Compatible discrete Maxwell constraints and finite algorithms

Fix nonnegative integers $n_0,n_1,n_2,n_3$ and ordered coordinate bases, and put $C^k:={\mathbb{R}}^{n_k}$ for $k=0,1,2,3$. The dual space $(C^k{)}^{\ast }$ is the vector space of linear functionals $C^k\to \mathbb{R}$, with the dual basis to the chosen coordinate basis. Let $D_0\in {\mathbb{R}}^{n_1\times n_0}$, $D_1\in {\mathbb{R}}^{n_2\times n_1}$, and $D_2\in {\mathbb{R}}^{n_3\times n_2}$ be linear maps satisfying $D_1{D}_0=0$ and $D_2{D}_1=0$. This data is a finite cochain complex. The transpose $D_k^{\mathsf{T}}$ denotes the representing matrix of the dual linear map in the dual bases. Let $M_1\in {\mathbb{R}}^{n_1\times n_1}$ and $M_2\in {\mathbb{R}}^{n_2\times n_2}$ be symmetric positive-definite matrices: $v^{\mathsf{T}}{M}_{k}v>0$ for each nonzero $v\in C^k$. They represent linear maps $M_1:C^1\to (C^1{)}^{\ast }$ and $M_2:C^2\to (C^2{)}^{\ast }$ in the chosen bases; in positive dimensions their inverses exist by the finite-dimensional positive-matrix result cited below, and the unique map on a zero-dimensional space is its own inverse.

The finite state space is $\mathcal{X}:=C^2\oplus (C^1{)}^{\ast }$, with state $(b,d)$, where $b$ is a column of $C^2$ coordinates and $d$ is a column in the dual basis of $(C^1{)}^{\ast }$. In this item, “phase space” means this finite-dimensional state space only; no symplectic form is asserted. On an open interval $J\subseteq \mathbb{R}$, a semidiscrete trajectory is a continuously differentiable map $t\mapsto (b(t),d(t))\in \mathcal{X}$ satisfying

$\dot{b}(t)=-D_1{M}_1^{-1}d(t),\dot{d}(t)=D_1^{\mathsf{T}}{M}_{2}b(t)\mathrm{.}$

Its two discrete constraints are $D_{2}b(t)=0$ in $C^3$ and $D_0^{\mathsf{T}}d(t)=0$ in $(C^0{)}^{\ast }$. The associated quadratic energy is

$\mathcal{E}(b,d):=\frac{1}{2}(b^{\mathsf{T}}{M}_{2}b+d^{\mathsf{T}}{M}_1^{-1}d)\mathrm{.}$

All objects here are finite-dimensional real linear-algebra data. The term “Maxwell” names this algebraic model and does not identify it with a continuum Maxwell or Einstein–Maxwell solution.

Closed Maxwell current and gauge boundary term

Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity

For a finite oriented cell complex, signed incidence matrices give one source of such maps: an entry is $+1$ or $-1$ according as the oriented lower cell occurs in the oriented boundary with agreeing or opposite orientation, and is $0$ when it does not occur. In a simplicial realization, the boundary of an ordered simplex is $\mathrm{\partial }[v_0,\dots ,v_k]={\sum }_{i=0}^k(-1{)}^i[v_0,\dots ,{\widehat{v}}_i,\dots ,v_k]$. Each codimension-two face obtained by deleting vertices $i<j$ occurs in ${\mathrm{\partial }}^2$ twice, with coefficients $(-1{)}^i(-1{)}^{j-1}$ and $(-1{)}^j(-1{)}^i$, which are opposites. Hence ${\mathrm{\partial }}^2=0$; taking transposes of the boundary matrices gives the two zero-composition identities for the cochain maps. This definition also admits any real matrices satisfying the displayed identities. Positive-definite mass maps are additional data and are not determined by incidence alone.

The cochain sequence is displayed below. The arrows are exactly $D_0,D_1,D_2$ in the ordered coordinate bases; the zero-composition identities are hypotheses of the definition and are not proved by the diagram.

C^0 \arrow[r,"D_0"] & C^1 \arrow[r,"D_1"] & C^2 \arrow[r,"D_2"] & C^3

The incidence identities support exact finite constraint calculations. They do not supply a family of meshes, commuting projections, approximation estimates, uniform energy or CFL bounds, or a continuum-limit theorem. The source’s cochain-sampling and Whitney-form results have those additional geometric and approximation scopes; none is claimed here.

Use the finite cochain data fixed by the preceding definition item, namely finite real coordinate spaces $C^0,C^1,C^2,C^3$, maps $D_0,D_1,D_2$ with $D_1{D}_0=D_2{D}_1=0$, and symmetric positive-definite mass maps $M_1,M_2$. Inputs are rational matrices $D_0,D_1,D_2,M_1,M_2$, rational initial coordinates $b_0\in C^2$ and $d_{1\mathrm{/}2}\in (C^1{)}^{\ast }$, a positive rational step size $\tau$, and an integer $N\ge 0$. The matrices and states use the fixed ordered bases and dual bases from that definition. All arithmetic is exact over $\mathbb{Q}$; floating-point arithmetic is not part of this algorithm.

First compute $R=M_1^{-1}$ once by fixed-order, no-row-swap Gauss–Jordan elimination of the augmented matrix $[M_1\mid I_{n_1}]$. For each pivot column $j=1,\dots ,n_1$, let $p$ be the current entry in position $(j,j)$. If $p=0$, return an inconsistency error. Otherwise divide row $j$ by $p$ and, for every $k\mathrm{\ne }j$, replace row $k$ by row $k$ minus its current column-$j$ entry times row $j$. After all columns are processed, let $R$ be the right block. For $n_1=0$, take the unique empty matrix for $R$ and perform no elimination.

Set $(b,d):=(b_0,d_{1\mathrm{/}2})$ and record this initial state. For each $n=0,\dots ,N-1$, in the stated order compute

$b_{\mathrm{n}\mathrm{e}\mathrm{w}}:=b-\tau D_{1}Rd,d_{\mathrm{n}\mathrm{e}\mathrm{w}}:=d+\tau D_1^{\mathsf{T}}{M}_2{b}_{\mathrm{n}\mathrm{e}\mathrm{w}}\mathrm{.}$

Append $(b_{\mathrm{n}\mathrm{e}\mathrm{w}},d_{\mathrm{n}\mathrm{e}\mathrm{w}})$ to the output list and then set $(b,d):=(b_{\mathrm{n}\mathrm{e}\mathrm{w}},d_{\mathrm{n}\mathrm{e}\mathrm{w}})$. After the loop, return $R$ together with the list of the $N+1$ states $(b_n,d_{n+1\mathrm{/}2})$ for $0\le n\le N$. The error branch is unreachable on admissible inputs; the separate finite-step termination lemma proves this and the exact inverse output.

Finite cochain Maxwell phase space

Finite arithmetic termination and cost ↗

The update uses the state order $b_n,d_{n+1\mathrm{/}2}$ and this ordering fixes the sign of the cross term in its invariant. Positive definiteness makes every pivot in the fixed no-row-swap schedule nonzero; the separate lemma proves this by Schur-complement induction. The source’s Whitney finite-element leapfrog has its own spaces, mass matrices, boundary conditions and CFL hypotheses. This algorithm makes no claim of being its spatial discretization or of giving a continuum stability or convergence result.

Use finite-dimensional real cochain data as in the definition item and fix $\tau >0$. For a state $(b,d)\in \mathcal{X}$, define the real one-step map $T_{\tau }:\mathcal{X}\to \mathcal{X}$ by

$b^{+}:=b-\tau D_1{M}_1^{-1}d,d^{+}:=d+\tau D_1^{\mathsf{T}}{M}_2{b}^{+},T_{\tau }(b,d):=(b^{+},d^{+})\mathrm{.}$

On rational admissible matrices, rational initial coordinates, and rational $\tau >0$, this is the one-step update used by the exact rational algorithm; the algorithm's input domain is the rational one stated in its Definition. For every real input state, $D_2{b}^{+}=D_{2}b$ and $D_0^{\mathsf{T}}{d}^{+}=D_0^{\mathsf{T}}d$, so an input satisfying both zero constraints maps to a state satisfying both. Every continuously differentiable semidiscrete solution conserves $\mathcal{E}(b,d)$. For any sequence $(b_n,d_{n+1\mathrm{/}2}{)}_{n\ge 0}$ with $(b_{n+1},d_{n+3\mathrm{/}2})=T_{\tau }(b_n,d_{n+1\mathrm{/}2})$, define

$Q_{\tau }(b,d):=b^{\mathsf{T}}{M}_{2}b+d^{\mathsf{T}}{M}_1^{-1}d-\tau b^{\mathsf{T}}{M}_2{D}_1{M}_1^{-1}d,Q_n:=Q_{\tau }(b_n,d_{n+1\mathrm{/}2})\mathrm{.}$

Then $Q_n$ is exactly invariant under $T_{\tau }$. Equip coordinate and dual-coordinate columns with their standard Euclidean inner products, so $⟨x,y⟩=x^{\mathsf{T}}y$ and $\parallel x{\parallel }_2=(x^{\mathsf{T}}x{)}^{1\mathrm{/}2}$. Regard $S:=M_2^{1\mathrm{/}2}{D}_1{M}_1^{-1\mathrm{/}2}$ as a real linear map from the dual-coordinate columns of $(C^1{)}^{\ast }$ to the coordinate columns of $C^2$. Set $\sigma :=0$ when its domain $(C^1{)}^{\ast }$ is zero; otherwise set $\sigma :={\max }_{\parallel y{\parallel }_2=1}\parallel Sy{\parallel }_2$. For a nonzero domain this is the largest singular value of $S$. The quadratic form $Q_{\tau }$ is positive definite on $\mathcal{X}$ whenever $\tau \sigma <2$.

Finite cochain Maxwell phase space

Finite compatible leapfrog evolution

Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity

The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector

Cauchy–Schwarz: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \parallel u\parallel \parallel v\parallel$, with equality exactly for linearly dependent vectors

The sign in $Q_n$ follows from the specified order $b_n,d_{n+1\mathrm{/}2}$. This is a finite algebraic result; the cited finite-element sources use their own field spaces, material operators and boundary conditions. No mesh-uniform stability or continuum-convergence theorem is claimed.

For rational admissible input matrices and initial state, positive rational step size $\tau$, and finite $N\ge 0$, the specified finite leapfrog algorithm returns after finitely many exact rational arithmetic operations, with the stated $N+1$ states and the exact rational inverse $M_1^{-1}$. A dense implementation uses $O(n_1^3)$ rational arithmetic operations for the one-time elimination and $O(n_1^2+n_1{n}_2+n_2^2)$ per update. Retaining all input matrices $D_0,D_1,D_2,M_1,M_2$, the computed inverse, and all $N+1$ states uses $O(n_1{n}_0+n_1^2+n_1{n}_2+n_2^2+n_2{n}_3+(N+1)(n_1+n_2))$ rational entries. These are arithmetic-operation and storage counts only; they give no bound on rational bit growth, floating-point roundoff, linear-solve residuals, or continuum accuracy.

Finite cochain Maxwell phase space

Finite compatible leapfrog evolution