A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients

1 Introduction

Partial differential equations (PDE) with sign-changing coefficients are used to model physical phenomena of meta-materials, for example, wave transmission problems between classical materials and meta-materials. Therefore, there is an emerging interest in the development of numerical methods for such PDEs. However, the bilinear forms for problems with sign-changing coefficients are not coercive, so standard techniques relying on coercivity cannot be applied. The lack of coercivity indeed poses fundamental challenges to study the well-posedness of such PDEs, as well as the development of numerical methods.

A popular approach in the study of numerical methods for PDEs with sign-changing coefficients is the “ $T$ -coercivity” approach. This approach utilizes a linear operator $T$ that recovers the coercivity of the bilinear form. The existence of such an operator $T$ is a natural assumption because the existence of $T$ is equivalent to the well-posedness of the PDE in the sense of Banach–Nečas–Babuška in the functional analysis framework [7]. The application of the $T$ operator to develop numerical methods was first proposed in [2]. Since then, the $T$ -coercivity approach has been used in the development of numerical methods for Poisson-type problems [7], Helmholtz-type problems [8], and Maxwell-type problems [2, 5, 4].

In numerical methods that use $T$ -coercivity, the $T$ operator cannot be used directly, because the domain and the range of $T$ are not the discrete spaces in general. For this reason, $T_{h}$ is introduced, a discrete approximation of $T$ , which has the coercivity recovery property in the discrete setting. The existence of such $T_{h}$ is non-trivial and difficult to guarantee because the form of $T$ is unknown in general. For general geometry and meshes sufficient conditions for the existence of $T_{h}$ are proposed in [7]

, but these include non-quantitative constants such as a norm of the discrete interpolation of

T

and the ratio of the positive and negative coefficients (the contrast). It is furthermore shown in [7] that locally symmetric meshes across the transmission interface, where the coefficient changes sign, improve the quality of numerical solutions. However, the generation of such meshes is a non-trivial restriction in mesh generation for complex interfaces.

Hybridization (or static condensation) was originally introduced for mixed finite element methods to reduce computational cost [15], and it was further analyzed in [1]. More recently, hybridization was combined with the discontinuous Galerkin (DG) method. This resulted in the hybridizable discontinuous Galerkin (HDG) method which was shown to be more efficient than standard DG methods. The HDG method was systematically presented in [9] for elliptic partial differential equations. Since its introduction, it has been extended to many different problems. These include HDG methods for the Helmholtz problem, e.g., [12, 17] and the Maxwell problem, e.g., [6, 14].

In this paper, we develop a numerical method for Poisson-type problems with a sign-changing coefficient that avoids the $T$ -coercivity approach. The numerical method we present in this paper is always well-posed without any conditions on the domain geometry or the ratio between the negative and positive coefficients. This is achieved by employing two key ideas for the problem: a mixed formulation of the problem and using a hybridized discontinuous Galerkin (HDG) method with sign-changing stabilization parameters. We will see that the sign-changing stabilization parameter and the discontinuous test function space in discontinuous Galerkin methods allow us to obtain stable numerical methods without any non-quantitative assumptions on mesh size or coefficients.

We also carry out an error analysis of the HDG method for this problem. The error analysis of HDG methods typically uses the Aubin–Nitsche duality argument with the full elliptic regularity assumption. However, this assumption is not feasible in our setting, because the PDE is not elliptic. By revealing a stabilized saddle point problem structure of HDG methods, we circumvent this obstacle and we can show an error estimate without using the duality argument. We note that such analysis has been applied also to HDG methods for standard Poisson-type problems to avoid the full elliptic regularity assumption in the a priori error analysis (see

[11]).

The outline of this paper is as follows. In Section 2 we introduce the Poisson problem with a sign-changing coefficient. We introduce the HDG method and discuss its well-posedness in Section 3. An a priori error analysis is presented in Section 4. In Section 5 we show that a superconvergence result can be obtained when a suitable regularity assumption is available. Our analysis is verified by numerical examples in Section 6, while conclusions are drawn in Section 7.

2 The Poisson problem with a sign-changing coefficient

Let $Ω \subset R^{d}$ , $d \geq 2$ , be a polygonal domain that is divided into two subdomains $Ω_{+}$ and $Ω_{-}$ such that $¯ ¯¯ ¯ Ω = {¯ ¯¯ ¯ Ω}_{+} \cup {¯ ¯¯ ¯ Ω}_{-}$ and $Ω_{+} \cap Ω_{-} = \emptyset$ . The boundaries of $Ω$ , $Ω_{+}$ and $Ω_{-}$ are denoted by, respectively, $\partial Ω$ , $\partial Ω_{+}$ and $\partial Ω_{-}$ . The interface separating the domains is denoted by $Γ_{I} := {¯ ¯¯ ¯ Ω}_{+} \cap {¯ ¯¯ ¯ Ω}_{-}$ and we define furthermore $Γ_{+} := \partial Ω_{+} ∖ Γ_{I}$ and $Γ_{-} := \partial Ω_{-} ∖ Γ_{I}$ . We assume that $Γ_{D}$ and $Γ_{N}$ are disjoint subsets of $\partial Ω$ such that $\partial Ω =_{D} \cup_{N}$

. The outward unit normal vector field on

\partial Ω

is denoted by

n

Let $σ$ be a scalar function defined as

σ := {\begin{matrix} σ_{+} & on Ω_{+}, σ_{-} & on Ω_{-}, \end{matrix}

(1)

where $σ_{+} > 0$ and $σ_{-} < 0$ are constants. The contrast is defined as $κ_{σ} := σ_{-} / σ_{+}$ . Throughout this paper we assume that

0 < σ_{min} \leq σ_{+}, - σ_{-} \leq σ_{max} < + \infty .

(2)

We consider the following partial differential equation (PDE) for the scalar $u : Ω \to R$ : equationparentequation


$∇⋅\delσ∇u$	$= f$	$in Ω,$	(3a)
$u$	$= u_{D}$	$on Γ_{D},$	(3b)
$- σ \nabla u \cdot n$	$= u_{N}$	$on Γ_{N},$	(3c)

where $u_{D} : Γ_{D} \to R$ and $u_{N} : Γ_{N} \to R$ are given boundary data and $f : Ω \to R$ is a given source term. This is not a standard second order PDE, because the sign of $σ$ is indefinite. Let $H10(Ω):=\cbrv∈H1(Ω):v=0 on ∂Ω$ where $H^{1} (Ω)$ is the standard Sobolev space of $L^{2}$ functions such that the $L^{2}$ -norm of their gradient is bounded. The variational formulation for the problem (when $u_{D} = u_{N} = 0$ ) is given by: Find $u \in H_{0}^{1} (Ω)$ such that

∫Ωσ∇u⋅∇v\difx=∫Ωfv\difx∀v∈H10(Ω).

(4)

It is known that this problem is not always well-posed, for example, when $κ_{σ} = - 1$ , the problem is ill-posed. The conditions of well-posedness of the problem depends on the values of $σ$ and the geometry of $Ω_{+}$ and $Ω_{-}$ . For more information on the well-posedness of this problem we refer to [7, 3].

3 The HDG method

Introducing the auxiliary variable $q = - σ \nabla u$ , Eq. 3 can be written as a system of first-order equations: equationparentequation


$σ^{- 1} q + \nabla u$	$= 0$	$in Ω,$	(5a)
$\nabla \cdot q$	$= - f$	$in Ω,$	(5b)
$u$	$= u_{D}$	$on Γ_{D},$	(5c)
$- σ \nabla u \cdot n$	$= u_{N}$	$on Γ_{N} .$	(5d)

In this section we introduce the HDG method for Eq. 5.

3.1 Preliminaries

Let $T_{h}$ be a triangulation of $Ω$ and $F_{h}$ be the set of faces in the triangulation $T_{h}$ with co-dimension 1. By $T_{h}^{+}$ we denote the subset of $T_{h}$ such that its elements are in $Ω_{+}$ . $T_{h}^{-}$ is defined similarly. We assume that $T_{h}$ is a conforming triangulation with respect to $Γ_{I}$ , i.e., there is a subset $F_{h}^{i}$ of $F_{h}$ such that $Γ_{I} = \cup_{F \in F_{h}^{i}} ¯ ¯¯ ¯ F$ . For later reference we define $F_{h}^{+}$ and $F_{h}^{-}$ as the subsets of interior facets of $F_{h}$ such that the facets are in $Ω_{+}$ and $Ω_{-}$ , respectively. We also define $F_{h}^{\partial}$ as the facets on $\partial Ω$ , i.e., $\partial Ω = \cup_{F_{h}^{\partial}} ¯ ¯¯ ¯ F$ . In summary, $F_{h}$ is a disjoint union of $F_{h}^{\partial}$ , $F_{h}^{+}$ , $F_{h}^{-}$ , $F_{h}^{i}$ . We also define $F_{h}^{D}$ as the subset of $F_{h}$ such that $_{D} = \cup_{F \in F_{h}^{D}} ¯ ¯¯ ¯ F$ .

Let $K \in T_{h}$ , denote the diameter of $K$ by $h_{K}$ , and let $h = {max}_{K \in T_{h}} h_{K}$ . For scalar functions $u, v \in L^{2} (K)$ and vector functions $q, r \in L^{2} (K; R^{d})$ we denote, respectively, $\delu,vK:=∫Kuv\difx$ and $\delq,rK:=∫Kq⋅r\difx$ . Furthermore, for two functions $r$ and $v$ with well-defined traces on $\partial K$ , we define $⟨r⋅n,v⟩∂K:=∫∂Kr⋅nKv\difs$ , where $n_{K}$ is the unit outward normal vector field on $\partial K$ . Additionally, we define $\delu,vD:=∑K⊂D\delu,vK$ , $\delq,rD:=∑K⊂D\delq,rK$ , and $⟨ r \cdot n, v ⟩_{\partial T_{h}} := \sum_{K \in T_{h}} ⟨ r \cdot n_{K}, v ⟩_{\partial K}$ , with similar definitions on $T_{h}^{+}$ and $T_{h}^{-}$ .

We use the standard notation for Sobolev spaces based on the $L^{2}$ -norm, i.e., $H^{s} (D)$ , $s \geq 0$ , denotes the Sobolev space based on the $L^{2}$ -norm with $s$ -differentiability on the domain $D$ . We refer to [13] for a rigorous definition of this space. The norm on $H^{s} (D)$ is denoted by $\norm⋅s,D$ . When $s = 0$ we drop the subscript $s$ .

To define the HDG method let

V_{h} (K) = P_{k} (K; R^{d}), W_{h} (K) = P_{k} (K), M_{h} (F) = P_{k} (F),

where $P_{k} (D)$ and $P_{k} (D; R^{d})$ are the spaces of scalar and $d$ -dimensional vector valued polynomials of degree at most $k$ on a domain $D$ . We will use the following finite element spaces:

$V_{h}$	$=\cbr[0]r∈L2(Th;Rd):r\|K∈Vh(K),∀K∈Th,$	(6)
$W_{h}$	$=\cbr[0]v∈L2(Th):v\|K∈Wh(K),∀K∈Th,$	(7)
$M_{h}$	$=\cbr[0]μ∈L2(Fh):μ\|F∈Mh(F),∀F∈Fh,μ\|FDh=0.$	(8)

3.2 The discrete formulation

For a sufficiently regular solution $(u, q)$ of the mixed problem Eq. 5, and for $¯ u$ the restriction of $u$ on $F_{h}$ , we can derive a system of variational equations from Eq. 5 with test functions in $V_{h} \times W_{h} \times M_{h}$ as equationparentequation


$\del[0]σ−1q,rΩ+\del[0]∇u,rΩ−⟨u−¯u,r⋅n⟩∂Th$	$= 0$	$r \in V_{h},$	(9a)
$\del[0]q,∇vΩ−⟨q⋅n+τ(u−¯u),v⟩∂Th$	$=\del[0]f,vΩ$	$v \in W_{h},$	(9b)
${⟨ q \cdot n + τ (u - ¯ u), ¯ v ⟩}_{\partial T_{h} ∖ F_{h}^{D}}$	$= {⟨ u_{N}, ¯ v ⟩}_{F_{h}^{N}}$	$¯ v \in M_{h} .$	(9c)

Here we use $τ$ to denote a piecewise constant function adapted to $T_{h}$ ; $τ$ in the integration on $\partial K$ is the trace of $τ |_{K}$ . Our HDG method for Eq. 5 is the discrete counterpart of this system of variational equations, i.e., we find $(q_{h}, u_{h}, {¯ u}_{h}) \in V_{h} \times W_{h} \times M_{h}$ such that equationparentequation


$\del[0]σ−1qh,rΩ+\del[0]∇uh,rΩ−⟨uh−¯uh,r⋅n⟩∂Th$		$r \in V_{h},$	(10a)
$\del[0]qh,∇vΩ−⟨qh⋅n+τ(uh−¯uh),v⟩∂Th$	$=−⟨τuD,v⟩FDh+\del[0]f,vΩ$	$v \in W_{h},$	(10b)
${⟨ q_{h} \cdot n + τ (u_{h} - ¯ u_{h}), ¯ v ⟩}_{\partial T_{h}}$	$= {⟨ u_{N}, ¯ v ⟩}_{F_{h}^{N}}$	$¯ v \in M_{h} .$	(10c)

Here equations Eq. 9a, Eq. 9b and the discrete equations Eq. 10a, Eq. 10b look inconsistent. This is because we impose the restriction $μ |_{F_{h}^{D}} = 0$ on $M_{h}$ so that we can take the same trial and test function spaces in our numerical method. This choice will be useful to reveal a stabilized saddle point structure of our numerical method, and also allow us to obtain optimal error estimates without the Aubin–Nitsche duality argument. To clarify the consistency between Eq. 9a, Eq. 9b and Eq. 10a, Eq. 10b, we point out that Eq. 10a, Eq. 10b with ${¯ u}_{h} \in {~ M}_{h}$ are equivalent to the discretized forms of Eq. 9a, Eq. 9b with ${~ M}_{h}$ defined by

~Mh=\cbr[0]μ∈L2(Fh):μ|F∈Mh(F) ∀F∈Fh, ⟨μ,λ⟩FDh=⟨uD,λ⟩FDh ∀λ∈Mh(FDh).

For later use we define

ah(q,r):=\del[0]σ−1q,rΩ,bh(v,¯v;r):=\del[0]∇v,rΩ−⟨v−¯v,r⋅n⟩∂Th,ch(u,¯u;v,¯v):=⟨τ(u−¯u),v−¯v⟩∂Th.

(11)

Then the sum of the left-hand sides of Eq. 10 can be rewritten as

B_{h} (q_{h}, u_{h}, {¯ u}_{h}; r, v, ¯ v) := a_{h} (q_{h}, r) + b_{h} (u_{h}, {¯ u}_{h}; r) + b_{h} (v, ¯ v; q_{h}) - c_{h} (u_{h}, {¯ u}_{h}; v, ¯ v) .

(12)

For brevity, but without loss of generality, we assume $u_{D} = u_{N} = 0$ and $Γ_{D} \neq \emptyset$ in the remainder of this paper.

In the following lemma we prove well-posedness of the HDG method Eq. 10.

Lemma 1 (Well-posedness)

Let $k \geq 0$ . If $τ > 0$ on $Ω_{+}$ and $τ < 0$ on $Ω_{-}$ , there exists a unique solution $(q_{h}, u_{h}, {¯ u}_{h}) \in V_{h} \times W_{h} \times M_{h}$ to Eq. 10. $_{□}$

Proof

To show well-posedness of Eq. 10 assume that $f = 0$ . We will show that $q_{h}$ , $u_{h}$ , ${¯ u}_{h}$ vanish.

Let $χ_{D}$

be the characteristic function that has the value 1 in domain

D

and 0 elsewhere. Then let

r = χ_{Ω_{+}} q_{h} - χ_{Ω_{-}} q_{h}

v = - χ_{Ω_{+}} u_{h} + χ_{Ω_{-}} u_{h}

¯ v = χ_{_{+}} {¯ u}_{h} - χ_{_{-}} {¯ u}_{h}

in Eq. 10, and add all the equations. The evaluation of Eq. 10 with the first components of these test functions gives

\del[0]σ−1qh,qhΩ++⟨τ(uh−¯uh),uh−¯uh⟩∂T+h=0.

Similarly, the evaluation of Eq. 10 with the second components of the above test functions gives

−\del[0]σ−1qh,qhΩ−−⟨τ(uh−¯uh),uh−¯uh⟩∂T−h=0.

Since $σ^{- 1}$ and $τ$ are positive and negative on $Ω_{+}$ and $Ω_{-}$ , respectively, we can conclude that $q_{h} = 0$ on $Ω$ and $u_{h} = {¯ u}_{h}$ on $F_{h}^{+} \cup F_{h}^{-} \cup F_{h}^{i}$ . This also implies that $u_{h}$ is continuous on $Ω$ .

Since $q_{h} = 0$ and $u_{h} - {¯ u}_{h} = 0$ it follows from Eq. 10a that $\nabla u_{h} = 0$ , i.e., $u_{h}$ is constant on $Ω$ . Then $u_{h} = 0$ on $Ω$ because $u_{h} = {¯ u}_{h} = 0$ on $Γ_{D}$ . $_{■}$

We consider the following norms on $V_{h}$ and $W_{h} \times M_{h}$ :

	$\norm[0]r2Vh$	$=\del[0]σ−1r,rΩ+−\del[0]σ−1r,rΩ−,$
	$\norm[0](v,¯v)2Wh×Mh$	$=\del[0]v,vΩ+⟨\|τ\|\del[0]v−¯v,v−¯v⟩∂Th,$

where $| τ |$ is the absolute value of $τ$ . Additionally, we define $X_{h} := V_{h} \times W_{h} \times M_{h}$ with norm

\norm[0](r,v,¯v)Xh:=\del[1]\norm[0]r2Vh+\norm[0](v,¯v)2Wh×Mh12.

Furthermore, in the remainder of this paper we let $C > 0$ be a constant independent of the mesh size $h$ .

Lemma 2

There exists $C_{Ω} > 0$ which depends only on $Ω$ and $Γ_{D}$ such that for $v \in W_{h}$ there exists $w \in H^{1} (Ω; R^{d})$ satisfying $\nabla \cdot w = v$ and $\normw1,Ω≤CΩ\normvΩ$ with $w \cdot n = 0$ on $Γ_{N}$ . $_{□}$

Proof

We will use a result in [16, p.176] which claims that for any given $f \in L^{2} (Ω)$ and $a \in H^{\frac{1}{2}} (\partial Ω)$ satisfying $∫∂Ωa⋅n\difs=∫Ωf\difx$ , there exists $w \in H^{1} (Ω; R^{d})$ such that $\nabla \cdot w = f$ , $w = a$ on $\partial Ω$ , and $\normwH1(Ω)≤C\normfL2(Ω)+\normaH12(∂Ω)$ .

We first note that if $z \in L^{2} (Ω)$ has mean value zero, then there exists $w \in H_{0}^{1} (Ω; R^{d})$ such that $\nabla \cdot w = z$ and $\normwH1(Ω)≤C\normzL2(Ω)$ by taking $a = 0$ in the above result.

Consider now a decomposition $v = v_{0} + v_{m}$ with $vm=χΩ1|Ω|∫Ωv\difx$ where $| Ω |$ is the Lebesgue measure of $Ω$ , and where $v_{0}$ is the component of $v$ with mean-value zero. It is clear that $\normv0L2(Ω)+\normvmL2(Ω)≤C\normvL2(Ω)$ . As just noted, there exists $w_{0} \in H_{0}^{1} (Ω; R^{d})$ such that $\nabla \cdot w_{0} = v_{0}$ and $\normw0H1(Ω)≤C\normv0L2(Ω)$ . Since $Ω$ is a polygonal/polyhedral domain, one can find a function $ϕ \in H^{1 / 2} (\partial Ω)$ whose support resides on $Γ_{D}$ , $∫ΓDϕ⋅n\difs=c0>0$ , and $\normϕH12(∂Ω)=1$ . We can find $ϕ_{0}$ , a renormalization of $ϕ$ , satisfying $∫ΓDϕ0⋅n\difs=∫Ωvm\difx$ , and it holds that

\normϕ0H12(∂Ω)≤\normvmL1(Ω)c0≤C\normvmL2(Ω),

with a constant that depends on $| Ω |$ and $c_{0}$ . Applying the result in [16, p.176] there exists $w_{1} \in H^{1} (Ω; R^{d})$ such that $\nabla \cdot w_{1} = v_{m}$ , $w_{1} = ϕ_{0}$ on $\partial Ω$ , and $\normw1H1(Ω)≤C\normϕ0H1/2(∂Ω)+\normvmL2(Ω)≤C\normvmL2(Ω)$ . Then $w = w_{0} + w_{1}$ is a function in $H^{1} (Ω; R^{d})$ satisfying the desired conditions. $_{■}$

Theorem 1 (inf-sup condition)

Suppose that $k \geq 0$ . If $τ > 0$ on $Ω_{+}$ and $τ < 0$ on $Ω_{-}$ , and $γ_{0} \leq | τ_{K} | \leq γ_{1}$ for every $K \in T_{h}$ , with positive constants $γ_{0}$ and $γ_{1}$ independent of $h$ , then there exists a positive constant $β$ independent of $h$ such that

inf(p,z,¯z)∈Xhsup(r,v,¯v)∈XhBh(p,z,¯z;r,v,¯v)\norm[0](p,z,¯z)Xh\norm[0](r,v,¯v)Xh≥β>0.

(13)

$_{□}$

Proof

We first prove the following weak inf-sup condition: There exist $C_{0}, C_{1} > 0$ independent of $h$ such that

inf(z,¯z)∈Wh×Mhsupr∈Vhbh(z,¯z;r)\normrVh≥C0\normzΩ−C1⟨|τ|(z−¯z),z−¯z⟩12∂Th.

(14)

We show this by proving an equivalent condition, i.e., there exist $C_{0}^{'}, C_{1}^{'}, C_{2}^{'} > 0$ independent of $h$ such that for any $0 \neq (z, ¯ z) \in W_{h} \times M_{h}$ there exists $r \in V_{h}$ such that $\normrVh≤C′2\normzΩ$ and

bh(z,¯z;r)≥C′0\normz2Ω−C′1⟨|τ|(z−¯z),z−¯z⟩12∂Th\normzΩ.

(15)

By Lemma 2 there exists $w \in H^{1} (Ω; R^{d})$ with $w \cdot n |_{Γ_{N}} = 0$ such that $\nabla \cdot w = - z$ and $\normw1,Ω≤CΩ\normzΩ$ with $C_{Ω}$ depending only on $Ω$ and $Γ_{D}$ . We use $Π_{h} w$ to denote the element-wise $L^{2}$ orthogonal projection of $w$ into $V_{h}$ . It then holds that

\del[0]w,rK=\del[0]Πhw,rK∀r∈Pk−1(K;Rd),

for all $K \in T_{h}$ . Then

−\normz2Ω=\del[0]z,∇⋅wΩ=−\del[0]∇z,wΩ+⟨z,w⋅n⟩∂Th=−\del[0]∇z,ΠhwΩ+⟨z−¯z,w⋅n⟩∂Th=−bh\del[0]z,¯z;Πhw+⟨z−¯z,(w−Πhw)⋅n⟩∂Th,

(16)

where the third equality holds because $w \cdot n$ and $¯ z$ are single-valued on facets, $¯ z = 0$ on $Γ_{D}$ by the definition of $M_{h}$ , and $w \cdot n = 0$ on $Γ_{N}$ .

The element-wise trace inequality and the Cauchy-Schwarz inequality give

(17)

Combining Eq. 16 and Eq. 17 we obtain

bh\del[0]z,¯z;Πhw≥\normz2Ω−C′⟨|τ|(z−¯z),z−¯z⟩12∂Th\normzΩ.

(18)

Moreover,

\normΠhwΩ≤\normwΩ≤\normw1,Ω≤CΩ\normzΩ

(19)

holds, so the weak inf-sup condition Eq. 15 follows.

To prove Eq. 13 suppose that $r_{0} \in V_{h}$ is an element satisfying Eq. 15 for given $z$ . For given $(p, z, ¯ z) \in X_{h}$ we take $r = χ_{Ω_{+}} p - χ_{Ω_{-}} p + ε r_{0}$ , $v = - χ_{Ω_{+}} z + χ_{Ω_{-}} z$ , $¯ v = - χ_{_{+}} ¯ z + χ_{_{-}} ¯ z$ in Eq. 12, with $ε > 0$ to be determined later. Then by Eq. 15 and Young’s inequality,

	$B_{h} (p, z, ¯ z; r, v, ¯ v)$	$=\normp2Vh+ε\del[0]σ−1p,r0Ω+εbh(z,¯z;r0)$
		$+ {⟨ \| τ \| (z - ¯ z), z - ¯ z ⟩}_{\partial T_{h}}$
		$≥\normp2Vh−12\normp2Vh−12ε2(C′2)2\normz2Ω$
		$+εC′0\normz2Ω−εC′1⟨\|τ\|(z−¯z),z−¯z⟩12∂Th\normzΩ$
		$+ {⟨ \| τ \| (z - ¯ z), z - ¯ z ⟩}_{\partial T_{h}}$
		$≥\normp2Vh−12\normp2Vh−12ε2(C′2)2\normz2Ω$
		$+εC′0\normz2Ω−12⟨\|τ\|(z−¯z),z−¯z⟩∂Th$
		$−12ε2(C′1)2\normz2Ω+⟨\|τ\|(z−¯z),z−¯z⟩∂Th.$

Choosing $ε$ sufficiently small, we obtain

Bh(p,z,¯z;r,v,¯v)≥C\del[1]\normp2Vh+\normz2Ω+⟨|τ|(z−¯z),z−¯z⟩∂Th.

Finally, it is not difficult to show, by the choice of $(r, v, ¯ v)$ , that

\norm(r,v,¯v)Xh≤C\norm(p,z,¯z)Xh.

This proves Eq. 13. $_{■}$

4 Error analysis

In this section we present the a priori error estimates of our HDG method. For this we recall an interpolation result (cf. [10, Theorem 2.1]): For $K \in T_{h}$ suppose that $τ |_{\partial K}$ is a piecewise constant function which is nonnegative if $K \subset Ω_{+}$ and nonpositive if $K \subset Ω_{-}$ such that $τ |_{\partial K} \neq 0$ . Then there exists a bounded interpolation operator $Π := (Π_{V}, Π_{W})$ from $H^{1} (K; R^{d}) \times H^{1} (K)$ to $V_{h} (K) \times W_{h} (K)$ such that equationparentequation


$\del[0]ΠVr,~rK$	$=\del[0]r,~rK$	$∀~r∈\sbr[0]Pk−1(K)d,$	(20a)
$\del[0]ΠWv,~vK$	$=\del[0]v,~vK$	$\forall ~ v \in P_{k - 1} (K),$	(20b)
${⟨ Π_{V} r \cdot n + τ Π_{W} v, λ ⟩}_{F}$	$= {⟨ r \cdot n + τ v, λ ⟩}_{F}$	$\forall λ \in P_{k} (F),$	(20c)

for $F \subset \partial K$ and $(r, v) \in H^{1} (K; R^{d}) \times H^{1} (K)$ , and

	$\normr−ΠVrK$	$\leq C h_{K}^{k_{r} + 1} \| r \|_{k_{r} + 1, K} + h_{K}^{k_{v} + 1} τ_{K}^{*} \| v \|_{k_{v} + 1, K},$		(21)
	$\normv−ΠWvK$	$\leq C h_{K}^{k_{v} + 1} \| v \|_{k_{v} + 1, K} + \frac{h_{K}^{k_{r} + 1}}{τ_{K}^{max}} \| \nabla \cdot r \|_{k_{r}, K},$		(22)

hold with $0 \leq k_{v}, k_{r} \leq k$ . Here,

	$τ_{K}^{max} = {\begin{matrix} {max}_{F \subset \partial K} τ \|_{F} & if τ \geq 0, {max}_{F \subset \partial K} (- τ \|_{F}) & if τ \leq 0, \end{matrix}$
	$τ_{K}^{} = {\begin{matrix} {max}_{F \subset \partial K ∖ F^{}} τ \|_{F} & if τ \geq 0, {max}_{F \subset \partial K ∖ F^{*}} (- τ \|_{F}) & if τ \leq 0, \end{matrix}$

where $F^{*}$ is the face where $τ_{K}^{max}$ is attained, and the implicit constants are independent of $K$ and $τ$ . We remark that the proof in [10] is only for nonnegative $τ$ . The proof, however, can easily be adapted to nonpositive $τ$ , so we omit its proof here. We will also require $P_{M}$ , the facet-wise $L^{2}$ projection from $L^{2} (F_{h} ∖ F_{h}^{D})$ to $M_{h}$ .

For brevity of notation, we will denote the difference between an unknown $ϕ$ and its approximation $ϕ_{h}$ by $e_{ϕ} := ϕ - ϕ_{h}$ . It will be convenient to also split the error into interpolation and approximation errors:

$e_{q}$	$= e_{q}^{I} + e_{q}^{h} := (q - Π_{V} q) + (Π_{V} q - q_{h}),$
$e_{u}$	$= e_{u}^{I} + e_{u}^{h} := (u - Π_{W} u) + (Π_{W} u - u_{h}),$	(23)
$e_{¯ u}$	$= e^{I}$

	$B_{h} (p, z, ¯ z; r, v, ¯ v)$	$=\normp2Vh+ε\del[0]σ−1p,r0Ω+εbh(z,¯z;r0)$
		$+ {⟨ \| τ \| (z - ¯ z), z - ¯ z ⟩}_{\partial T_{h}}$
		$≥\normp2Vh−12\normp2Vh−12ε2(C′2)2\normz2Ω$
		$+εC′0\normz2Ω−εC′1⟨\|τ\|(z−¯z),z−¯z⟩12∂Th\normzΩ$
		$+ {⟨ \| τ \| (z - ¯ z), z - ¯ z ⟩}_{\partial T_{h}}$
		$≥\normp2Vh−12\normp2Vh−12ε2(C′2)2\normz2Ω$
		$+εC′0\normz2Ω−12⟨\|τ\|(z−¯z),z−¯z⟩∂Th$
		$−12ε2(C′1)2\normz2Ω+⟨\|τ\|(z−¯z),z−¯z⟩∂Th.$

A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients

Authors

Superconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems

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Computer-assisted analysis of the sign-change structure for elliptic problems

A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method

Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators

1 Introduction

2 The Poisson problem with a sign-changing coefficient

3 The HDG method

3.1 Preliminaries

3.2 The discrete formulation

Lemma 1 (Well-posedness)

Proof

Lemma 2

Proof

Theorem 1 (inf-sup condition)

Proof

4 Error analysis

A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients

Authors

Related Research

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An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients

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A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method

Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators

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1 Introduction

2 The Poisson problem with a sign-changing coefficient

3 The HDG method

3.1 Preliminaries

3.2 The discrete formulation

Lemma 1 (Well-posedness)

Proof

Lemma 2

Proof

Theorem 1 (inf-sup condition)

Proof

4 Error analysis