Well-posedness of the finite FE-HMM micro solves

Every continuous periodic P1 nodal hat ${\varphi }_i$ on $\mathcal{M}$ is in $H^1(Y)$ by the conforming piecewise-polynomial theorem in [F5]. For each coordinate $0\le j<2$ and Fourier mode $k\in {\mathbb{Z}}^2$, apply [F8] on each triangle to ${\varphi }_i(y)e^{-2\pi ik\cdot y}$. Interior-edge terms cancel by matching traces; opposite boundary-edge terms cancel because ${\varphi }_i$ is periodic and $k$ is integral. Thus $\widehat{{\mathrm{\partial }}_j{\varphi }_i}(k)=2\pi i{k}_j{\widehat{\varphi }}_i(k)$. Parseval [F9] applied to ${\varphi }_i$ and its two weak derivatives shows that symmetric Fourier partial sums converge in $H^1(Y)$; they are real smooth periodic trigonometric polynomials, so ${\varphi }_i\in H_{\mathrm{\#}}^1(Y)$. The periodic nodal hats form a basis of the full periodic P1 space because a continuous P1 function is determined by its nodal values, with opposite-edge vertices identified. Each hat is nonnegative and positive on an incident triangle, so $m_i={\int }_Y{\varphi }_i>0$. A nodal combination ${\sum }_i{c}_i{\varphi }_i$ has zero mean exactly when ${\sum }_i{m}_i{c}_i=0$; solving this constraint for $c_1$ shows that the ${\psi }_i$ defined in the algorithm form a basis of $W_{\mathcal{M}}$ (and when $N=1$ that space is zero). For fixed $K,p$, on $V_{\mathrm{\#}}$ define $c_K(u,v)={\int }_Y{b}_K\mathrm{\nabla }u\cdot \mathrm{\nabla }v$ and $F_{K,p}(v)=-{\int }_Y{b}_{K}p\cdot \mathrm{\nabla }v$. The bounds [F3] give $\mathrm{\mid }{c}_K(u,v)\mathrm{\mid }\le \mathrm{\Lambda }\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$, $c_K(v,v)\ge \lambda \mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2$, and $\mathrm{\mid }{F}_{K,p}(v)\mathrm{\mid }\le \mathrm{\Lambda }\mathrm{\mid }p\mathrm{\mid }\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$. By [F2], [F4] gives a unique solution $w_{K,p}$. The finite-dimensional gradient inner product is positive definite on $W_{\mathcal{M}}$ because it is a subspace of the mean-zero space $V_{\mathrm{\#}}$; hence it is complete and the same boundedness and coercivity estimates give the unique $w_{K,p}^h$ by [F4]. Testing its equation with each basis function ${\psi }_i$ gives exactly the matrix system $G^{K}c=r_{\cdot ,p}^K$ from the algorithm. Its matrix is symmetric positive definite since, for each nonzero coefficient vector, the corresponding $z_h\mathrm{\ne }0$ in $W_{\mathcal{M}}$ has ${\int }_Y{b}_K\mathrm{\mid }\mathrm{\nabla }{z}_h{\mathrm{\mid }}^2>0$.

Each solution is linear in $p$ by uniqueness. For either space $X=V_{\mathrm{\#}}$ or $X=W_{\mathcal{M}}$, the variational equation says $w_{K,p}^X$ minimizes $J_{K,p}(z)={\int }_Y{b}_K\mathrm{\mid }p+\mathrm{\nabla }z{\mathrm{\mid }}^2$ over $X$: expanding at the solution cancels the cross term and leaves $\int b_K\mathrm{\mid }\mathrm{\nabla }(z-w){\mathrm{\mid }}^2\ge 0$. For $p,q$, testing the $q$ equation with $w_{K,p}^X$ shows $p\cdot A_K^{X}q={\int }_Y{b}_K(p+\mathrm{\nabla }{w}_{K,p}^X)\cdot (q+\mathrm{\nabla }{w}_{K,q}^X),$ so the tensor is symmetric. Since ${\int }_Y\mathrm{\nabla }{w}_{K,p}^X=0$ by [F2], Jensen gives $J_{K,p}(w_{K,p}^X)\ge \lambda \mathrm{\mid }p{\mathrm{\mid }}^2$; the competitor zero gives it at most $\mathrm{\Lambda }\mathrm{\mid }p{\mathrm{\mid }}^2$.

The interior macro nodal functions listed in the algorithm form a basis of $V_H$, since a conforming P1 function is determined by its vertex values and the boundary values are fixed to zero. By [F5] the macro space is a finite-dimensional subspace of $H_0^1(Q)$. For $U\in V_H$, [F1] and step 2.1 imply $B_H(U,U)={\sum }_K\mathrm{\mid }K\mathrm{\mid }\mathrm{\nabla }U{\mathrm{\mid }}_K\cdot A_K^h\mathrm{\nabla }U{\mathrm{\mid }}_K\ge \lambda \mathrm{\parallel }\mathrm{\nabla }U{\mathrm{\parallel }}_{L^2(Q)}^2\mathrm{.}$ This is positive for every nonzero $U$; [F6] ensures the gradient seminorm is a norm on $V_H$. The load is bounded by $\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }U{\mathrm{\parallel }}_2\le \mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }U{\mathrm{\parallel }}_2$. Thus finite-dimensional Lax--Milgram [F4] gives a unique macro solution; if the space is zero, its sole element is the unique solution by definition. Testing with each interior nodal function gives $H_{rs}=B_H({\phi }_s,{\phi }_r)$ and $F_r={\int }_{Q}f{\phi }_r$, so $Hx=F$ is equivalent to the macro variational equation. In particular $H$ is symmetric positive definite. The micro and macro systems are therefore nonsingular.

Consider any matrix $A$ passed to ExactSolve; steps 1.1 and 3.1 show it is real, symmetric, positive definite, and hence nonsingular. After $k-1$ elimination stages its coefficient matrix is row-equivalent to $A$ and has block form $\left(\begin{array}{cc}R& C\\ 0& S\end{array}\right)$, where $R$ is upper triangular with nonzero diagonal. The full matrix and $R$ are nonsingular, so $S$ is nonsingular: otherwise a nonzero $z$ with $Sz=0$ and the solution $y$ of $Ry=-Cz$ would give a nonzero kernel vector $(y,z)$. Thus the first column of $S$ has a nonzero entry, and the prescribed least-row search finds a pivot. Row swaps and row additions preserve the solution set; after finitely many stages the system is upper triangular with nonzero diagonal. Back substitution is finite and correct by [F7], so ExactSolve terminates without its singular branch and returns the unique solution of the assembled matrix equation. The micro systems in step 1.1 are exactly the restricted cell variational equations on a basis; the macro system in step 3.1 is exactly the macro variational equation on a basis. This proves the solver assertion for both.