Periodic FE-HMM with simplicial macro and periodic P1 micro meshes

This procedure defines a real-arithmetic finite-dimensional FE-HMM discretization. Its output is $U_H$ unless an ExactSolve call returns singular-system, in which case that token is returned; Well-posedness of the finite FE-HMM micro solves ↗ proves the exceptional branch is unreachable for the stated inputs. It makes no bit-complexity claim, no coefficient-oracle claim, and no claim about numerically computing its exact integral entries; those entries are the mathematical values of the displayed integrals. Fix $Q=(0,1{)}^2$ and coefficient $a$ from Uniformly elliptic periodic scalar diffusion data. Use the diagonal of the square from $(0,0)$ to $(1,1)$ as a two-triangle initial matching macro mesh. Label that shared diagonal $0$ in both triangles and each exterior edge $1$. It is a compatible labeled mesh in the sense of Compatible 2D newest-vertex-bisection meshes. The macro input is any finite admissible descendant triangulation $\mathcal{T}$ of this mesh. Its cells are closed affine triangles as in Matching affine simplicial meshes and shape regularity.

Choose $ϵ>0$, a micro side length $\delta >0$, $f\in L^2(Q)$, and a finite matching triangulation $\mathcal{M}$ of the unit square whose opposite boundary edges and vertices match under translation by the PDE coordinate vectors $e_1,e_2$ of Periodic correctors and effective tensor. The periodic micro space is $W_{\mathcal{M}}:=\{z\in V_{\mathrm{\#}}:z\text{ is continuous and affine on every triangle of }\mathcal{M}\}\mathrm{.}$ Here $z$ means an $H_{\mathrm{\#}}^1(Y)$ class with a continuous periodic piecewise-affine representative; the periodic closure condition identifies opposite traces. Thus $W_{\mathcal{M}}$ is the finite-dimensional mean-zero subspace of the periodic P1 space, as proved in Well-posedness of the finite FE-HMM micro solves ↗. Let $V_H$ be the conforming zero-boundary continuous piecewise-affine macro space on $\mathcal{T}$, using Broken and conforming polynomial spaces on a matching mesh. Its inclusion in $H_0^1(Q)$ is supplied by Continuous matching piecewise polynomials belong to W1p and Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare.

Identify vertices of $\mathcal{M}$ that match under the opposite-edge translations, and order the resulting periodic vertex classes by the lexicographically least representative of each class. Let ${\varphi }_1,\dots ,{\varphi }_N$ be the continuous periodic P1 nodal functions for these classes, with ${\varphi }_i=1$ at class $i$ and $0$ at the other classes. Set $m_i={\int }_Y{\varphi }_{i}dy$. If $N=1$, use the empty basis of $W_{\mathcal{M}}$; otherwise use ${\psi }_i:={\varphi }_i-\frac{m_i}{m_1}{\varphi }_1(2\le i\le N)$ as the ordered basis of $W_{\mathcal{M}}$. The nodal basis spans the full periodic P1 space, and these functions form a basis of its mean-zero subspace.

Use the following exact solver on the systems assembled below. Given a real matrix $A\in {\mathbb{R}}^{n\times n}$ and $b\in {\mathbb{R}}^n$, ExactSolve(A,b) returns the empty vector if $n=0$. Otherwise use temporary solver labels $1,\dots ,n$, with solver coordinate $i$ corresponding to library coordinate $i-1$ of ${\mathbb{R}}^n$ (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$), and form the augmented matrix $[A\mid b]$. For $k=1,\dots ,n$, choose the least row index $p\ge k$ whose current entry in column $k$ is nonzero; if there is no such row, return singular-system. Swap the entire augmented rows $p$ and $k$. For each $i=k+1,\dots ,n$, set $r=A_{ik}\mathrm{/}{A}_{kk}$, replace the entire augmented row $i$ by row $i-r{\mathrm{r}\mathrm{o}\mathrm{w}}_k$, and set its column-$k$ entry to zero. After this loop, compute in order $k=n,n-1,\dots ,1$ by $x_k=\frac{b_k-{\sum }_{j=k+1}^n{A}_{kj}{x}_j}{A_{kk}},$ using entries of the resulting upper-triangular augmented matrix. Return the vector $x\in {\mathbb{R}}^n$ whose library coordinate $x(k-1)$ equals $x_k$. All arithmetic and zero tests are exact over $\mathbb{R}$; the systems assembled below are positive definite.

Order macro triangles by lexicographic order of their barycenters. For each such $K$, use its barycenter $x_K$ as the single quadrature point with positive weight $\mathrm{\mid }K\mathrm{\mid }$, and set $b_K(y):=a\text{\hspace{0.17em}⁣}(\frac{x_K}{ϵ}+\frac{\delta }{ϵ}y)(y\in Y)\mathrm{.}$ The affine change-of-variables theorem A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions makes this a measurable coefficient class independent of the representative of $a$, with $\lambda \le b_K\le \mathrm{\Lambda }$ almost everywhere. For each PDE coordinate vector $e_j$ of Periodic correctors and effective tensor, in order $j=1,2$, assemble $G_{i\mathrm{\ell }}^K:={\int }_Y{b}_K\mathrm{\nabla }{\psi }_{\mathrm{\ell }}\cdot \mathrm{\nabla }{\psi }_{i}dy,r_{i,j}^K:=-{\int }_Y{b}_K{e}_j\cdot \mathrm{\nabla }{\psi }_{i}dy$ over the micro-basis indices $2\le i,\mathrm{\ell }\le N$. If that basis is empty, set $w_{K,j}^h=0$; otherwise call ExactSolve(G^K,r^K_{\cdot,j}), index its returned coordinates by $2,\dots ,N$, and set $w_{K,j}^h={\sum }_{\mathrm{\ell }=2}^N{x}_{\mathrm{\ell }}{\psi }_{\mathrm{\ell }}$. Then set $A_K^h{e}_j:={\int }_Y{b}_K(y)(e_j+{\mathrm{\nabla }}_y{w}_{K,j}^h(y))dy\mathrm{.}$ If the micro call returns singular-system, propagate that output; Well-posedness of the finite FE-HMM micro solves ↗ proves this branch is not reached. Otherwise identify solver index $i-1$ with micro-basis index $i$ for $2\le i\le N$ and use the returned coefficient as $x_i$ in the displayed sum. The singular-system branch is included to make the solver instruction total.

Order the interior vertices of $\mathcal{T}$ lexicographically and let ${\phi }_1,\dots ,{\phi }_M$ be the corresponding nodal basis of $V_H$. If $M=0$, set $U_H=0$. Otherwise assemble $H_{rs}:={\sum }_{K\in \mathcal{T}}\mathrm{\mid }K\mathrm{\mid }\mathrm{\nabla }{\phi }_s{\mathrm{\mid }}_K\cdot A_K^h\mathrm{\nabla }{\phi }_r{\mathrm{\mid }}_K,F_r:={\int }_{Q}f{\phi }_{r}dx,$ call ExactSolve(H,F). If it returns singular-system, propagate that output; the well-posedness lemma proves this branch is not reached. Otherwise set $U_H:={\sum }_{s=1}^M{x}_s{\phi }_s$, with solver index $s$ identified with macro-basis index $s$. This basis system is exactly the variational system $B_H(U,V)={\sum }_{K\in \mathcal{T}}\mathrm{\mid }K\mathrm{\mid }\mathrm{\nabla }U{\mathrm{\mid }}_K\cdot A_K^h\mathrm{\nabla }V{\mathrm{\mid }}_K={\int }_{Q}fV(V\in V_H)\mathrm{.}$ The macro matrix is positive definite and its singular-system branch is unreachable by Well-posedness of the finite FE-HMM micro solves ↗. Output $U_H$. The centroid quadrature is positive and exact for constants, which is the only quadrature property used in the error theorem.

The local continuum comparator uses the same coefficient $b_K$ on the unit micro cell with the full periodic space $V_{\mathrm{\#}}$; it is defined and proved well posed in Well-posedness of the finite FE-HMM micro solves ↗. The period ratio is $r=\delta \mathrm{/}ϵ$. No accuracy is asserted for arbitrary $r$; micro approximation and sampling/resonance are separated in FE-HMM macro, micro, and sampling error separation.