A Unified Theory of Non-overlapping Robin–Schwarz Methods: Continuous and Discrete, Including Cross Points

Abstract

Non-overlapping Schwarz methods with generalized Robin transmission conditions were originally introduced by B. Després for time-harmonic wave propagation problems and have largely developed over the past thirty years. The aim of the paper is to provide both a review of the available formulations and methods as well as a consistent theory applicable to more general cases than studied until to date. An abstract variational framework is provided reformulating the original problem by the well-known form involving a scattering operator and an interface exchange operator, and the equivalence between the formulations is discussed thoroughly. The framework applies to a series of wave propagation problems throughout the de Rham complex, such as the scalar Helmholtz equation, Maxwell’s equations, a dual formulation of the Helmholtz equation in H(div), as well as any conforming finite element discretization thereof, and it applies also to coercive problems. Three convergence results are shown. The first one (using compactness) and the second one (based on absorption) generalize Després’ early findings and apply as well to the FETI-2LM formulation (a discrete method introduced by de La Bourdonnaye, Farhat, Macedo, Magoulés, and Roux). The third result, oriented on the work by Collino, Ghanemi, and Joly, establishes a convergence rate and covers cases with cross points, while not requiring any regularity of the solution. The key ingredient is a global interface exchange operator, proposed originally by X. Claeys and further developed by Claeys and Parolin, here worked out in full generality. The third type of convergence theory is applicable at the discrete level as well, where the exchange operator is allowed to be even local. The resulting scheme can be viewed as a generalization of the 2-Lagrange-multiplier method introduced by S. Loisel, and connections are drawn to another technique proposed by Gander and Santugini.

Appendices

Proof of Theorem 4.5

Beforehand, observe that since none of the operators T, ${\mathcal {X}}$ couples subdomain dofs or trace dofs that correspond to two different global dofs, we can treat one global interface dof at a time. So without loss of generality, we may assume that ${\mathcal {D}}_\varGamma$ consists of a single global interface dof k, such that our goal is a proof for $\dim ({\mathcal {Z}}) = \ell _k$.

Let ${\mathcal {E}}_k'$ be the edges of a fixed minimal spanning tree of the connectivity graph ${\mathcal {C}}_k$ (see Fig. 18, left) and let us collect the remaining edges ${\mathcal {E}}_k \setminus {\mathcal {E}}_k'$ in a sequence $(e_1\,,\ldots ,e_m)$. By classical graph theory, $\#{{\mathcal {E}}_k'} = \#{{\mathcal {N}}_k} - 1$ and for each remaining edge $e_i$, $i=1,\ldots ,m$, there exists an associated cycle ${\mathcal {L}}_i$ consisting of edges from ${\mathcal {E}}_k' \cup \{ e_1,\ldots ,e_i \}$, such that the number of independent cycles in ${\mathcal {C}}_k$ is given by $\ell _k = m$ (see Fig. 18, middle).

As one observes, $\mu \in \varLambda ^*$ has two values per facet (one for each subdomain), so accordingly, two values per edge of the connectivity graph. Since each subdomain corresponds to a node of the graph, we can think of these values as being associated with the endpoints of the edges. Under that perspective, the operator $T^{\textsf{T}}$ sums up the all values per node, while the operator $(I + {\mathcal {X}}^{\textsf{T}})$ sums up the two values per edge.

For each edge $e_i$, we define an element ${{\widehat{\mu }}}_i \in {\mathcal {Z}}$ by putting values $\pm 1$ along the associated cycle ${\mathcal {L}}_i$ as illustrated in Fig. 18 (right) and zero elsewhere. Apparently, $T^{\textsf{T}}{{\widehat{\mu }}}_i = 0$ because the two non-zero values associated with each node within the cycle have opposite sign. At the same time, the two values associated with each edge within the cycle sum up to zero as well, so $(I + {\mathcal {X}}^{\textsf{T}}) {{\widehat{\mu }}}_i = 0$. Altogether, ${{\widehat{\mu }}}_i \in {\mathcal {Z}}$. Moreover, the element ${{\widehat{\mu }}}_i$ is linearly independent from $\{ {{\widehat{\mu }}}_j \}_{j=1}^{i-1}$ because the latter are not supported on the edge $e_i$. Therefore $\dim ({\mathcal {Z}}) \ge m$.

In order to see that $\dim ({\mathcal {Z}}) \le m$, we let $\mu \in {\mathcal {Z}}$ be arbitrary but fixed. Recall the sequence $(e_1,\ldots ,e_m)$ of remaining edges and let us start with the edge $e_m$. Since $(I - {\mathcal {X}}^{\textsf{T}})\mu = 0$, the two values of $\mu$ on edge $e_m$ must have opposite sign. Therefore, we can find a coefficient $\alpha _m$ such that $\mu _m := \mu - \alpha _m {{\widehat{\mu }}}_m$ has vanishing values on edge $e_m$. We proceed inductively. For $i > 1$, suppose that $\mu _i \in {\mathcal {Z}}$ has vanishing values on all edges $e_i,\ldots ,e_m$. Then, since $(I - {\mathcal {X}}^{\textsf{T}})\mu _i = 0$, the two values of $\mu _i$ on edge $e_{i-1}$ must have opposite sign. So there exists a coefficient $\alpha _{i-1}$ such that $\mu _{i-1} := \mu _i - \alpha _{i-1} {{\widehat{\mu }}}_{i-1}$ has vanishing values on edge $e_{i-1}$. Since ${{\widehat{\mu }}}_{i-1}$ has vanishing values on all the edges $e_i,\ldots ,e_m$, the function $\mu _{i-1}$ vanishes on all the edges $e_{i-1},\ldots ,e_m$ and $\mu _{i-1} \in {\mathcal {Z}}$. The inductive process stops with $\mu _1 \in {\mathcal {Z}}$ vanishing entirely on all remaining edges $e_1,\ldots ,e_m$, and so the only possible non-zero values of $\mu _1$ are located at the edges of the minimal spanning tree. This spanning tree, however, must have nodes with just one edge attached. The condition $T^{\textsf{T}}\mu _1 = 0$ implies that the values of $\mu _1$ at these node and the attached edges is zero. Using the condition $(I + {\mathcal {X}}^{\textsf{T}}) \mu _1 = 0$ along the edges allows to show, eventually, that $\mu _1 = 0$. Therefore, $\dim ({\mathcal {Z}}) = m$ and ${\mathcal {Z}} = \text {span}\{ {{\widehat{\mu }}}_1, \ldots , {{\widehat{\mu }}}_m \}$.

Fig. 18

Illustration of the proof Theorem 4.5Left: minimal spanning tree, $\circ$ nodes with just one edge attached Middle: black edges: minimal spanning tree, colored: sequence of remaining edges with associated cycles Right: Values of ${{\widehat{\mu }}}_5$ associated with the cycle of $e_5$

Invertibility of Generalized Robin Problems

In this section, we investigate the invertibility of the augmented operator $A + \alpha T^{\textsf{T}}M T$, cf. Assumption A5. As for classical wave propagation, the two building blocks are the Fredholm property and the injectivity, which altogether ensure a bounded inverse, see e.g. [56, 77]. In the discrete case, invertibility is usually proved either via injectivity alone or via an inf-sup condition derived from the continuous counterpart, cf. e.g. [56].

We begin with the injectivity (Sect. B.1), visit some general tools on the Fredholm property (Sect. B.2) and apply these for standard as well as generalized Robin problems (Sect. B.3 and Sect. B.4). As will be noted, for some constellations in the case of Maxwell’s equations, the Fredholm property remains an open problem.

1.1 Injectivity

We start with the assumption that the operator M is block-diagonal, which allows to treat one subdomain at a time.

Assumption C1

The operator M from Ass. A3 has the block-diagonal form $M = \textrm{diag}(M_i)_{i=1}^N$, where each operator $M_i :\varLambda _i \rightarrow \varLambda _i^*$ is real-valued, symmetric, non-negative, and definite, i.e., $\langle M_i \lambda _i, {\overline{\lambda }}_i \rangle = 0 \implies \lambda _i = 0$ for all $\lambda _i \in \varLambda _i$.

Note that Assumption A6 implies Ass. C1 but not vice versa.

Assumption C2

In accordance with Assumption A8, the following holds:

• (i) In the coercive case (with $\alpha = 1$):

  $$\begin{aligned} \big [ A_i v_i = 0 \text { and } T_i v_i = 0 \big ] \implies v_i = 0 \qquad \forall v_i \in U_i\,. \end{aligned}$$

• (ii) In the wave propagation case (with $\alpha = \text {i}$):

  $$\begin{aligned} \big [ (A_{i,0} - A_{i,2})v_i = 0 \text { and } A_{i,1} v_i = 0 \text { and } T_i v_i = 0 \big ] \implies v_i = 0 \qquad \forall v_i \in U_i\,. \end{aligned}$$

Assumption C2 can be seen as an abstract version of Holmgreen’s theorem: $A_i v = 0$ (in case (i)) implies that the Neumann trace on the interface is zero, $T_i v = 0$ means that the Dirichlet trace on the interface is zero. Inside the subdomain, v fulfills the homogeneous PDE, so v must vanish entirely; see also [39, Thm. 1] or [40, Thm. A.1]. In the typical coercive cases, $A_i$ has a finite-dimensional kernel (constant functions, rigid body modes), which is fixed by the Dirichlet condition. For Maxwell’s equations, Part (ii) of Ass. C2 is widely known as the continuation principle, cf. e.g. [77, Sect. 4.6].

In the discrete case, the following proposition, which is essentially [11, Lemma 5.1], allows to derive Ass. C2 from the assumptions on the original operator ${\widehat{A}}$.

Proposition B.1

Let Assumption A8 hold, assume that $\ker ({\widehat{A}}) = \{ 0 \}$ (cf. Sect. 2.2), and let, in addition, Assumption B1 be fulfilled. Then Assumption C2 holds as well.

Proof

Assume without loss of generality, that we are in Case (ii) of Assumption A8 (the proof of Case (i) is analogous) and that $(A_{i,0} - A_{i,2}) v_i = 0$ and $A_{i,1} v_i = 0$ and $T_i v_i = 0$. Assumption B1 guarantees that $\ker (T_i) \subseteq U_{i,B}$, and so for $T_i v_i = 0$ there exists a function ${\widehat{v}} \in {\widehat{U}}$ such that $v_i = R_i {\widehat{v}}$ and $R_j {\widehat{v}} = 0$ for all $j \ne i$. From this and our initial assumptions, we can conclude that ${\widehat{A}} {\widehat{v}} = 0$, which implies ${\widehat{v}} = 0$ and therefore $v_i = 0$. $\square$

Proposition B.2

(injectivity) Let Assumptions A8, C1, and C2 hold. Then the operator ${{\widetilde{A}}}_i$ is injective.

Proof

For arbitrary but fixed $v \in U_i$ with ${{\widetilde{A}}}_i v = 0$, we show that $v = 0$.

Coercive case ($\alpha = 1$): ${{\widetilde{A}}}_i = A_i + T_i^{\textsf{T}}M_i T_i$ with $A_i$ real-valued, symmetric, and non-negative. Due to our assumptions,

$$\begin{aligned} \langle A_i v, {\overline{v}} \rangle + \langle M_i T_i v, T_i {\overline{v}} \rangle = 0, \end{aligned}$$

and since both terms are non-negative, both must vanish. From the assumptions on $A_i$ and $M_i$, this implies $A_i v = 0$ and $T_i v = 0$. Assumption C2(i) guarantees that $v = 0$.

Wave propagation case ($\alpha = \text {i}$):

$$\begin{aligned} {{\widetilde{A}}}_i = A_{i,0} + \text {i}{{\widetilde{A}}}_{i,1} - A_{i,2}\,, \qquad \text {where } {{\widetilde{A}}}_{i,1} = A_{i,1} + T_i^{\textsf{T}}M_i T_i\,. \end{aligned}$$

Due to our assumptions,

$$\begin{aligned} \langle (A_{i,0} - A_{i,2}) v, {\overline{v}} \rangle + \text {i}\Big ( \underbrace{ \langle A_{i,1} v, {\overline{v}} \rangle }_{ \ge 0 } + \underbrace{ \langle M_i T_i v, T_i {\overline{v}} \rangle }_{ \ge 0 } \Big ) = 0. \end{aligned}$$

Both the real and imaginary part must vanish. Since all the summands in the imaginary part are non-negative,

$$\begin{aligned} \langle A_{i,1} v, {\overline{v}} \rangle = 0 \qquad \text {and} \qquad \langle M_i T_i v, T_i v \rangle = 0. \end{aligned}$$

From the assumptions on $A_{i,1}$ and on $M_i$, it follows that $A_{i,1} v = 0$ and $T_i v = 0$. Recalling that ${{\widetilde{A}}}_i v = 0$, this implies also that $(A_{i,0} - A_{i,2})v = 0$. Assumption C2(ii) guarantees that $v = 0$. $\square$

1.2 Technical Tools for Fredholm Operators

Lemma B.3

Let V be a real or complexified Hilbert space and $A :V \rightarrow V^*$ a bounded linear operator that fulfills a generalized Gårding inequality with respect to an isomorphism ${\textsf{F}} :V \rightarrow V$ and a compact bounded linear operator $C :V \rightarrow V^*$, i.e., there exists a constant $\gamma > 0$ such that

$$\begin{aligned} \big | \langle A v, {\textsf{F}} {\overline{v}} \rangle + \langle C v, {\overline{v}} \rangle \big | \ge \gamma \Vert v \Vert _V^2 \qquad \forall v \in V, \end{aligned}$$

(B.1)

Then A is Fredholm with index zero.

Proof

We find that the (possibly complex-valued) operator ${\textsf{F}}^{\textsf{T}}A + C$ is positive bounded from below in the sense that

$$\begin{aligned} \big | \langle ({\textsf{F}}^{\textsf{T}}A + C) v, {\overline{v}} \rangle \big | \ge \gamma \Vert v \Vert _V^2 \qquad \forall v \in V. \end{aligned}$$

Since ${\textsf{F}}^{\textsf{T}}A + C$ is obviously bounded, a suitable version of the Lax-Milgram lemma (see e.g. [77, Lemma 2.21]) implies that ${\textsf{F}}^{\textsf{T}}A + C$ is an isomorphism. In particular, ${\textsf{F}}^{\textsf{T}}A + C$ is Fredholm with index zero. Since C is compact, a standard result (see e.g. [74, Thm 2.26]) implies that ${\textsf{F}}^{\textsf{T}}A$ is Fredholm with index zero. Since ${\textsf{F}}^{\textsf{T}}$ is an isomorphism, another standard argument (see e.g. [74, Thm. 2.21]) yields that A itself is Fredholm with index zero. $\square$

Lemma B.4

Let V be a complexified Hilbert space and $A :V \rightarrow V^*$ a bounded linear operator of the form

$$\begin{aligned} A = A_0 + \text {i}\sigma A_1 - A_2\,, \end{aligned}$$

with $\sigma \in \{ +1, -1 \}$ and with linear, bounded, real-valued, symmetric, and non-negative operators $A_i :V \rightarrow V^*$. Moreover, assume real-valued projection operators ${\textsf{N}}$ and ${\textsf{R}} :V \rightarrow V$ with ${\textsf{N}} + {\textsf{R}} = I$ such that

• (i) $A_0 {\textsf{N}} = 0$,

• (ii) $A_2 {\textsf{R}}$ is compact,

• (iii) either (a) $A_1 {\textsf{N}}$ is compact or (b) $A_1 {\textsf{R}}$ is compact, and

• (iv) there exists a constant $c > 0$ such that $\langle (A_0 + A_1 + A_2) v, {\overline{v}} \rangle \ge c\, \Vert v \Vert _V^2$ for all $v \in V$.

Then A is Fredholm with index zero. The same holds if (iii) and (iv) are replaced by the alternative conditions

• (iii’) ${\textsf{N}}^{\textsf{T}}A_1 {\textsf{R}}$ is compact, and

• (iv’) there exists a constant $c > 0$ such that $\langle (A_0 + A_2) v, {\overline{v}} \rangle \ge c\, \Vert v \Vert _V^2$ for all $v \in V$.

Remark B.5

The case ${\textsf{N}} = 0$ is admitted. In that case, only (ii) and (iv) are required.

Proof of Lemma B.4

We define ${\textsf{F}} := {\textsf{R}} - {\textsf{N}}$, which is an isomorphism:

$$\begin{aligned} {\textsf{F}}^2 = (2{\textsf{R}} - I)^2 = 4{\textsf{R}}^2 - 4{\textsf{R}} + I = I. \end{aligned}$$

Using property (i) and the relations ${\textsf{F}} = 2 {\textsf{R}} - I = I - 2 {\textsf{N}}$, we find that

$$\begin{aligned} \langle A v, {\textsf{F}} w \rangle&= \langle A_0 v, ({\textsf{R}} - {\textsf{N}}) w \rangle - \langle A_2 v, ({\textsf{R}} - {\textsf{N}}) w \rangle + \text {i}\sigma \langle A_1 v, ({\textsf{R}} - {\textsf{N}}) w \rangle \\&= \langle A_0 v, ({\textsf{R}} + {\textsf{N}}) w \rangle + \langle A_2 v, (I - 2 {\textsf{R}}) w \rangle + \text {i}\sigma \langle A_1 v, (2 {\textsf{R}} - I) w \rangle \\&= \langle (A_0 + A_2) v, w \rangle - \text {i}\sigma \langle A_1 v, w \rangle - 2 \langle A_2 v, {\textsf{R}} w \rangle + 2 \text {i}\sigma \langle A_1 v, {\textsf{R}} w \rangle , \end{aligned}$$

which will be used for Case (b). Alternatively, we have

$$\begin{aligned} \langle A v, {\textsf{F}} w \rangle&= \langle A_0 v, ({\textsf{R}} + {\textsf{N}}) w \rangle + \langle A_2 v, (I - 2 {\textsf{R}}) w \rangle + \text {i}\sigma \langle A_1 v, (I - 2 {\textsf{N}}) w \rangle \\&= \langle (A_0 + A_2) v, w \rangle + \text {i}\sigma \langle A_1 v, w \rangle - 2 \langle A_2 v, {\textsf{R}} w \rangle - 2 \text {i}\sigma \langle A_1 v, {\textsf{N}} w \rangle , \end{aligned}$$

which will be used for Case (a). We define $C :V \rightarrow V^*$ by

$$\begin{aligned} \langle C v, w \rangle := {\left\{ \begin{array}{ll} 2 \langle A_2 v, {\textsf{R}} w \rangle + 2 \text {i}\sigma \langle A_1 v, {\textsf{N}} w \rangle &{} \text {in Case~(a),}\\ 2 \langle A_2 v, {\textsf{R}} w \rangle - 2 \text {i}\sigma \langle A_1 v, {\textsf{R}} w \rangle &{} \text {in Case~(b),} \end{array}\right. } \end{aligned}$$

which is a compact operator by property (ii) and (iii). Then,

$$\begin{aligned} \big | \langle (A + C) v, {\textsf{F}} {\overline{v}} \rangle \big | = \big | \underbrace{\langle (A_0 + A_2) v, {\overline{v}} \rangle }_{\in {\mathbb {R}}_0^+} {} + \text {i}\underbrace{\delta \sigma }_{= \pm 1} \underbrace{\langle A_1 v, {\overline{v}} \rangle }_{\in {\mathbb {R}}_0^+} \big | = \langle (A_0 + A_1 + A_2) v, {\overline{v}} \rangle \ge c\, \Vert v \Vert _V^2\,, \end{aligned}$$

with $\delta = 1$ in Case (a) and $\delta = -1$ is Case (b). An application of Lemma B.3 shows that ${{\widetilde{A}}}$ is Fredholm. Under the conditions (iii’) and (iv’), we can use

$$\begin{aligned} \langle A v, {\textsf{F}} w \rangle&= \langle A_0 v, ({\textsf{R}} + {\textsf{N}}) w \rangle + \langle A_2 v, (I - 2{\textsf{R}}) w \rangle + \text {i}\sigma \langle A_1 ({\textsf{R}} + {\textsf{N}}) v, ({\textsf{R}} - {\textsf{N}}) w \rangle \\&= \langle (A_0 + A_2) v , w \rangle - 2 \langle A_2 v, {\textsf{R}} w \rangle + \text {i}\sigma \big ( \langle A_1 {\textsf{R}} v, {\textsf{R}} w \rangle - \langle A_1 {\textsf{N}} v, {\textsf{N}} w \rangle \\&\qquad - \langle A_1 {\textsf{R}} v, {\textsf{N}} w \rangle + \langle A_1 {\textsf{N}} v, {\textsf{R}} w \rangle \big ). \end{aligned}$$

We define $C :V \rightarrow V^*$ by $\langle C v, w \rangle := 2 \langle A_2 v, {\textsf{R}} w \rangle + \text {i}\sigma \langle A_1 {\textsf{R}} v, {\textsf{N}} w \rangle - \text {i}\sigma \langle A_1 {\textsf{N}} v, {\textsf{R}} w \rangle$, which is a compact operator by properties (ii) and (iii’). Due to (iv’)

$$\begin{aligned} \big | \langle (A + C) v, {\textsf{F}} {\overline{v}} \rangle \big |&= \big | \underbrace{ \langle (A_0 + A_2) v, {\overline{v}} \rangle }_{\in {\mathbb {R}}_0^+} + \text {i}\sigma \! \underbrace{ \big ( \langle A_1 {\textsf{R}} v, {\textsf{R}} {\overline{v}} \rangle \! - \! \langle A_1 {\textsf{N}} v, {\textsf{N}} {\overline{v}} \rangle \big )}_{ \in {\mathbb {R}} } \big | \\&\ge \langle (A_0 + A_2) v, {\overline{v}} \rangle \ge c \Vert v \Vert _V^2\,, \end{aligned}$$

and so again Lemma B.3 implies that ${{\widetilde{A}}}$ is Fredholm. $\square$

1.3 Standard Robin Problems

For the following, let us assume that $\varOmega \subset {\mathbb {R}}^3$ is a bounded Lipschitz domain and $\varGamma _D$, $\varGamma _N$, $\varGamma _R \subset \partial \varOmega$ disjoint surfaces such that $\partial \varOmega = \overline{\varGamma _D \cup \varGamma _N \cup \varGamma _R}$ and such that $\varGamma _R$ has non-vanishing surface measure. Note, however, that $\varGamma _D$ and/or $\varGamma _N$ are allowed to be empty. Moreover, any of the sets $\partial \varGamma _D$, $\partial \varGamma _N$, $\partial \varGamma _R$ (unless empty) should fulfill the requirements stated in [59, Sect. 2], in particular being the union of closed curves that are piecewise $C^1$. Later on, it is further assumed that $\varOmega$ is a curvilinear Lipschitz polyhedron.

1.3.1 The Primal Helmholtz Equation

Let ${\widehat{U}} := H^1_D(\varOmega ) = \{ v \in H^1(\varOmega ) :v = 0 \text { on } \varGamma _D \}$ and let the operator ${\widehat{A}} :{\widehat{U}} \rightarrow {\widehat{U}}^*$ be given by ${\widehat{A}} = {\widehat{A}}_0 + \text {i}{\widehat{A}}_1 - {\widehat{A}}_2$ with

$$\begin{aligned} \langle {\widehat{A}}_0 {\widehat{u}}, {\widehat{v}} \rangle = \int _\varOmega \nabla {\widehat{u}} \cdot \nabla {\widehat{v}} \, dx, \qquad \langle {\widehat{A}}_1 {\widehat{u}}, {\widehat{v}} \rangle = \int _{\varGamma _R} \eta \, {\widehat{u}}\, {\widehat{v}} \, ds, \qquad \langle {\widehat{A}}_2 {\widehat{u}}, {\widehat{v}} \rangle = \int _\varOmega \kappa ^2 {\widehat{u}}\, {\widehat{v}} \, dx. \end{aligned}$$

We set $\widehat{{\textsf{N}}} = 0$ and $\widehat{{\textsf{R}}} = I$. Since ${\widehat{A}}_2$ is compact and since

$$\begin{aligned} \langle ({\widehat{A}}_0 + {\widehat{A}}_1 + {\widehat{A}}_2) {\widehat{v}}, \overline{{\widehat{v}}} \rangle \ge \min (1, \kappa _{\min }^2) \Vert {\widehat{v}} \Vert _{H^1(\varOmega )}^2\,, \end{aligned}$$

where $\kappa _{\min }$ is a positive lower bound for the coefficient $\kappa$ in $\varOmega$, Lemma B.4 guarantees that ${\widehat{A}}$ is Fredholm.

1.3.2 The Dual Helmholtz Equation

Let ${\widehat{U}} := \{ \textbf{v}\in \textbf{H}_{\varGamma _N}(\mathop {{\text {div}}}, \varOmega ) :\textbf{v}_{n|\varGamma _R} \in L^2(\varGamma _R) \}$, where $\textbf{v}_n$ denotes the normal trace of $\textbf{v}$ in $H^{-1/2}(\partial \varOmega )$. The restriction $\textbf{v}_{n|\varGamma _R}$ is well-defined in $H^{-1/2}_{00}(\varGamma _R)$. We use the norm $\Vert \textbf{v}\Vert _{{\widehat{U}}}^2 = \Vert \textbf{v}\Vert _{\textbf{L}^2(\varOmega )}^2 + \Vert \mathop {{\text {div}}}\textbf{v}\Vert _{L^2(\varOmega )}^2 + \Vert \textbf{v}_{n|\varGamma _R} \Vert _{L^2(\varGamma _R)}^2$. The operator ${\widehat{A}} :{\widehat{U}} \rightarrow {\widehat{U}}^*$ is given by ${\widehat{A}} = {\widehat{A}}_0 + \text {i}{\widehat{A}}_1 - {\widehat{A}}_2$ with

$$\begin{aligned} \langle {\widehat{A}}_0 {{\widehat{\textbf{u}}}}, {{\widehat{\textbf{v}}}} \rangle = \int _\varOmega \kappa ^{-2} \mathop {{\text {div}}}{{\widehat{\textbf{u}}}}\, \mathop {{\text {div}}}{{\widehat{\textbf{v}}}} \, dx, \quad \langle {\widehat{A}}_1 {{\widehat{\textbf{u}}}}, {{\widehat{\textbf{v}}}} \rangle = \int _{\varGamma _R} \eta ^{-1}\, {{\widehat{\textbf{u}}}}_n {{\widehat{\textbf{v}}}}_n \, ds, \quad \langle {\widehat{A}}_2 {{\widehat{\textbf{u}}}}, {{\widehat{\textbf{v}}}} \rangle = \int _\varOmega {{\widehat{\textbf{u}}}} \cdot {{\widehat{\textbf{v}}}} \, dx. \end{aligned}$$

Due to the regular decomposition result in [59], there exist bounded, linear, and real-valued projections $\underline{{\textsf{N}}} :\textbf{H}_{\varGamma _N}(\mathop {{\text {div}}}, \varOmega ) \rightarrow \textbf{H}_{\varGamma _N}(\mathop {{\text {div}}}0, \varOmega )$ and $\underline{{\textsf{R}}} :\textbf{H}_{\varGamma _N}(\mathop {{\text {div}}}, \varOmega ) \rightarrow \textbf{H}^1_{\varGamma _N}(\varOmega )$ with $\underline{{\textsf{N}}} + \underline{{\textsf{R}}} = I$ in $\textbf{H}_{\varGamma _N}(\mathop {{\text {div}}}, \varOmega )$. For ${{\widehat{\textbf{v}}}} \in {\widehat{U}}$,

$$\begin{aligned} {{\widehat{\textbf{v}}}}_{n|\varGamma _R} = (\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}} )_{n|\varGamma _R} + (\underline{{\textsf{N}}} {{\widehat{\textbf{v}}}})_{n|\varGamma _R} \in L^2(\varGamma _R). \end{aligned}$$

Assuming that $\varOmega$ is a curvilinear polyhedron (cf. [6]), the outer normal $\textbf{n}$ is piecewise smooth. Therefore, since $\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}} \in \textbf{H}^1_{\varGamma _N}(\varOmega )$, we find that $(\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}})_{n|\varGamma _R} = (\underline{{\textsf{R}}}{{\widehat{\textbf{v}}}})_{|\varGamma _R} \cdot \textbf{n}\in H^{1/2}_{\text {pw}}(\varGamma _R) \subset L^2(\varGamma _R)$. This shows that $(\underline{{\textsf{N}}} {{\widehat{\textbf{v}}}})_{n|\varGamma _R} \in L^2(\varGamma _R)$ as well. Hence, we can restrict $\underline{{\textsf{R}}}$, $\underline{{\textsf{N}}}$ to operators $\widehat{{\textsf{R}}}$, $\widehat{{\textsf{N}}} :{\widehat{U}} \rightarrow {\widehat{U}}$, and we meet the prerequisites of Lemma B.4:

• $\widehat{{\textsf{R}}}$, $\widehat{{\textsf{N}}}$ are projectors and $\widehat{{\textsf{R}}} + \widehat{{\textsf{N}}} = I$ in ${\widehat{U}}$,

• ${\widehat{A}}_0 \widehat{{\textsf{N}}} = 0$ since $\underline{{\textsf{N}}}$ maps to $\textbf{H}_{\varGamma _N}(\mathop {{\text {div}}}0, \varOmega )$,

• ${\widehat{A}}_2 \widehat{{\textsf{R}}}$ is compact since $\underline{{\textsf{R}}}$ maps to $\textbf{H}^1_{\varGamma _N}(\varOmega )$ which is compactly embedded in $\textbf{L}^2(\varOmega )$,

• ${\widehat{A}}_1 \widehat{{\textsf{R}}}$ is compact since $(\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}})_{n|\varGamma _R} \in H^{1/2}_{\text {pw}}(\varGamma _R)$ which is compactly embedded in $L^2(\varGamma _R)$,

• $\langle ({\widehat{A}}_0 + {\widehat{A}}_1 + {\widehat{A}}_2) {{\widehat{\textbf{v}}}}, \overline{{{\widehat{\textbf{v}}}}} \rangle \ge \min (1, \kappa _{\max }^{-2}, \eta _{\max }^{-1}) \big ( \Vert \mathop {{\text {div}}}{{\widehat{\textbf{v}}}} \Vert _{L^2(\varOmega )}^2 + \Vert {{\widehat{\textbf{v}}}} \Vert _{\textbf{L}^2(\varOmega )}^2 + \Vert {{\widehat{\textbf{v}}}}_{n|\varGamma _R} \Vert _{L^2(\varGamma _R)}^2 \big )$,

where $\kappa _{\max }$, $\eta _{\max }$ are finite upper bounds for the coefficients $\kappa$, $\eta$ in $\varOmega$.

1.3.3 Maxwell’s Equations

To avoid complications, it is again assumed that $\varOmega$ is a curvilinear Lipschitz polyhedron (cf. [6]). Let ${\widehat{U}} := \{ \textbf{v}\in \textbf{H}_{\varGamma _D}(\mathop {{\text {\textbf{curl}}}}, \varOmega ) :\textbf{v}_{\tau |\varGamma _R} \in L^2(\varGamma _R) \}$, where $\textbf{v}_\tau = \textbf{v}\times \textbf{n}$ denotes the tangential trace of $\textbf{v}$ in $\textbf{H}^{-1/2}(\partial \varOmega )$ and the restriction $\textbf{v}_{\tau |\varGamma _R}$ is well-defined, for details see [6, 77]. We use the norm $\Vert \textbf{v}\Vert _{{\widehat{U}}}^2 = \Vert \textbf{v}\Vert _{\textbf{L}^2(\varOmega )}^2 + \Vert \mathop {{\text {\textbf{curl}}}}\textbf{v}\Vert _{\textbf{L}^2(\varOmega )}^2 + \Vert \textbf{v}_{\tau |\varGamma _R} \Vert _{\textbf{L}^2(\varGamma _R)}^2$. The operator ${\widehat{A}} :{\widehat{U}} \rightarrow {\widehat{U}}^*$ is given by ${\widehat{A}} = {\widehat{A}}_0 + \text {i}{\widehat{A}}_1 - {\widehat{A}}_2$ with

$$\begin{aligned}&\langle {\widehat{A}}_0 {{\widehat{\textbf{u}}}}, {{\widehat{\textbf{v}}}} \rangle = \! \int _\varOmega \!\! \mu ^{-1} \mathop {{\text {\textbf{curl}}}}{{\widehat{\textbf{u}}}} \cdot \mathop {{\text {\textbf{curl}}}}{{\widehat{\textbf{v}}}} \, dx, \quad \langle {\widehat{A}}_1 {{\widehat{\textbf{u}}}}, {{\widehat{\textbf{v}}}} \rangle = \! \int _{\varGamma _R} \!\!\! \omega \eta ^{-1}\, {{\widehat{\textbf{u}}}}_\tau \cdot {{\widehat{\textbf{v}}}}_\tau \, ds, \quad \\&\langle {\widehat{A}}_2 {{\widehat{\textbf{u}}}},{{\widehat{\textbf{v}}}} \rangle = \! \int _\varOmega \!\! \omega ^2 \varepsilon {{\widehat{\textbf{u}}}} \cdot {{\widehat{\textbf{v}}}} \, dx. \end{aligned}$$

There exist bounded, linear, and real-valued projectors $\underline{{\textsf{N}}} :\textbf{H}_{\varGamma _D}(\mathop {{\text {\textbf{curl}}}}, \varOmega ) \rightarrow \textbf{H}_{\varGamma _D}(\mathop {{\text {\textbf{curl}}}}0, \varOmega )$ and $\underline{{\textsf{R}}} :\textbf{H}_{\varGamma _D}(\mathop {{\text {\textbf{curl}}}}, \varOmega ) \rightarrow \textbf{H}^1_{\varGamma _D}(\varOmega )$ with $\underline{{\textsf{R}}} + \underline{{\textsf{N}}} = I$ in $\textbf{H}_{\varGamma _D}(\mathop {{\text {\textbf{curl}}}}, \varOmega )$, see [59]. For ${{\widehat{\textbf{v}}}} \in {\widehat{U}}$,

$$\begin{aligned} {{\widehat{\textbf{v}}}}_{\tau |\varGamma _R} = (\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}} )_{\tau |\varGamma _R} + (\underline{{\textsf{N}}} {{\widehat{\textbf{v}}}})_{\tau |\varGamma _R} \in \textbf{L}^2(\varGamma _R). \end{aligned}$$

Due to the assumptions on $\varOmega$, the normal $\textbf{n}$ is piecewise smooth. Therefore, since $\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}} \in \textbf{H}^1_{\varGamma _D}(\varOmega )$, we find that $(\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}})_{\tau |\varGamma _R} = (\underline{{\textsf{R}}}{{\widehat{\textbf{v}}}})_{|\varGamma _R} \times \textbf{n}\in \textbf{H}^{1/2}_{\text {pw}}(\varGamma _R) \subset \textbf{L}^2(\varGamma _R)$. This shows that $(\underline{{\textsf{N}}} {{\widehat{\textbf{v}}}})_{\tau |\varGamma _R} \in \textbf{L}^2(\varGamma _R)$ as well. Hence, we can restrict $\underline{{\textsf{N}}}$, $\underline{{\textsf{R}}}$ to operators $\widehat{{\textsf{N}}}$, $\widehat{{\textsf{R}}} :{\widehat{U}} \rightarrow {\widehat{U}}$, and we meet the prerequisites of Lemma B.4:

• $\widehat{{\textsf{N}}}$, $\widehat{{\textsf{R}}}$ are projectors and $\widehat{{\textsf{N}}} + \widehat{{\textsf{R}}} = I$ in ${\widehat{U}}$,

• ${\widehat{A}}_0 \widehat{{\textsf{N}}} = 0$ since $\underline{{\textsf{N}}}$ maps to $\textbf{H}_{\varGamma _D}(\mathop {{\text {\textbf{curl}}}}0, \varOmega )$,

• ${\widehat{A}}_2 \widehat{{\textsf{R}}}$ is compact since $\underline{{\textsf{R}}}$ maps to $\textbf{H}^1_{\varGamma _D}(\varOmega )$ which is compactly embedded in $\textbf{L}^2(\varOmega )$,

• ${\widehat{A}}_1 \widehat{{\textsf{R}}}$ is compact since $(\underline{{\textsf{R}}} {{\widehat{\textbf{v}}}})_{\tau |\varGamma _R} \in \textbf{H}^{1/2}_{\text {pw}}(\varGamma _R)$ which is compactly embedded in $\textbf{L}^2(\varGamma _R)$,

• $\langle ({\widehat{A}}_0 + {\widehat{A}}_1 + {\widehat{A}}_2) {{\widehat{\textbf{v}}}}, \overline{{{\widehat{\textbf{v}}}}} \rangle \ge \min (\mu _{\max }^{-1}, \omega ^2 \varepsilon _{\min }^2, \omega \eta _{\min }) \big ( \Vert \mathop {{\text {\textbf{curl}}}}{{\widehat{\textbf{v}}}} \Vert _{L^2(\varOmega )}^2 + \Vert {{\widehat{\textbf{v}}}} \Vert _{\textbf{L}^2(\varOmega )}^2 + \Vert {{\widehat{\textbf{v}}}}_{\tau |\varGamma _R} \Vert _{\textbf{L}^2(\varGamma _R)}^2 \big )$,

where $\mu _{\max }$, $\varepsilon _{\min }$, $\eta _{\min }$ are finite upper/positive lower bounds for the coefficients $\mu$, $\varepsilon$, $\eta$ in $\varOmega$.

1.4 Generalized Robin Problems

In this section, we investigate whether the subdomain operator

$$\begin{aligned} {{\widetilde{A}}}_i := A_i + \alpha T_i^{\textsf{T}}M_i T_i \end{aligned}$$

is Fredholm with index zero.

1.4.1 Wave Propagation Case

In accordance with Assumption A8 (with $\alpha = \text {i}$), we assume that there exist projectors ${\textsf{N}}_i$, ${\textsf{R}}_i$ such that

• (i) $A_{i,0} {\textsf{N}}_i = 0$,

• (ii) $A_{i,2} {\textsf{R}}_i$ is compact,

• (iii) either (a) $A_{i,1} {\textsf{N}}_i$ is compact or (b) $A_{i,1} {\textsf{R}}_i$ is compact, and

• (iv) there exists a constant $c_i > 0$ such that $\langle (A_{i,0} + A_{i,1} + A_{i,2}) v, {\overline{v}} \rangle \ge c \Vert v \Vert _{U_i}^2$ for all $v \in U_i$.

such that Lemma B.4 implies that $A_i$ is Fredholm with index zero. With the definitions

$$\begin{aligned} {{\widetilde{A}}}_i = A_{i,0} + \text {i}{{\widetilde{A}}}_{i,1} - A_{i,2}\,, \qquad \text {where } {{\widetilde{A}}}_{i,1} = A_{i,1} + T_i^{\textsf{T}}M_i T_i\,, \end{aligned}$$

Lemma B.4 would imply that ${{\widetilde{A}}}_i$ is Fredholm with index zero as well if, in addition to the above:

• (v) either (a) ${{\widetilde{A}}}_{i,1} {\textsf{N}}_i$ is compact or (b) ${{\widetilde{A}}}_{i,1} {\textsf{R}}_i$ is compact, and

• (vi) there exists a constant $c_i > 0$ such that $\langle (A_{i,0} + {{\widetilde{A}}}_{i,1} + A_{i,2}) v, {\overline{v}} \rangle \ge c \Vert v \Vert _{U_i}^2$ for all $v \in U_i$.

The inequality (vi) follows from (iv) because $M_i$ is non-negative. We are left with the question whether the operator $T_i^{\textsf{T}}M_i T_i {\textsf{N}}_i$ (in case (a)) or $T_i^{\textsf{T}}M_i T_i {\textsf{R}}_i$ (in case (b)) is compact:

• For the primal Helmholtz formulation, we can use ${\textsf{N}}_i = 0$.

• If the trace operator $T_i$ itself is compact (cf. Theorem 6.4), we are also done.

• For the dual Helmholtz formulation, (iii) holds with case (b), $\textrm{range}(T_i {\textsf{R}}_i) \subset H^{1/2}_{\text {pw}}(\varGamma _i)$, where $\varGamma _i$ is the chosen interface, possibly split into facets. The latter space is compactly embedded in $L^2(\varGamma _i)$, and thus also compactly embedded in any chosen trace space $\varLambda _i$ (which requires at most $H^{-1/2}$-regularity). Therefore, $T_i {\textsf{R}}_i :U_i \rightarrow \varLambda _i$ is compact, and so (v) holds with case (b).

The Maxwell case is to a large extent open, at least if $T_i$ is not compact. However, one exceptional situation shall be mentioned: If $A_{i,1} = 0$ and $M_i$ is orthogonal with respect to the regular decomposition, i.e., ${\textsf{N}}_i^{\textsf{T}}T_i^{\textsf{T}}M_i T_i {\textsf{R}}_i = 0$, then using the alternative conditions (iii’), (iv’) in Lemma B.4, one can show that ${{\widetilde{A}}}_i$ is Fredholm with index zero. For more results see [83, Sect. 3.4.2].

1.4.2 Coercive Case

In accordance with Assuption A8 (with $\alpha = 1$), we assume that there exists a compact operator¹⁶$C_i$ such that $A_i + C_i$ is bounded positively from below, such that $A_i$ is Fredholm with index zero. Since, by assumption $M_i$ is non-negative,

$$\begin{aligned} \langle ({{\widetilde{A}}}_i + C_i) v, {\overline{v}} \rangle = \langle (A_i + C_i) v, {\overline{v}} \rangle + \langle M_i T_i v, T_i {\overline{v}} \rangle \ge \gamma _i \Vert v \Vert _{U_i}^2, \end{aligned}$$

for some constant $\gamma _i > 0$ such that ${{\widetilde{A}}}_i$ is Fredholm.