Backward Euler FEM with interval-averaged forms and load

Input a real evolution triple $(V,H,V^{\ast })$ and a finite-dimensional conforming subspace $V_h\subset V$ with its inherited $H$ inner product and $V$ norm; a finite-element use instantiates it with a zero-boundary conforming polynomial space. Input also a final time $T>0$, a finite partition $0=t_0<t_1<\cdots <t_N=T$, and a real bilinear form family $a_h(t;\cdot ,\cdot ):V_h\times V_h\to \mathbb{R}$ such that $t\mapsto a_h(t;v_h,w_h)$ is measurable for each fixed $v_h,w_h\in V_h$. There must be constants $0\le M<\mathrm{\infty }$, $\alpha >0$ and one null set $E\subset (0,T)$ such that $\mathrm{\mid }{a}_h(t;v_h,w_h)\mathrm{\mid }\le M\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_V,a_h(t;v_h,v_h)\ge \alpha \mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V^2(v_h,w_h\in V_h,t\in (0,T)\setminus E)\mathrm{.}$ The same exceptional set and constants apply to all pairs. Input a strongly measurable load $g\in L^2(0,T;V_h^{\ast })$ and an initial value $U^0\in V_h$. A finite-element load $f\in L^2(0,T;V^{\ast })$ is supplied as its restriction $g(t)=f(t){\mathrm{\mid }}_{V_h}$. For $I_j=(t_{j-1},t_j)$ and ${\tau }_j=t_j-t_{j-1}$, form the scalar bilinear and Bochner load averages ${\bar{a}}_j(v_h,w_h)={\tau }_j^{-1}{\int }_{I_j}{a}_h(t;v_h,w_h)dt,{\bar{g}}_j={\tau }_j^{-1}{\int }_{I_j}g(t)dt\mathrm{.}$ For $j=1,\dots ,N$, solve for $U^j\in V_h$ from ${\tau }_j^{-1}(U^j-U^{j-1},v_h{)}_H+{\bar{a}}_j(U^j,v_h)=⟨{\bar{g}}_j,v_h{⟩}_{V_h^{\ast },V_h}(v_h\in V_h)\mathrm{.}$ Return the nodes $U^0,\dots ,U^N$, the piecewise-constant function $\bar{U}(t)=U^j$ on each open interval $(t_{j-1},t_j)$ (endpoint values do not affect the $L^2$ function), and the continuous piecewise affine reconstruction $\widehat{U}$ joining consecutive nodes. The initial value is the separately supplied $U^0$; the initial jump is $U^1-U^0$, and subsequent jumps are $U^j-U^{j-1}$. Invalid spaces, partitions, or form/data hypotheses return invalid-input; the well-definedness theorem proves that each indicated solve has a unique answer under the stated hypotheses. Because coefficients and loads are averaged, this procedure is defined on their almost-everywhere equivalence classes and does not sample a representative at a point. For a time-dependent form this is the averaged-coefficient backward Euler method; with piecewise-constant time trials it is also the $dG(0)$ step.