Compatible finite elements, constraints and coercive error control

Assume Countable Choice as in the real L^2 and weak-derivative conventions. Let I be a nonempty open interval in R, and let Omega be a nonempty bounded Lipschitz polyhedron in one SI laboratory frame under Framework-local SI units, the vacuum constants and the exact/measured status of their values. Let T be a finite matching affine tetrahedral mesh with realization closure(Omega), with mesh and orientation conventions as in Lowest-order Whitney and Nedelec de Rham spaces. Use the real graph space H(curl;Omega) of The real H(curl) graph space on a polyhedron and define H_0(curl;Omega) by the closure of C_c^\infty(Omega;R^3) in that graph norm, as in Zero tangential trace H(curl) space. Define H_0^1(Omega) as the closure of C_c^\infty(Omega;R) in W^{1,2}(Omega;R), using the norm and closure conventions of Integer-order Sobolev spaces and their norms.

Let W_h^0 be the continuous piecewise P1 scalar space, where P1(K) is the real scalar-polynomial space of total degree at most one on tetrahedron K. On K with barycentric coordinates lambda_0,...,lambda_3, the lowest-order Whitney edge space is spanned by phi_ij=lambda_i grad(lambda_j)-lambda_j grad(lambda_i) for its six oriented edges. Globally N_h^1 consists of these piecewise fields with one common oriented circulation integral_e u dot t_e ds on each shared edge e, where t_e is its chosen unit tangent; it is H(curl)-conforming. The face space N_h^2 consists of piecewise RT0 fields, RT0(K)={x -> a+c x:a in R3,c in R}, with one common oriented flux integral_F u dot n_F dS on each shared face F, where n_F is its oriented unit normal; it is H(div)-conforming in the graph space The real H(div) space. N_h^3 consists of cellwise constants. These spaces are the full spaces in Lowest-order Whitney and Nedelec de Rham spaces. Set S_h := W_h^0 intersect H_0^1(Omega), V_h := N_h^1 intersect H_0(curl;Omega), and M_h := N_h^2. The finite-dimensional map grad:S_h -> V_h is well-defined: for q in S_h, its weak gradient is L^2, curl(grad q)=0 distributionally, and a sequence from C_c^\infty converging to q in W^{1,2} has gradients converging to grad q in H(curl). On each tetrahedron K with vertices v_0,...,v_3, write q_i:=q(v_i) for the nodal values of the continuous piecewise-affine representative; then

$\mathrm{\nabla }q{\mathrm{\mid }}_K={\sum }_{0\le i<j\le 3}(q_j-q_i){\varphi }_{ij}\mathrm{.}$

This identity puts grad q in the local Whitney edge span and gives its oriented circulation q_j-q_i along [v_i,v_j]; the shared endpoint values agree between cells because q is continuous. Take the canonical full-space nodal and edge reference bases from Lowest-order Whitney and Nedelec de Rham spaces. Since S_h and V_h are finite-dimensional subspaces of the full nodal and edge spaces, choose any bases chi_1,...,chi_ng of S_h and phi_1,...,phi_ne of V_h obtained by dimensionless real linear combinations of those reference bases. Choose psi_1,...,psi_nb for M_h dual to all oriented face-flux degrees of freedom, and theta_1,...,theta_nc for N_h^3 dual to the cell-integral degrees of freedom, with theta_l=|K_l|^-1 on cell K_l and zero on the other cells. Thus chi_j is dimensionless, phi_i has units m^-1, psi_alpha has units m^-2, and theta_l has units m^-3. Define the real coordinate matrices C,D,G by curl(phi_i)=sum_alpha C_{alpha i} psi_alpha, div(psi_alpha)=sum_l D_{l alpha} theta_l, and grad(chi_j)=sum_i G_{ij} phi_i. These matrices are dimensionless representations of the oriented differentials; D is the face-to-cell incidence matrix in the face-moment basis, and changes of basis in S_h,V_h give the corresponding matrices for their PEC subspaces. Because curl grad=0 and div curl=0 on the full finite-element complex, DC=0 and CG=0. Here D denotes only the magnetic face-to-cell incidence matrix.

Write Sym_3(R) for the real symmetric 3 by 3 matrices. Let epsilon,mu:Omega -> Sym_3(R) be measurable, essentially bounded, time-independent coefficient fields with constants 0<a_epsilon<=b_epsilon<infinity and 0<a_mu<=b_mu<infinity such that almost everywhere

$a_{\epsilon }\mathrm{\mid }v{\mathrm{\mid }}^2\le v\cdot \epsilon (x)v\le b_{\epsilon }\mathrm{\mid }v{\mathrm{\mid }}^2,a_{\mu }\mathrm{\mid }v{\mathrm{\mid }}^2\le v\cdot \mu (x)v\le b_{\mu }\mathrm{\mid }v{\mathrm{\mid }}^2(v\in {\mathbb{R}}^3)\mathrm{.}$

Their units are F m^-1 and H m^-1. Pointwise inversion makes mu^-1 measurable, symmetric and essentially bounded, with eigenvalues between b_mu^-1 and a_mu^-1. These coefficient fields are explicit local model data; no material-realization or constitutive derivation is asserted. Let sigma:Omega -> Sym_3(R) be another measurable, essentially bounded, time-independent field with sigma(x) positive semidefinite almost everywhere and units S m^-1. Let J_ext:I -> L^2(Omega;R^3) be continuous in time and have units A m^-2.

Define the real matrices and current-load vector by

$(M_e{)}_{ij}:={\int }_{\mathrm{\Omega }}{\varphi }_i\cdot \epsilon {\varphi }_{j}dx,(M_b{)}_{\alpha \beta }:={\int }_{\mathrm{\Omega }}{\psi }_{\alpha }\cdot {\mu }^{-1}{\psi }_{\beta }dx,S_{ij}:={\int }_{\mathrm{\Omega }}{\varphi }_i\cdot \sigma {\varphi }_{j}dx,j_i(t):={\int }_{\mathrm{\Omega }}{\varphi }_i\cdot J_{\mathrm{e}\mathrm{x}\mathrm{t}}(x,t)dx\mathrm{.}$

All integrals are finite because the mesh is finite, its basis functions are piecewise polynomial, and the coefficient fields are essentially bounded. M_e and M_b are symmetric positive definite on nonzero basis spaces: their quadratic forms are the positive coefficient-weighted L^2 norms of the represented fields; if a basis space is zero-dimensional, the positive-definiteness condition is vacuous. S is symmetric positive semidefinite. The oriented matrices satisfy DC=CG=0. The edge coefficients e_i have units V and the edge basis functions units m^-1; the face coefficients b_alpha have units Wb and the face basis functions units m^-2. The incidence matrices C,D,G are dimensionless, M_e has units F, M_b has units H^-1, S has units S, and j has units A.

For C1 coefficient vectors e:I -> R^ne and b:I -> R^nb, define E_h(t):=sum_i e_i(t)phi_i and B_h(t):=sum_alpha b_alpha(t)psi_alpha, and put H_h:=mu^-1 B_h in L^2. The perfect-electric-conductor (PEC) semidiscrete model is the weak Galerkin system

${\int }_{\mathrm{\Omega }}(\epsilon {\mathrm{\partial }}_t{E}_h+\sigma E_h+J_{\mathrm{e}\mathrm{x}\mathrm{t}})\cdot v_{h}dx={\int }_{\mathrm{\Omega }}{H}_h\cdot \mathrm{curl}{v}_{h}dx(v_h\in V_h),{\mathrm{\partial }}_t{B}_h=-\mathrm{curl}{E}_h\text{in }{M}_h\mathrm{.}$

Here H_h is required only to be L^2; no H(curl) regularity or boundary trace for it is asserted. Testing the first identity against the edge basis and expanding the second in the face basis gives exactly M_e dot e=C^T M_b b-Se-j and dot b=-Ce, the finite compatible matrix system. The quadratic semidiscrete energy is the finite Joule-valued quantity one-half times the integral of E_h dot epsilon E_h plus B_h dot mu^-1 B_h.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Finite-energy PEC and impedance boundary formulation

Finite compatible matrix system

The real H(curl) graph space on a polyhedron

The real H(div) space

Lowest-order Whitney and Nedelec de Rham spaces

Zero tangential trace H(curl) space

Integer-order Sobolev spaces and their norms

Discrete de Rham cohomology and contractible-mesh exactness

Assume Countable Choice and the semidiscrete model of PEC Whitney edge-face Maxwell semidiscretization on a nonempty open time interval I. For every t0 in I, initial vectors e(t0)=e0 and b(t0)=b0, and continuous forcing J_ext:I -> L^2(Omega;R3), the finite compatible matrix system for the edge and face coefficients has a unique C1 solution on all of I. Define

$W_h(t):=\frac{1}{2}{\int }_{\mathrm{\Omega }}(E_h(x,t)\cdot \epsilon (x)E_h(x,t)+B_h(x,t)\cdot \mu (x{)}^{-1}{B}_h(x,t))dx,q(t):=-G^T{M}_{e}e(t)\mathrm{.}$

Then for every t in I,

$\frac{d}{dt}{W}_h(t)=-e(t{)}^{T}Se(t)-e(t{)}^{T}j(t),Db(t)=Db(t_0),\dot{q}(t)=G^T(Se(t)+j(t))\mathrm{.}$

The first identity is the exact semidiscrete field-energy balance in joules; e^T S e and e^T j have units watts, representing discrete conductivity loss and source work. The conserved vector D b has units webers and is the face-to-cell magnetic-divergence coordinate. The coordinate q has units coulombs and is the weak discrete-divergence quantity defined by the finite matrix system; dot q has units amperes. This theorem does not infer a nodal charge-deposition method from it.

PEC Whitney edge-face Maxwell semidiscretization

Energy, existence and algebraic constraints

Assume Countable Choice. Let Omega be a nonempty bounded Lipschitz polyhedron in R3 in one SI laboratory frame. Use the real space V:=H_0(curl;Omega), where H_0(curl;Omega) is the graph closure of compactly supported smooth vector fields as defined by the mathematical supplier. Let epsilon,mu:Omega -> Sym_3(R) be static measurable essentially bounded fields with units F m^-1 and H m^-1 and constants 0<a_epsilon<=b_epsilon<infinity and 0<a_mu<=b_mu<infinity such that, almost everywhere,

$a_{\epsilon }\mathrm{\mid }v{\mathrm{\mid }}^2\le v\cdot \epsilon (x)v\le b_{\epsilon }\mathrm{\mid }v{\mathrm{\mid }}^2,a_{\mu }\mathrm{\mid }v{\mathrm{\mid }}^2\le v\cdot \mu (x)v\le b_{\mu }\mathrm{\mid }v{\mathrm{\mid }}^2(v\in {\mathbb{R}}^3)\mathrm{.}$

Take the lossless branch sigma=0 of the semidiscrete Maxwell model. Fix a reference length ell_0>0 in metres, and equip V with the equivalent, dimensionally homogeneous graph norm

$\mathrm{\parallel }v{\mathrm{\parallel }}_{V,{\mathrm{\ell }}_0}^2:=\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2+{\mathrm{\ell }}_0^2\mathrm{\parallel }\mathrm{curl}v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$

Let s>0 be a real frequency-shift parameter in s^-1, let F:V -> R be a bounded real linear functional scaled so that F(v) has units W s^-1 when v has electric-field units V m^-1, and define

$a_s(u,v):={\int }_{\mathrm{\Omega }}({\mu }^{-1}\mathrm{curl}u\cdot \mathrm{curl}v+s^2\epsilon u\cdot v)dx\mathrm{.}$

Let {T_n}{n>=1} be a shape-regular family of matching affine tetrahedral meshes of closure(Omega), with h_n:=max{K in T_n} diam(K) tending to zero. For each mesh set V_n:=N_{h_n}^1 intersect H_0(curl;Omega), using the PEC Whitney edge space of PEC Whitney edge-face Maxwell semidiscretization. Let

$\alpha :=\min \{(b_{\mu }{\mathrm{\ell }}_0^2{)}^{-1},s^2{a}_{\epsilon }\},M:=\max \{(a_{\mu }{\mathrm{\ell }}_0^2{)}^{-1},s^2{b}_{\epsilon }\}\mathrm{.}$

Then there is a unique u in V satisfying a_s(u,v)=F(v) for every v in V. For each n there is a unique Galerkin solution u_n in V_n satisfying a_s(u_n,v_n)=F(v_n) for every v_n in V_n. For every n,

$\mathrm{\parallel }u-u_n{\mathrm{\parallel }}_{V,{\mathrm{\ell }}_0}\le \frac{M}{\alpha }{\inf }_{v_n\in V_n}\mathrm{\parallel }u-v_n{\mathrm{\parallel }}_{V,{\mathrm{\ell }}_0},$

and u_n converges to u in the V,ell_0 norm as n tends to infinity. Both alpha and M have units (H m)^-1, so their ratio is dimensionless.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Finite-energy PEC and impedance boundary formulation

PEC Whitney edge-face Maxwell semidiscretization

The real H(curl) graph space on a polyhedron

Lowest-order Whitney and Nedelec de Rham spaces

Zero tangential trace H(curl) space

$L^2$ with the integral pairing is a Hilbert space

The Lax--Milgram theorem

Uniformly bounded commuting projection for lowest-order Nedelec spaces

Let T be a finite matching affine tetrahedral mesh whose realization is closure(Omega), with Omega a nonempty bounded Lipschitz polyhedron. Let W_h^0,N_h^1,N_h^2,N_h^3 be the full lowest-order Whitney de Rham spaces of Lowest-order Whitney and Nedelec de Rham spaces, and use their real L^2 inner products. Define

${\mathcal{H}}_h^1:=\ker (\mathrm{curl}{\mathrm{\mid }}_{N_h^1})\cap (\mathrm{\nabla }{W}_h^0{)}^{\perp },{\mathcal{H}}_h^2:=\ker (\mathrm{div}{\mathrm{\mid }}_{N_h^2})\cap (\mathrm{curl}{N}_h^1{)}^{\perp },{\beta }_k:={\dim }_{\mathbb{R}}{H}_k(\mathrm{\mid }\mathcal{T}\mathrm{\mid };\mathbb{R})(k=1,2)\mathrm{.}$

Under Countable Choice, dim_R H_h^1=beta_1 and dim_R H_h^2=beta_2 for the full complex. These are the harmonic complements for the full, absolute finite-element complex.

For the separate PEC-relative family, assume the full Axiom of Choice, including its Countable Choice and Dependent Choice consequences, and let {T_h} be a shape-regular family of matching affine tetrahedral meshes of the same Omega with h=max_K diam(K)->0. Assume also the exact full-Dirichlet H(curl) regular decomposition and the relative commuting-projection hypotheses stated in Facts & Assumptions below. For each mesh write V_h:=N_h^1 intersect H_0(curl;Omega), S_h:=W_h^0 intersect H_0^1(Omega), K_h:=ker(curl|_{V_h}), B_h:=grad S_h, and H_h^rel:=K_h intersect B_h^\perp, with orthogonality in real L^2. Put

$X_h:=\{v_h\in V_h:(v_h,\mathrm{\nabla }{q}_h{)}_{L^2}=0\text{ for every }{q}_h\in S_h\},Z_h:=V_h\cap K_h^{\perp }\mathrm{.}$

Then K_h=B_h direct-sum H_h^rel, every sequence v_h in X_h bounded in H(curl;Omega) has an L^2-convergent subsequence, and there is a constant C independent of h such that ||v_h||_2<=C||curl v_h||_2 for every v_h in Z_h; the constant C has units metres when v_h is an electric field. If H_h^rel contains a nonzero vector for a mesh, that vector lies in X_h and has zero curl, so no curl Poincare inequality holds on that X_h. The full-complex absolute Betti count in the first clause is not asserted for this relative PEC kernel.

PEC Whitney edge-face Maxwell semidiscretization

Discrete de Rham cohomology and contractible-mesh exactness

A bounded regular decomposition with both summands satisfying full Dirichlet conditions

Uniformly bounded commuting projection for lowest-order Nedelec spaces

Discrete compactness and uniform Poincare inequality for Nedelec fields

The diagram gives the moment-coordinate identification for the full complex. The vertical arrows send a piecewise scalar to its vertex values, an edge field to its oriented edge circulations, a face field to its oriented face fluxes, and a cellwise constant to its cell integrals. The bottom arrows are oriented simplicial coboundaries. The commuting squares mean that the edge, face, and cell moments of grad, curl, and div are the signed sums on the corresponding oriented boundaries. This is the finite-complex interface proved by the cited mathematical supplier; no proof step below relies on the drawing.

\begin{tikzcd}
W_h^0 \arrow[r,"\nabla"] \arrow[d,"\cong"'] &
N_h^1 \arrow[r,"\operatorname{curl}"] \arrow[d,"\cong"'] &
N_h^2 \arrow[r,"\operatorname{div}"] \arrow[d,"\cong"'] &
N_h^3 \arrow[d,"\cong"] \\
C^0(\mathcal T;\mathbb R) \arrow[r,"\delta^0"] &
C^1(\mathcal T;\mathbb R) \arrow[r,"\delta^1"] &
C^2(\mathcal T;\mathbb R) \arrow[r,"\delta^2"] &
C^3(\mathcal T;\mathbb R)
\end{tikzcd}