= Solution
The <finite element method> replaces an infinite-dimensional variational problem by a finite-dimensional one, usually using functions that are polynomial on small mesh elements. <Ritz method> and <Galerkin method> describe how the discrete equations are selected; neither intrinsically requires piecewise polynomials, though those spaces make assembly local and sparse.
For a concrete realization of the two-point problem, choose homogeneous <Dirichlet boundary conditions> at both endpoints of $(0,1)$. The differential expression alone is not a complete boundary value problem, so this boundary choice is an explicit assumption. Let $p\in W^{1,\infty}$ with $p\geq p_0>0$, $q\in L^\infty$ with $q\geq0$, and $f\in L^2$. Use the <zero-boundary Sobolev space> $V=H_0^1(0,1)$ with norm $\|v\|_V=\|v'\|_2$. By the <Poincare inequality> this is equivalent to its full first-order Sobolev norm. Multiplying by a test function and applying <integration by parts> gives
$$
a(u,v)=\ell(v)\quad(v\in V),\qquad
a(u,v)=\int_0^1(pu'v'+quv)\,dx,\quad
\ell(v)=\int_0^1fv\,dx.
$$
The endpoint terms vanish by the essential boundary condition. The form is a <symmetric bilinear form>, bounded by
$$
|a(u,v)|\leq M\|u\|_V\|v\|_V,\qquad
M=\|p\|_\infty+C_P^2\|q\|_\infty,
$$
and it is a <coercive bilinear form>:
$$
a(v,v)\geq p_0\|v\|_V^2.
$$
The load is a bounded linear functional. The <Lax-Milgram theorem> gives a unique weak solution and a bound $\|u\|_V\leq\|\ell\|_{V^*}/p_0$. Under the stated coefficient regularity it solves the differential equation in the usual weak sense.
The <Ritz method> minimizes the energy
$$
J(v)=\tfrac12a(v,v)-\ell(v)
$$
over $V$, or over a conforming finite-dimensional subspace $V_h\subset V$. Its first variation is $a(v,w)-\ell(w)$; symmetry and coercivity make the functional strictly convex. In fact, if $u$ is the weak solution,
$$
J(v)-J(u)=\tfrac12a(v-u,v-u)\geq0.
$$
Thus the energy minimizer is exactly the weak solution. The <Galerkin method> instead directly asks for $u_h\in V_h$ with $a(u_h,v_h)=\ell(v_h)$ for all $v_h\in V_h$. For this symmetric coercive problem, <Ritz-Galerkin equivalence for a symmetric coercive form> shows that the discrete minimizer and Galerkin solution coincide. For a nonsymmetric problem, a Galerkin formulation can still apply but the same quadratic minimization generally cannot represent its full <bilinear form>; appropriate coercivity or inf-sup conditions must then justify the chosen spaces.
Subtract the continuous and discrete weak equations to obtain <Galerkin orthogonality> $a(u-u_h,v_h)=0$. Since $a$ defines the <energy norm> $\|v\|_a=\sqrt{a(v,v)}$, the <Pythagorean identity> gives
$$
\|u-v_h\|_a^2=\|u-u_h\|_a^2+\|u_h-v_h\|_a^2,
\qquad
\boxed{\|u-u_h\|_a=\inf_{v_h\in V_h}\|u-v_h\|_a.}
$$
In a general reference norm, the <Céa lemma> yields
$$
\boxed{\|u-u_h\|_V\leq\frac{M}{p_0}
\inf_{v_h\in V_h}\|u-v_h\|_V.}
$$
The proof uses $a(e,e)=a(e,u-v_h)$ together with coercivity and continuity. Dense approximation spaces therefore imply convergence; the estimate reduces a numerical error problem to an approximation problem.
For implementation, partition the interval by nodes $0=x_0<\cdots<x_N=1$. Choose continuous piecewise-linear functions and the interior <piecewise-linear hat functions> $\phi_j$ as a basis, with the two boundary coefficients prescribed. Write $u_h=\sum_jU_j\phi_j$. The coefficients satisfy
$$
\boxed{KU=F,\qquad K_{ij}=\int_0^1(p\phi_j'\phi_i'+q\phi_j\phi_i)\,dx,
\quad F_i=\int_0^1f\phi_i\,dx.}
$$
The <stiffness matrix> is symmetric positive definite: $U^TKU=a(u_h,u_h)>0$ for nonzero interior coefficients. The local supports make it tridiagonal in this one-dimensional linear-element case.
On an element of length $h_e$, for constant element coefficients $p_e,q_e$, the two-node <matrix> and constant-load vector are
$$
K_e=\frac{p_e}{h_e}
\begin{pmatrix}1&-1\\-1&1\end{pmatrix}
+\frac{q_eh_e}{6}
\begin{pmatrix}2&1\\1&2\end{pmatrix},
\qquad
F_e=\frac{f_eh_e}{2}\begin{pmatrix}1\\1\end{pmatrix}.
$$
For variable coefficients, integrate $p,q,f$ against the same shape functions rather than silently replacing them by constants. Element contributions are added at shared nodes, and boundary values are eliminated or lifted into the right side. A sparse direct solve or a suitable iterative method gives the nodal coefficients. Numerical quadrature should preserve the required accuracy and, with positive coefficients and weights, the positive energy structure.
A <finite element interpolation estimate> gives $\inf_{v_h}\|u-v_h\|_{H^1}\leq Ch\|u\|_{H^2}$ for these elements, where $h$ is the largest element size and the solution has the stated regularity. The <Céa lemma> then gives \b[first-order energy error]. If the dual elliptic problem has $H^2$ regularity, the <Aubin–Nitsche duality argument> gives \b[second-order $L^2$ error]: for $e=u-u_h$, solve $a(v,z)=(e,v)_{L^2}$, then use $a(e,z)=a(e,z-I_hz)$ to gain one further factor of $h$. Higher-order elements improve rates when the solution is sufficiently smooth.
Nonzero Dirichlet data are handled by a boundary lifting and an affine trial space. <Neumann boundary conditions> enter naturally through the integrated boundary term; Robin terms modify both the form and the load. With pure Neumann conditions and $q=0$, constants lie in the kernel: existence needs the load/flux compatibility condition, uniqueness needs a mean constraint, and the preceding Dirichlet coercivity argument cannot simply be reused. \b[Conforming approximation, boundary conditions and coercivity are the ingredients connecting the finite-element construction to a justified Ritz or Galerkin solution.]
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