnecessary for the <Galerkin method>. For any bounded, coercive form, the <Céa lemma> gives
$$
\boxed{\|u-u_h\|_X\leq\frac L\gamma
\inf_{v_h\in X_h}\|u-v_h\|_X.}
$$
To prove it, put $e=u-u_h$. <Galerkin orthogonality> yields $a(e,e)=a(e,u-v_h)$, so coercivity and boundedness give $\gamma\|e\|_X^2\leq L\|e\|_X\|u-v_h\|_X$. Divide when $e\neq0$ and take the <infimum>. Thus approximation capability plus uniform coercivity gives convergence. In contrast, minimizing $\tfrac12a(v,v)-\ell(v)$ for a nonsymmetric form differentiates its symmetric part and generally does not solve the original weak equation.
A concrete symmetric example is the one-dimensional reaction-diffusion problem from Question 5. Continuous <piecewise-linear hat functions> give the diffusion <stiffness matrix> $S$ with diagonal $2/h$ and neighbouring entries $-1/h$, and the <mass matrix> $M$ with diagonal $2h/3$ and neighbouring entries $h/6$. Therefore $K=S+M$ and $F_i=ih^2$. Its <positive-definite matrix> is the coordinate form of the continuous energy. The exact smooth solution $u(x)=x-\sinh(x)/\sinh1$ makes this an explicit test of the method, not just an abstract existence result.
For a nonsymmetric example, take positive $\varepsilon$, constant real $\beta$, and
$$
-\varepsilon u''+\beta u'+u=f,\qquad u(0)=u(1)=0.
$$
Its form is $a(u,v)=\varepsilon\int u'v'+\beta\int u'v+\int uv$. Since $\int v'v=0$, it has coercivity constant $\min(\varepsilon,1)$ in the full $H^1$ norm, although it is not symmetric when $\beta\neq0$. The <Galerkin method> remains well posed. On the uniform hat basis, $K=\varepsilon S+\beta C+M$, where $C_{i,i+1}=1/2$, $C_{i,i-1}=-1/2$ and $C_{ii}=0$. The <skew-symmetric matrix> $C$ cancels from $c^TKc$, proving nonsingularity. A Ritz energy would omit precisely this advection contribution. Small $\varepsilon$ may nevertheless make the approximation constant large and produce poorly resolved layers; algebraic solvability is not an accuracy guarantee.
Quantitative convergence uses a <finite element interpolation estimate>. On a <shape-regular mesh>, for piecewise-linear conforming elements and $u\in H^2(\Omega)$ in the usual low-dimensional setting,
$$
\|u-I_hu\|_{H^1}\leq Ch\|u\|_{H^2},\qquad
\|u-I_hu\|_{L^2}\leq Ch^2\|u\|_{H^2}.
$$
The <Céa lemma> therefore gives an $O(h)$ energy error. The <Aubin–Nitsche duality argument> improves the <L2 norm> error when the dual elliptic problem has $H^2$ regularity: solve $a(v,z)=(e,v)_{L^2}$ and use
$$
\|e\|_{L^2}^2=a(e,z-I_hz)
\leq Ch\|e\|_{H^1}\|z\|_{H^2}
\leq Ch\|e\|_{H^1}\|e\|_{L^2}.
$$
Consequently
$$
\boxed{\|u-u_h\|_{H^1}=O(h),\qquad
\|u-u_h\|_{L^2}=O(h^2),}
$$
with constants depending on the solution and regularity bounds. The second estimate is conditional on dual regularity; corners or insufficient data regularity can reduce the rate.
The same <Galerkin method> handles evolution equations. For the homogeneous <heat equation>, the <semidiscrete finite element heat equation> is $M\dot c+Sc=0$. Because $M$ is a <positive-definite matrix> and $S$ is a <positive semidefinite matrix>,
$$
\frac{d}{dt}(c^TMc)=-2c^TSc\leq0.
$$
The decreasing <quadratic form> is exactly the <L2 norm> squared of the finite-element function. This separates spatial <stability of a numerical method> from the subsequent time integrator, just as in the <Schrödinger equation> example. \b[Conformity, coercivity and approximation estimates together explain convergence.]
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