Crank–Nicolson and cG(1) for an autonomous parabolic form

Input 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 $0=t_0<\cdots <t_N=T$, a time-independent real bounded coercive bilinear form $a$ on $V$, a load $f\in L^2(0,T;V^{\ast })$, and $U^0\in V_h$. Put ${\bar{f}}_j={\tau }_j^{-1}{\int }_{t_{j-1}}^{t_j}f(t)dt$. For $j=1,\dots ,N$, solve ${\tau }_j^{-1}(U^j-U^{j-1},v_h{)}_H+a\text{\hspace{0.17em}⁣}(\frac{U^j+U^{j-1}}{2},v_h)=⟨{\bar{f}}_j,v_h⟩(v_h\in V_h)\mathrm{.}$ Return the nodal values and their continuous piecewise-linear interpolant. There are no interior time jumps; the initial value is the supplied $U^0$. The coefficient form is time-independent, while the load is averaged, so no point value of an $L^2(V^{\ast })$ equivalence class is used. Invalid inputs return invalid-input; the well-definedness theorem proves that each indicated step has a unique solution under the stated hypotheses.

For each interval, the same equation is the continuous-Galerkin $cG(1)$ equation with continuous piecewise-linear trial functions and constant test functions: integrating $U^{\mathrm{\prime }}$ gives $U^j-U^{j-1}$, and integrating the linear trial function in the time-independent form gives its midpoint value. Thus this item asserts the $cG(1)$ equivalence only; it does not assert rates for higher-degree $cG$ or discontinuous-Galerkin methods.