A midpoint-quadrature P1 solve has an explicit Strang error budget

Assume Countable Choice. Put $V:=H_0^1(0,1;\mathbb{R})$ with its $H^1$ norm, and let $V_h\subseteq V$ be a finite-dimensional conforming piecewise-linear space on an interval partition with maximum cell width $h$. Let $u\in C^2([0,1])\cap V$ and assume its nodal interpolant $I_{h}u$ belongs to $V_h$. Let $A\in C^2([0,1])$ satisfy $0<\alpha \le A(x)\le M$, and write $C_A:=\mathrm{\parallel }{A}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(0,1)}$. Define $a(v,w)={\int }_0^{1}A{v}^{\mathrm{\prime }}{w}^{\mathrm{\prime }}$ and let $a_h$ use one midpoint quadrature sample per cell. Let $F$ and $F_h$ be bounded real linear exact and discrete loads, with $F(v)=a(u,v)$ for every $v\in V$, and put ${\delta }_{F,h}:={\sup }_{w_h\mathrm{\ne }0}\mathrm{\mid }F(w_h)-F_h(w_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_{H^1}$, with ${\delta }_{F,h}=0$ if $V_h=\{0\}$. Then the exact and discrete solutions satisfy

$\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_{H^1}\le (1+2M\mathrm{/}\alpha )\sqrt{2}h\mathrm{\parallel }{u}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}+\frac{2}{\alpha }(\frac{C_A{h}^2}{24}(\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_2+h\mathrm{\parallel }{u}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }})+{\delta }_{F,h})\mathrm{.}$

The explicit ${\delta }_{F,h}$ is the data/source oscillation term; no rate or vanishing of it is asserted without an additional load approximation hypothesis. Here coercivity follows directly because every sampled coefficient satisfies $A(m_K)\ge \alpha$; this special argument does not infer stability from quadrature exactness.