In order to solve (2.1) by a combination of DPG method and finite elements, we
formulate the problem in a heterogeneous way, using different variational forms in different
parts of the domain. For ease of illustration, we restrict ourselves to two Lipschitz sub-domains
${\mathrm{\Omega}}_{1}$, ${\mathrm{\Omega}}_{2}$ (again of polygonal/polyhedral form, each with one connected component)
with boundaries $\partial {\mathrm{\Omega}}_{1}$, $\partial {\mathrm{\Omega}}_{2}$, as specified in Figure 1.
There, also a notation for the boundary pieces is introduced. In particular, Γ denotes
the interface between the sub-domains. The picture indicates that both sub-domains touch the
boundary of Ω (where the homogeneous Dirichlet condition is imposed), but this is not
essential. For instance, one sub-domain, ${\mathrm{\Omega}}_{2}$, can be of annular type so that, in this case,
$\partial \mathrm{\Omega}\subset \partial {\mathrm{\Omega}}_{2}$ and $\mathrm{\Gamma}=\partial {\mathrm{\Omega}}_{1}$.
Other combinations can be analyzed without difficulty, also including Neumann conditions.
Nevertheless, since our analysis centers around proving coercivity of bilinear forms,
we need positivity of the combined advection-reaction term on a sub-domain that does not touch
the Dirichlet boundary.

#### Assumption 2.1.

For $i=1,2$, the following condition holds: If $\mathrm{meas}({\mathrm{\Gamma}}_{i})=0$, then there exists $\beta >0$ such that
$\frac{1}{2}\mathrm{div}\bm{\beta}+\gamma \ge \beta $ a.e. in ${\mathrm{\Omega}}_{i}$.

Figure 1 Decomposition of the domain Ω into sub-domains. $\overline{\mathrm{\Omega}}={\overline{\mathrm{\Omega}}}_{1}\cup {\overline{\mathrm{\Omega}}}_{2}$, ${\mathrm{\Omega}}_{1}\cap {\mathrm{\Omega}}_{2}=\mathrm{\varnothing}$, ${\overline{\mathrm{\Omega}}}_{1}\cap {\overline{\mathrm{\Omega}}}_{2}=\mathrm{\Gamma}$, ${\overline{\mathrm{\Omega}}}_{1}\cap \partial \mathrm{\Omega}={\overline{\mathrm{\Gamma}}}_{1}$, ${\overline{\mathrm{\Omega}}}_{2}\cap \partial \mathrm{\Omega}={\overline{\mathrm{\Gamma}}}_{2}$.

*Standard and broken Sobolev spaces.*
Essential for the DPG method is to use broken test spaces. Therefore, at this early stage
we consider a partitioning ${\mathcal{\mathcal{T}}}_{1}$ of ${\mathrm{\Omega}}_{1}$ into (regular non-intersecting) finite elements
*T* such that ${\overline{\mathrm{\Omega}}}_{1}=\bigcup \{\overline{}T;T\in {\mathcal{\mathcal{T}}}_{1}\}$,
and with skeleton $\mathcal{\mathcal{S}}:=\{\partial T;T\in {\mathcal{\mathcal{T}}}_{1}\}$.

Before describing the variational formulation, we introduce the Sobolev spaces we need.
For a domain $\omega \subset \mathrm{\Omega}$, we use the standard spaces ${L}_{2}(\omega )$, ${H}^{1}(\omega )$,
${H}_{0}^{1}(\omega )$, and $\mathbf{H}(\mathrm{div},\omega )$.
The trace operator acting on ${H}^{1}(\omega )$ will be denoted simply by ${(\cdot )|}_{\partial \omega}$.
Then we define the trace space ${H}^{1/2}(\partial \omega ):={{H}^{1}(\omega )|}_{\partial \omega}$
and its dual space
${H}^{-1/2}(\partial \omega ):={({H}^{1/2}(\partial \omega ))}^{\prime}$
with canonical norms.
The duality pairing on $\partial \omega $ is ${\u3008\cdot ,\cdot \u3009}_{\partial \omega}$
and extends the ${L}_{2}(\partial \omega )$ bilinear form. Correspondingly,
${(\cdot ,\cdot )}_{\omega}$ is the ${L}_{2}(\omega )$ bilinear form.

We also need ${H}_{D}^{1}({\mathrm{\Omega}}_{i})$ consisting of ${H}^{1}$-functions with
vanishing trace on ${\mathrm{\Gamma}}_{i}$ ($i=1,2$).
Vector-valued spaces and functions will be denoted by bold symbols.
Connected with ${\mathcal{\mathcal{T}}}_{1}$ we use the product spaces ${H}^{1}({\mathcal{\mathcal{T}}}_{1})$ and $\mathbf{H}(\mathrm{div},{\mathcal{\mathcal{T}}}_{1})$
with corresponding product norms.

Now, related with ${\mathcal{\mathcal{T}}}_{1}$ are the skeleton trace spaces

${H}^{1/2}(\mathcal{\mathcal{S}}):=\{\widehat{u}\in {\mathrm{\Pi}}_{T\in {\mathcal{\mathcal{T}}}_{1}}{H}^{1/2}(\partial T);\text{there exists}w\in {H}^{1}(\mathrm{\Omega})\text{such that}\widehat{u}{|}_{\partial T}=w{|}_{\partial T}\text{for all}T\in {\mathcal{\mathcal{T}}}_{1}\},$${H}^{-1/2}(\mathcal{\mathcal{S}}):=\{\widehat{\sigma}\in {\mathrm{\Pi}}_{T\in {\mathcal{\mathcal{T}}}_{1}}{H}^{-1/2}(\partial T);\text{there exists}\bm{q}\in \mathbf{H}(\mathrm{div},\mathrm{\Omega})\text{such that}\widehat{\sigma}{|}_{\partial T}=(\bm{q}\cdot {\mathbf{n}}_{T}){|}_{\partial T}\text{for all}T\in {\mathcal{\mathcal{T}}}_{1}\}$

and

${H}_{00}^{1/2}(\mathcal{\mathcal{S}}):=\{\widehat{u}\in {H}^{1/2}(\mathcal{\mathcal{S}});\widehat{u}{|}_{\partial {\mathrm{\Omega}}_{1}}=0\},$${H}_{D}^{1/2}(\mathcal{\mathcal{S}}):=\{\widehat{u}\in {H}^{1/2}(\mathcal{\mathcal{S}});\widehat{u}{|}_{{\mathrm{\Gamma}}_{1}}=0\}.$

Here, ${\mathbf{n}}_{T}$ is the exterior unit normal vector on $\partial T$, and
${(\bm{q}\cdot {\mathbf{n}}_{T})|}_{\partial T}$ indicates the standard way of defining normal traces
of $\mathbf{H}(\mathrm{div},T)$-functions.
The notation ${\widehat{u}|}_{\partial {\mathrm{\Omega}}_{1}}=0$ (resp. ${\widehat{u}|}_{{\mathrm{\Gamma}}_{1}}=0$)
is to be understood in the sense that $\widehat{u}$ is a ${\mathcal{\mathcal{T}}}_{1}$-piecewise trace of an element of
${H}_{0}^{1}({\mathrm{\Omega}}_{1})$ (resp. of ${H}_{D}^{1}({\mathrm{\Omega}}_{1})$).
These trace spaces are equipped with the norms

${\parallel \widehat{u}\parallel}_{{H}^{1/2}(\mathcal{\mathcal{S}})}:=inf\{{\parallel w\parallel}_{{H}^{1}(\mathrm{\Omega})};w\in {H}^{1}(\mathrm{\Omega})\text{such that}\widehat{u}{|}_{\partial T}=w{|}_{\partial T}\text{for all}T\in {\mathcal{\mathcal{T}}}_{1}\},$${\parallel \widehat{\sigma}\parallel}_{{H}^{-1/2}(\mathcal{\mathcal{S}})}:=inf\{{\parallel \bm{q}\parallel}_{\mathbf{H}(\mathrm{div},\mathrm{\Omega})};\bm{q}\in \mathbf{H}(\mathrm{div},\mathrm{\Omega})\text{such that}\widehat{\sigma}{|}_{\partial T}=(\bm{q}\cdot {\mathbf{n}}_{T}){|}_{\partial T}\text{for all}T\in {\mathcal{\mathcal{T}}}_{1}\},$

and analogously for ${H}_{00}^{1/2}(\mathcal{\mathcal{S}})$ and ${H}_{D}^{1/2}(\mathcal{\mathcal{S}})$.
For functions $\widehat{u}\in {H}^{1/2}(\mathcal{\mathcal{S}})$, $\widehat{\sigma}\in {H}^{-1/2}(\mathcal{\mathcal{S}})$ (they are elements of product spaces)
and $\bm{\tau}\in \mathbf{H}(\mathrm{div},{\mathcal{\mathcal{T}}}_{1})$, $v\in {H}^{1}({\mathcal{\mathcal{T}}}_{1})$, we use the duality pairings

${\u3008\widehat{u},\bm{\tau}\cdot \mathbf{n}\u3009}_{\mathcal{\mathcal{S}}}:=\sum _{T\in {\mathcal{\mathcal{T}}}_{1}}{\u3008{\widehat{u}|}_{\partial T},\bm{\tau}\cdot {\mathbf{n}}_{T}\u3009}_{\partial T},{\u3008\widehat{\sigma},v\u3009}_{\mathcal{\mathcal{S}}}:=\sum _{T\in {\mathcal{\mathcal{T}}}_{1}}{\u3008{\widehat{\sigma}|}_{\partial T},v\u3009}_{\partial T}.$

*Heterogeneous formulation in ${\mathrm{\Omega}}_{\mathrm{1}}\mathrm{\cup}{\mathrm{\Omega}}_{\mathrm{2}}$.*
In ${\mathrm{\Omega}}_{1}$, where the DPG method will be used, we consider an ultra-weak variational formulation.
As mentioned before, this is just for illustration as any other formulation of
primal, mixed, dual-mixed or strong type can be used and analyzed analogously to our case;
cf. [12, Section 2.3].

The ultra-weak formulation requires additional independent unknowns

$\bm{\sigma}:=\underset{\xaf}{\alpha}\nabla u-\bm{\beta}u\text{on}{\mathrm{\Omega}}_{1},\widehat{u}:={{\mathrm{\Pi}}_{T\in {\mathcal{\mathcal{T}}}_{1}}u|}_{\partial T},\widehat{\sigma}:={{\mathrm{\Pi}}_{T\in {\mathcal{\mathcal{T}}}_{1}}(\bm{\sigma}\cdot {\mathbf{n}}_{T})|}_{\partial T}.$(2.2)

Then we test the defining relation of $\bm{\sigma}$
with ${\underset{\xaf}{\alpha}}^{-T}\bm{\tau}$ for $\bm{\tau}\in \mathbf{H}(\mathrm{div},{\mathcal{\mathcal{T}}}_{1})$, and equation (2.1) with $v\in {H}^{1}({\mathcal{\mathcal{T}}}_{1})$.
Integrating by parts element-wise and substituting
the corresponding terms by $\bm{\sigma}$, $\widehat{u}$, and $\widehat{\sigma}$, we obtain

${(u,{\mathrm{div}}_{\mathcal{\mathcal{T}}}\bm{\tau}+\bm{\beta}{\underset{\xaf}{\alpha}}^{-T}\bm{\tau}+\gamma v)}_{{\mathrm{\Omega}}_{1}}+{(\bm{\sigma},{\nabla}_{\mathcal{\mathcal{T}}}v+{\underset{\xaf}{\alpha}}^{-T}\bm{\tau})}_{{\mathrm{\Omega}}_{1}}-{\u3008\widehat{u},\bm{\tau}\cdot \mathbf{n}\u3009}_{\mathcal{\mathcal{S}}}-{\u3008\widehat{\sigma},v\u3009}_{\mathcal{\mathcal{S}}}={(f,v)}_{{\mathrm{\Omega}}_{1}}.$(2.3)

Here, ${\mathrm{div}}_{\mathcal{\mathcal{T}}}$ and ${\nabla}_{\mathcal{\mathcal{T}}}$ denote the ${\mathcal{\mathcal{T}}}_{1}$-piecewise divergence and gradient operators,
respectively.

In ${\mathrm{\Omega}}_{2}$, we use the standard primal formulation

${(\underset{\xaf}{\alpha}\nabla u-\bm{\beta}u,\nabla w)}_{{\mathrm{\Omega}}_{2}}+{(\gamma u,w)}_{{\mathrm{\Omega}}_{2}}-{\u3008{\mathbf{n}}_{{\mathrm{\Omega}}_{2}}\cdot (\underset{\xaf}{\alpha}\nabla u-\bm{\beta}u),w\u3009}_{\partial {\mathrm{\Omega}}_{2}}={(f,w)}_{{\mathrm{\Omega}}_{2}}$(2.4)

for $w\in {H}_{D}^{1}({\mathrm{\Omega}}_{2})$.

Solving (2.1) in Ω is equivalent to solving (in appropriate spaces)
(2.3) and (2.4) with homogeneous Dirichlet condition on $\partial \mathrm{\Omega}$
and transmission conditions on Γ.
These transmission conditions will be imposed in variational form. For the time being, we replace
${{\mathbf{n}}_{{\mathrm{\Omega}}_{2}}\cdot (\underset{\xaf}{\alpha}\nabla u-\bm{\beta}u)|}_{\mathrm{\Gamma}}$ by $-{\widehat{\sigma}|}_{\mathrm{\Gamma}}$ in (2.4).
Here, we slightly abuse the notation of $\widehat{\sigma}$ noting that
${\u3008\widehat{\sigma},v\u3009}_{\mathcal{\mathcal{S}}}={\u3008\widehat{\sigma},v\u3009}_{\mathrm{\Gamma}}$ for $v\in {H}^{1}({\mathrm{\Omega}}_{1})$ with ${v|}_{{\mathrm{\Gamma}}_{1}}=0$;
cf., e.g., [13, Section 2.2].

We formally distinguish between ${u}_{1}:={u|}_{{\mathrm{\Omega}}_{1}}$ and ${u}_{2}:={u|}_{{\mathrm{\Omega}}_{2}}$. Then our
preliminary heterogeneous variational formulation consists in finding

$(\bm{u},{u}_{2})=({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma},{u}_{2})\in U:={U}_{1}\times {H}_{D}^{1}({\mathrm{\Omega}}_{2})\mathit{\hspace{1em}}\text{with}\mathit{\hspace{1em}}{U}_{1}:={L}_{2}({\mathrm{\Omega}}_{1})\times {\mathbf{L}}_{2}({\mathrm{\Omega}}_{1})\times {H}_{D}^{1/2}(\mathcal{\mathcal{S}})\times {H}^{-1/2}(\mathcal{\mathcal{S}})$

such that ${\widehat{u}|}_{\mathrm{\Gamma}}={{u}_{2}|}_{\mathrm{\Gamma}}$ and

${({u}_{1},{\mathrm{div}}_{\mathcal{\mathcal{T}}}\bm{\tau}+\bm{\beta}{\underset{\xaf}{\alpha}}^{-T}\bm{\tau}+\gamma v)}_{{\mathrm{\Omega}}_{1}}+{(\bm{\sigma},{\nabla}_{\mathcal{\mathcal{T}}}v+{\underset{\xaf}{\alpha}}^{-T}\bm{\tau})}_{{\mathrm{\Omega}}_{1}}-{\u3008\widehat{u},\bm{\tau}\cdot \mathbf{n}\u3009}_{\mathcal{\mathcal{S}}}-{\u3008\widehat{\sigma},v\u3009}_{\mathcal{\mathcal{S}}}={(f,v)}_{{\mathrm{\Omega}}_{1}},$${(\underset{\xaf}{\alpha}\nabla {u}_{2}-\bm{\beta}{u}_{2},\nabla w)}_{{\mathrm{\Omega}}_{2}}+{(\gamma {u}_{2},w)}_{{\mathrm{\Omega}}_{2}}+{\u3008\widehat{\sigma},w\u3009}_{\mathrm{\Gamma}}={(f,w)}_{{\mathrm{\Omega}}_{2}}$(2.5)

for any $(\bm{v},w)\in V\times {H}_{D}^{1}({\mathrm{\Omega}}_{2})$ with

$\bm{v}=(v,\bm{\tau})\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}V:={H}^{1}({\mathcal{\mathcal{T}}}_{1})\times \mathbf{H}(\mathrm{div},{\mathcal{\mathcal{T}}}_{1}).$

This formulation can be used to define the combined DPG-FEM discretization,
but requires that ${\mathcal{\mathcal{T}}}_{1}$ is compatible across Γ with the finite element mesh
in ${\mathrm{\Omega}}_{2}$. We therefore replace the continuity constraint ${\widehat{u}|}_{\mathrm{\Gamma}}={{u}_{2}|}_{\mathrm{\Gamma}}$
by a variational coupling on Γ that is similar to a DG-bilinear form
involving jumps and fluxes across element boundaries. To this end, we abbreviate

$b(\bm{u},\bm{v}):={({u}_{1},{\mathrm{div}}_{\mathcal{\mathcal{T}}}\bm{\tau}+\bm{\beta}{\underset{\xaf}{\alpha}}^{-T}\bm{\tau}+\gamma v)}_{{\mathrm{\Omega}}_{1}}+{(\bm{\sigma},{\nabla}_{\mathcal{\mathcal{T}}}v+{\underset{\xaf}{\alpha}}^{-T}\bm{\tau})}_{{\mathrm{\Omega}}_{1}}-{\u3008\widehat{u},\bm{\tau}\cdot \mathbf{n}\u3009}_{\mathcal{\mathcal{S}}}-{\u3008\widehat{\sigma},v\u3009}_{\mathcal{\mathcal{S}}},$${c}_{2}({u}_{2},{w}_{2}):={(\underset{\xaf}{\alpha}\nabla {u}_{2}-\bm{\beta}{u}_{2},\nabla {w}_{2})}_{{\mathrm{\Omega}}_{2}}+{(\gamma {u}_{2},{w}_{2})}_{{\mathrm{\Omega}}_{2}},$(2.6)${L}_{1}(\bm{v}):={(f,v)}_{{\mathrm{\Omega}}_{1}},{L}_{2}({w}_{2}):={(f,{w}_{2})}_{{\mathrm{\Omega}}_{2}},$

and define the coupling bilinear form

$d(\bm{u},{u}_{2};\bm{w},{w}_{2}):={\u3008\widehat{\sigma},{w}_{2}\u3009}_{\mathrm{\Gamma}}+{\u3008\widehat{\chi},\widehat{u}-{u}_{2}\u3009}_{\mathrm{\Gamma}}+{\displaystyle \frac{1}{2}}{\u3008\bm{\beta}\cdot {\mathbf{n}}_{{\mathrm{\Omega}}_{1}}(\widehat{u}-{u}_{2}),\widehat{w}+{w}_{2}\u3009}_{\mathrm{\Gamma}}$(2.7)$\text{for}\hspace{1em}(\bm{u},{u}_{2}),(\bm{w},{w}_{2})\in U\hspace{1em}\text{with}\hspace{1em}\bm{u}=({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma}),\bm{w}=({w}_{1},\bm{\chi},\widehat{w},\widehat{\chi}).$

We will also need the bilinear form for ${\mathrm{\Omega}}_{1}$ that corresponds to ${c}_{2}(\cdot ,\cdot )$:

${c}_{1}({u}_{1},{w}_{1}):={(\underset{\xaf}{\alpha}\nabla {u}_{1}-\bm{\beta}{u}_{1},\nabla {w}_{1})}_{{\mathrm{\Omega}}_{1}}+{(\gamma {u}_{1},{w}_{1})}_{{\mathrm{\Omega}}_{1}}\mathit{\hspace{1em}}({u}_{1},{w}_{1}\in {H}^{1}({\mathrm{\Omega}}_{1})).$(2.8)

The construction of $d(\cdot ;\cdot )$ fulfills two objectives.
First, for a function $(\bm{u},{u}_{2})\in U$ with $\bm{u}=({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma})$
and continuity $\widehat{u}={u}_{2}$ on Γ, there holds
$d(\bm{u},{u}_{2};\bm{w},{w}_{2})={\u3008\widehat{\sigma},{w}_{2}\u3009}_{\mathrm{\Gamma}}$ for any $(\bm{w},{w}_{2})\in U$.
Therefore, this bilinear form simply replaces the duality ${\u3008\widehat{\sigma},w\u3009}_{\mathrm{\Gamma}}$
in (2.1).
Second, by selecting $(\bm{w},{w}_{2})=(\bm{u},{u}_{2})=({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma},{u}_{2})\in U$,
the bilinear form reduces to

$d(\bm{u},{u}_{2};\bm{u},{u}_{2})={\u3008\widehat{\sigma},\widehat{u}\u3009}_{\mathrm{\Gamma}}+\frac{1}{2}{\u3008\bm{\beta}\cdot {\mathbf{n}}_{{\mathrm{\Omega}}_{1}}\widehat{u},\widehat{u}\u3009}_{\mathrm{\Gamma}}-\frac{1}{2}{\u3008\bm{\beta}\cdot {\mathbf{n}}_{{\mathrm{\Omega}}_{1}}{u}_{2},{u}_{2}\u3009}_{\mathrm{\Gamma}}.$

The first term on the right-hand side is needed for consistency.
The last two terms are the ones that generate coercivity of the bilinear forms
${c}_{1}(\cdot ,\cdot )$ and ${c}_{2}(\cdot ,\cdot )$; cf. Lemma 3.1 below.

The final combined ultra-weak primal formulation of (2.1) then reads

$(\bm{u},{u}_{2})=({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma},{u}_{2})\in U\text{such that}$$b(\bm{u},\bm{v})={L}_{1}(\bm{v})\mathrm{\hspace{1em}}\text{for all}\bm{v}\in V,$(2.9a)${c}_{2}({u}_{2},{w}_{2})+d(\bm{u},{u}_{2};\bm{w},{w}_{2})={L}_{2}({w}_{2})\mathrm{\hspace{1em}}\text{for all}(\bm{w},{w}_{2})\in U.$(2.9b)

Note that, although in (2.9b) we test with functions $(\bm{w},{w}_{2})$ from *U*,
only ${w}_{2}$ and the restrictions to Γ of the $\widehat{w}$ and $\widehat{\chi}$-components of $\bm{w}$
are being used there. Therefore, (2.9a) represents problem (2.1) restricted
to ${\mathrm{\Omega}}_{1}$ without interface condition on Γ, whereas (2.9b) represents
the problem on ${\mathrm{\Omega}}_{2}$ and relates the Cauchy data ${\widehat{u}|}_{\mathrm{\Gamma}}$ and ${\widehat{\sigma}|}_{\mathrm{\Gamma}}$.

For reference, we explicitly specify the strong form of (2.9a):

$\bm{u}:=({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma})\in {U}_{1}\text{such that}B\bm{u}={L}_{1}.$(2.10)

By following [12], one can show that (2.9) is equivalent to (2.1),
so that, in particular, (2.9) has a unique solution; see also Remark 2.3 below.
However, since we will use different strategies for solving (2.9a) and (2.9b),
we need a slightly different representation.
To this end, we define the trial-to-test operator $\mathrm{\Theta}:{U}_{1}\to V$ by

${\u3008\mathrm{\Theta}\bm{u},\bm{v}\u3009}_{V}=b(\bm{u},\bm{v})\mathit{\hspace{1em}}\text{for all}\bm{v}\in V.$

Here, ${\u3008\cdot ,\cdot \u3009}_{V}$ denotes the canonical inner product in *V*.
Note that $\mathrm{\Theta}={\mathcal{\mathcal{R}}}^{-1}B$ with the Riesz operator $\mathcal{\mathcal{R}}:V\to {V}^{\prime}$.
Since *B* is defined on ${U}_{1}$ without boundary condition along Γ, it has a non-trivial kernel,
and so does Θ. Still, $\mathrm{\Theta}:{U}_{1}\to V$ is surjective. Therefore, denoting by
${\mathrm{\Theta}}_{\kappa}:=\kappa \mathrm{\Theta}$ the scaled trial-to-test operator (for $\kappa >0$ to be chosen)
yields the following equivalent formulation:
For given $\kappa >0$, find
$(\bm{u},{u}_{2})\in U$ such that

$a(\bm{u},{u}_{2};\bm{w},{w}_{2})=L(\bm{w},{w}_{2})\mathit{\hspace{1em}}\text{for all}(\bm{w},{w}_{2})\in U,$(2.11)

with

$a(\bm{u},{u}_{2};\bm{w},{w}_{2}):=b(\bm{u},{\mathrm{\Theta}}_{\kappa}\bm{w})+{c}_{2}({u}_{2},{w}_{2})+d(\bm{u},{u}_{2};\bm{w},{w}_{2})$

and

$L(\bm{w},{w}_{2}):={L}_{1}({\mathrm{\Theta}}_{\kappa}\bm{w})+{L}_{2}({w}_{2}).$

One of our main results is the following theorem.

#### Theorem 2.2.

*Let $\kappa \mathrm{>}\mathrm{0}$ be sufficiently large. Then the variational formulation (2.11) is well posed,
and equivalent to problem (2.1) in the following sense:
If $u\mathrm{\in}{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ solves (2.1), then
$\mathrm{(}\mathbf{u}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{=}\mathrm{(}{u}_{\mathrm{1}}\mathrm{,}\mathbf{\tau}\mathrm{,}\widehat{u}\mathrm{,}\widehat{\sigma}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}$, with ${u}_{i}\mathrm{:=}{u\mathrm{|}}_{{\mathrm{\Omega}}_{i}}$ ($i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$) and
$\mathbf{\tau}\mathrm{,}\widehat{u}\mathrm{,}\widehat{\sigma}$ defined by (2.2), satisfies $\mathrm{(}\mathbf{u}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{\in}U$ and solves (2.11).*

*Vice versa, if $\mathrm{(}\mathbf{u}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{=}\mathrm{(}{u}_{\mathrm{1}}\mathrm{,}\mathbf{\tau}\mathrm{,}\widehat{u}\mathrm{,}\widehat{\sigma}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{\in}U$ solves (2.11),
then **u* defined by ${u\mathrm{|}}_{{\mathrm{\Omega}}_{i}}\mathrm{:=}{u}_{i}$ ($i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$) satisfies $u\mathrm{\in}{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ and solves (2.1).

*Furthermore, the bilinear form $a\mathit{}\mathrm{(}\mathrm{\cdot}\mathrm{,}\mathrm{\cdot}\mathrm{)}$ is **U*-coercive, i.e.,

$a(\bm{u},{u}_{2};\bm{u},{u}_{2})\gtrsim {\parallel (\bm{u},{u}_{2})\parallel}_{U}^{2}\mathit{\hspace{1em}}\mathit{\text{for all}}(\bm{u},{u}_{2})\in U.$(2.12)

#### Proof.

By the assumptions on Ω, *f*, $\underset{\xaf}{\alpha}$, $\bm{\beta}$, and γ, problem (2.1)
is uniquely solvable. Furthermore, by the derivation of (2.11), if $u\in {H}_{0}^{1}(\mathrm{\Omega})$
solves (2.1), then $(\bm{u},{u}_{2})$ as specified in the statement solves (2.11).
This can be seen by integrating by parts and noting that
$d(\bm{u},{u}_{2};\bm{w},{w}_{2})={\u3008\widehat{\sigma},{w}_{2}\u3009}_{\mathrm{\Gamma}}$ since ${\widehat{u}|}_{\mathrm{\Gamma}}={{u}_{2}|}_{\mathrm{\Gamma}}$; cf. (2.7).

The coercivity of $a(\cdot ,\cdot )$ will be shown in Section 3.1 under the assumption
that $\kappa >0$ is large enough. It is also straightforward to show that
this bilinear form is bounded on $U\times U$,
as is the linear functional *L* on *U*. The Lax–Milgram lemma proves the
well-posedness of (2.11) for $\kappa >0$ being large enough.
∎

As previously mentioned, the continuity constraint ${\widehat{u}|}_{\mathrm{\Gamma}}={{u}_{2}|}_{\mathrm{\Gamma}}$ can also be
imposed strongly. In this case, the solution space is

${U}^{0}:=\{({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma};{u}_{2})\in U;\widehat{u}{|}_{\mathrm{\Gamma}}={u}_{2}{|}_{\mathrm{\Gamma}}\}$

and the coupling bilinear form reduces to

${d}^{0}(\bm{u},{w}_{2}):=d(\bm{u},{u}_{2};\bm{w},{w}_{2})={\u3008\widehat{\sigma},{w}_{2}\u3009}_{\mathrm{\Gamma}}\mathit{\hspace{1em}}\text{for all}(\bm{u},{u}_{2})=({u}_{1},\bm{\sigma},\widehat{u},\widehat{\sigma},{u}_{2}),(\bm{w},{w}_{2})\in {U}^{0}.$

The variational formulation becomes the following: For given $\kappa >0$, find $(\bm{u},{u}_{2})\in {U}^{0}$ such that

${a}^{0}(\bm{u},{u}_{2};\bm{w},{w}_{2})=L(\bm{w},{w}_{2})\mathit{\hspace{1em}}\text{for all}(\bm{w},{w}_{2})\in {U}^{0},$(2.13)

with

${a}^{0}(\bm{u},{u}_{2};\bm{w},{w}_{2}):=b(\bm{u},{\mathrm{\Theta}}_{\kappa}\bm{w})+{c}_{2}({u}_{2},{w}_{2})+{d}^{0}(\bm{u};{w}_{2})$(2.14)

and

$L(\bm{w},{w}_{2}):={L}_{1}({\mathrm{\Theta}}_{\kappa}\bm{w})+{L}_{2}({w}_{2}).$

Analogously to Theorem 2.2, one obtains the well-posedness of (2.13) and
coercivity of ${a}^{0}(\cdot ,\cdot )$.

#### Corollary 2.4.

*Let $\kappa \mathrm{>}\mathrm{0}$ be sufficiently large. Then the variational formulation (2.13) is well posed,
and equivalent to problem (2.1) in the following sense:
If $u\mathrm{\in}{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ solves (2.1), then
$\mathrm{(}\mathbf{u}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{=}\mathrm{(}{u}_{\mathrm{1}}\mathrm{,}\mathbf{\tau}\mathrm{,}\widehat{u}\mathrm{,}\widehat{\sigma}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}$, with ${u}_{i}\mathrm{:=}{u\mathrm{|}}_{{\mathrm{\Omega}}_{i}}$ ($i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$) and
$\mathbf{\tau}\mathrm{,}\widehat{u}\mathrm{,}\widehat{\sigma}$ defined by (2.2), satisfies $\mathrm{(}\mathbf{u}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{\in}{U}^{\mathrm{0}}$ and solves (2.13).*

*Vice versa, if $\mathrm{(}\mathbf{u}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{=}\mathrm{(}{u}_{\mathrm{1}}\mathrm{,}\mathbf{\tau}\mathrm{,}\widehat{u}\mathrm{,}\widehat{\sigma}\mathrm{,}{u}_{\mathrm{2}}\mathrm{)}\mathrm{\in}{U}^{\mathrm{0}}$ solves (2.13),
then **u* defined by ${u\mathrm{|}}_{{\mathrm{\Omega}}_{i}}\mathrm{:=}{u}_{i}$ ($i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$) satisfies $u\mathrm{\in}{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ and solves (2.1).

*Furthermore, for sufficiently large $\kappa \mathrm{>}\mathrm{0}$, the bilinear form ${a}^{\mathrm{0}}\mathit{}\mathrm{(}\mathrm{\cdot}\mathrm{,}\mathrm{\cdot}\mathrm{)}$ is ${U}^{\mathrm{0}}$-coercive, i.e.,*

$a(\bm{u},{u}_{2};\bm{u},{u}_{2})\gtrsim {\parallel (\bm{u},{u}_{2})\parallel}_{U}^{2}\mathit{\hspace{1em}}\mathit{\text{for all}}(\bm{u},{u}_{2})\in {U}^{0}.$

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