Spectral convergence and no nonzero pollution for PEC edge elements

Put $Y=L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ with inner product $b(u,v)=ϵ(u,v{)}_{L^2}$, and let $H_X$ be the $Y$-closure of $X$. The graph of distributional curl is closed: convergence of a field and its curl in $L^2$ passes to the distributional identity in The real H(curl) graph space on a polyhedron. Thus $H(\mathrm{curl})$ is Hilbert by [F6], and $H_0(\mathrm{curl})$ is a closed subspace by its closure definition. Since $K^{\perp }$ is closed in $Y$ by [F12] and the graph inclusion into $Y$ is continuous, $X=H_0(\mathrm{curl})\cap K^{\perp }$ is closed in the graph space. By [F1], the curl form $a(u,v)={\mu }^{-1}(\mathrm{curl}u,\mathrm{curl}v{)}_{L^2}$ is bounded and coercive on $X$ in its graph norm. The inclusion $J:X\to H_X$ is injective, compact by [F1], and has dense range by the definition of $H_X$; $H_X$ is an infinite-dimensional Hilbert space by [F2], [F11] and [F12]. Applying [F7] constructs the compact positive self-adjoint solution operator $T_X$ on $H_X$, whose nonzero eigenvalues are ${\nu }_j=1\mathrm{/}{\lambda }_j$ for the continuous curl-curl eigenvalues. Extend it to $Y$ as $T:=T_X{P}_{H_X}$, where $P_{H_X}$ is the orthogonal projection from [F12]. Composition with this bounded projection preserves compactness; orthogonality of the decomposition $Y=H_X\oplus H_X^{\perp }$ shows that $T$ is nonnegative and self-adjoint, with the same nonzero eigenvalues.

For a mesh ${\mathcal{T}}_h$, put $S_h=W_h^0\cap H_0^1(\mathrm{\Omega })$ and $G_h=\mathrm{\nabla }{S}_h$. On a tetrahedron with barycentric coordinates ${\lambda }_i$ and vertices $x_i$, the identity ${\sum }_{j\mathrm{\ne }i}{\varphi }_{ij}=-\mathrm{\nabla }{\lambda }_i$ follows from ${\sum }_j{\lambda }_j=1$. Since $x_{\mathrm{\ell }}={\sum }_i(x_i{)}_{\mathrm{\ell }}{\lambda }_i$, $e_{\mathrm{\ell }}=\mathrm{\nabla }{x}_{\mathrm{\ell }}={\sum }_i(x_i{)}_{\mathrm{\ell }}\mathrm{\nabla }{\lambda }_i$ is in the local Nedelec space. Hence each piecewise constant $\mathrm{\nabla }{q}_h$ lies in that space. For each oriented mesh edge $e=[x_i,x_j]$ and every tetrahedron $K$ containing it, its local circulation is ${\int }_e\mathrm{\nabla }(q_h{\mathrm{\mid }}_K)\cdot t_{e}ds=q_h(x_j)-q_h(x_i)\mathrm{.}$ The continuous piecewise-affine function $q_h$ has the same endpoint values in every incident tetrahedron, so the circulation is common to all cells containing $e$. Thus the local fields assemble with the required global edge conformity. Choose compactly supported smooth $q_j\to q_h$ in $H^1$ by the closure definition of $H_0^1$. Then $\mathrm{\nabla }{q}_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ and $\mathrm{\parallel }\mathrm{\nabla }{q}_j-\mathrm{\nabla }{q}_h{\mathrm{\parallel }}_{H(\mathrm{curl})}=\mathrm{\parallel }\mathrm{\nabla }{q}_j-\mathrm{\nabla }{q}_h{\mathrm{\parallel }}_{L^2}\to 0$, since both gradients have zero distributional curl. Thus $\mathrm{\nabla }{q}_h\in V_h$ and $G_h\subset K_h$. Because $ϵ>0$ is a constant scalar, the weighted and ordinary $L^2$ orthogonal complements coincide. Consequently $X_h=K_h^{\perp }\cap V_h$ is a subspace of the gradient orthogonal space in [F3].

On $X_h$, the curl form is positive definite: if $\mathrm{curl}{v}_h=0$, then $v_h\in K_h\cap K_h^{\perp }=\{0\}$. The space is finite-dimensional because the mesh has finitely many edge degrees of freedom. Choose an $a$-orthonormal basis ${\psi }_1,\dots ,{\psi }_N$ by [F10] and define $T_{h}f={\sum }_{i=1}^{N}b(f,{\psi }_i){\psi }_i$. For $v_h={\sum }_i{c}_i{\psi }_i$, this gives $a(T_{h}f,v_h)={\sum }_i{c}_{i}b(f,{\psi }_i)=b(f,v_h)$, so the formula defines the unique linear map $T_h:Y\to X_h$ satisfying the discrete source equation. For $f,g\in Y$, the same formula gives $b(T_{h}f,g)={\sum }_{i}b(f,{\psi }_i)b({\psi }_i,g)=a(T_{h}f,T_{h}g)=b(f,T_{h}g)$, so $T_h$ is self-adjoint. It also gives $b(T_{h}f,f)={\sum }_{i}b(f,{\psi }_i{)}^2=a(T_{h}f,T_{h}f)\ge 0$. Its finite-dimensional range makes it compact. If $T_h{e}_h=\nu e_h$ with $\nu \mathrm{\ne }0$, the defining equation gives $a(e_h,v_h)={\nu }^{-1}b(e_h,v_h)$ for every $v_h\in X_h$; conversely every discrete eigenpair with eigenvalue ${\lambda }_h$ satisfies $T_h{e}_h={\lambda }_h^{-1}{e}_h$. Thus its nonzero eigenvalues are ${\nu }_{h,j}=1\mathrm{/}{\lambda }_{h,j}$, counted with multiplicity by [F7].

Testing the defining equation with $u_h=T_{h}f$ gives ${\mu }^{-1}\mathrm{\parallel }\mathrm{curl}{u}_h{\mathrm{\parallel }}_2^2=b(f,u_h)\le \mathrm{\parallel }f{\mathrm{\parallel }}_Y\mathrm{\parallel }{u}_h{\mathrm{\parallel }}_Y$. By [F3] and the constant coefficients, $\mathrm{\parallel }{u}_h{\mathrm{\parallel }}_Y\le C\mathrm{\parallel }\mathrm{curl}{u}_h{\mathrm{\parallel }}_2$; hence $\mathrm{\parallel }{T}_{h}f{\mathrm{\parallel }}_{H(\mathrm{curl})}\le C^{\mathrm{\prime }}\mathrm{\parallel }f{\mathrm{\parallel }}_Y$, uniformly in $h$. For any bounded sequence $f_n\in Y$ and $h_n\to 0$, [F3] applied to $T_{h_n}{f}_n\in X_{h_n}$ supplies a strongly convergent subsequence in $Y$. Therefore $(T_h)$ is sequentially collectively compact and uniformly bounded in operator norm.

Fix $f\in Y$ and any sequence $h_n\to 0$. By step 2.1 and [F3], every subsequence of $u_n:=T_{h_n}f$ has a further subsequence converging strongly in $Y$ to some $u$. By [F6], after a further subsequence $u\in V$ and $\mathrm{curl}{u}_n\rightharpoonup \mathrm{curl}u$. For each $k\in K$, the projection supplier [F4] gives $k_{h_n}:={\pi }_{h_n,0}^{1}k\in V_{h_n}$ and $\mathrm{curl}{k}_{h_n}={\pi }_{h_n,0}^2\mathrm{curl}k=0$, so $k_{h_n}\in K_{h_n}$. Moreover $k_{h_n}\to k$ strongly in graph norm: given $\eta >0$, use the closure definition to choose $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ with $\mathrm{\parallel }k-\phi {\mathrm{\parallel }}_{H(\mathrm{curl})}<\eta$, then use [F5] to obtain $w_{h_n}=I_{h_n}\phi \in V_{h_n}$ with $w_{h_n}\to \phi$. Because ${\pi }_{h_n,0}^1$ is a uniformly bounded projection and equals the identity on $V_{h_n}$, $\mathrm{\parallel }{\pi }_{h_n,0}^{1}k-k{\mathrm{\parallel }}_{H(\mathrm{curl})}\le ({\sup }_h\mathrm{\parallel }{\pi }_{h,0}^1\mathrm{\parallel }+1)\mathrm{\parallel }k-w_{h_n}{\mathrm{\parallel }}_{H(\mathrm{curl})}$. Letting $n\to \mathrm{\infty }$ and then $\eta \to 0$ proves the convergence. Since $u_n\perp K_{h_n}$, taking limits in $b(u_n,k_{h_n})=0$ gives $b(u,k)=0$. As $k\in K$ was arbitrary, $u\in X$.

Let $v\in X$. By [F5], choose $v_{h_n}\in X_{h_n}$ with $v_{h_n}\to v$ in graph norm. The discrete equation is $a(u_n,v_{h_n})=b(f,v_{h_n})$. Along the subsequence of step 3.1, the curls of $u_n$ converge weakly and those of $v_{h_n}$ strongly, while both fields converge strongly in $L^2$. Passing to the limit gives $a(u,v)=b(f,v)$ for every $v\in X$. Coercivity in [F1] makes this continuous source problem unique, so $u=Tf$. Every subsequence has a further subsequence converging to $Tf$; therefore $T_{h_n}f\to Tf$ strongly in $Y$ for every $f\in Y$.

The operators $T$ and $T_h$ are compact, nonnegative and self-adjoint; steps 3.1 and 4.1 give the uniform bound, sequential collective compactness and pointwise convergence required by [F8]. Thus every isolated positive eigenvalue $\nu =1\mathrm{/}\lambda$ of $T$ is approximated by exactly its full multiplicity of eigenvalues ${\nu }_h$, with convergence of eigenspaces in symmetric $Y$-gap, and no positive spurious spectral limit occurs. Taking reciprocals ${\lambda }_h=1\mathrm{/}{\nu }_h$ preserves convergence and multiplicity. Because $ϵ$ is a positive constant, the $Y$-gap is equivalent to the ordinary $L^2$ gap. The uniform discrete curl Poincare estimate in [F3] gives ${\lambda }_h\ge c>0$, independent of $h$, by testing the eigenrelation with its eigenvector, so any finite limit of positive discrete eigenvalues is positive. The same reciprocal argument sends every positive finite limit of discrete ${\lambda }_h$ to a nonzero spectral limit of $T$, hence to a continuous eigenvalue by [F8]. This proves the theorem, including repeated clusters and the exclusion of nonzero pollution.