The mixed mini-element backward Euler Navier–Stokes scheme

Assume Countable Choice as inherited from the energy-space conventions. Interpret each interval forcing average as a Bochner integral; its existence follows from the stated L2 forcing and the Bochner integrability criterion (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Bochner integrability criterion). Let $D$ be as in Energy weak solutions of incompressible Navier–Stokes, let ${\mathcal{T}}_h$ be a conforming shape-regular affine simplicial mesh of $D$, and let $V_h,Q_h,Z_h$ be the MINI spaces and weak divergence kernel of The conforming P1-bubble/P1 mini element and Weak and pointwise divergence-free discrete velocities. For a partition $0=t_0<\cdots <t_N=T$, write ${\tau }_n=t_n-t_{n-1}$ and $f_h^n={\tau }_n^{-1}{\int }_{t_{n-1}}^{t_n}f(t)dt$. Given $u_h^0\in Z_h$, a backward Euler mini-element solution is a sequence $(u_h^n,p_h^n)\in V_h\times Q_h$ satisfying, for $n=1,\dots ,N$, ${(\frac{u_h^n-u_h^{n-1}}{{\tau }_n},v_h)}_{L^2}+a_{\nu }(u_h^n,v_h)+c(u_h^n;u_h^n,v_h)+b(v_h,p_h^n)=(f_h^n,v_h{)}_{L^2}(v_h\in V_h),$ $b(u_h^n,q_h)=0(q_h\in Q_h)\mathrm{.}$ All pressure variables use the zero-mean gauge $Q_h\subset L_0^2(D)$. Restriction to $v_h\in Z_h$ eliminates pressure. The equations are well-defined by Well-definedness of the finite-dimensional mini backward Euler equations ↗; this definition claims no solver termination or error rate.