PEC Whitney edge-face Maxwell semidiscretization

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