Convergence of adaptive discontinuous Galerkin methods (corrected version of [Math. Comp. 87 (2018), no. 314, 2611--2640])
Christian Kreuzer Emmanuil H. Georgoulis










Convergence of ]Convergence of adaptive 
discontinuous
  Galerkin methods
(corrected version of [Math. Comp. 87 (2018), no. 314, 2611–2640])


Ch. Kreuzer]Christian Kreuzer
Christian Kreuzer,
 Fakultät für Mathematik,
 Ruhr-Universität Bochum,
 Universitätsstrasse 150, D-44801 Bochum, Germany
 
http://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Kreuzer/index.html
christan.kreuzer@rub.de

The research of Christian Kreuzer was supported by the DFG research grant KR 3984/5-1.


E. H. Georgoulis]Emmanuil H. Georgoulis
Emmanuil H. Georgoulis, Department of Mathematics,
University of Leicester,
University Road, 
Leicester, LE1 7RH,
United Kingdom and Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou 157 80, Greece 
 
http://www.le.ac.uk/people/eg64
Emmanuil.Georgoulis@le.ac.uk

Emmanuil H. Georgoulis acknowledges support by the Leverhulme Trust.



[2010]Primary 65N30, 65N12, 65N50, 65N15

 
  We develop a general convergence theory for adaptive discontinuous
  Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG
  and LDG schemes as
  well as all practically relevant marking strategies. Another key feature of the
  presented result is, that it holds for penalty parameters only necessary for
  the standard analysis of the respective scheme. The analysis is
  based on a quasi interpolation into a newly developed limit space
  of the adaptively created non-conforming discrete spaces, 
  which enables to generalise the basic convergence result for
  conforming adaptive finite element methods by Morin, Siebert, and
  Veeser [A basic convergence result for conforming adaptive finite
  elements, Math. Models Methods Appl. Sci., 2008, 18(5),
  707–737].





[
    [
    December 30, 2023
=====================





§ INTRODUCTION



















 Discontinuous Galerkin finite element methods () have 
 enjoyed considerable attention during the last two decades,
 especially in the context of adaptive algorithms ([s]): the absence of any
 conformity requirements across element interfaces characterizing
 approximations allows for extremely general adaptive meshes
 and/or an easy implementation of variable local polynomial degrees in the
 finite element spaces.  There has been a substantial activity in
 recent years for 
the derivation of a posteriori bounds for discontinuous Galerkin
methods for elliptic problems 
<cit.>. 
Such a posteriori estimates are an essential building block in the
  context of adaptive algorithms, which typically consist of a loop 


    .

The convergence theory, however, for the `extreme' non-conformity
case of [s] had been a particularly challenging problem due to the
presence of a negative power of the mesh-size h stemming from the
discontinuity-penalization term. As a consequence, the error is not necessarily
monotone under refinement. 
Indeed, consulting the unprecedented developments of convergence and
optimality theory of conforming adaptive finite element methods ([s]) during
the last two decades, the strict reduction of some error quantity appears
to be fundamental for most of the results. 
In fact, Dörfler's marking 
strategy typically ensures that the error is uniformly reduced in
each iteration <cit.>
and leads to optimal convergence
rates <cit.>; 
compare also with the 
monographs <cit.> and the 
references therein. Showing that 
the error reduction is proportional to the estimator on the refined
elements, instance
optimality of an adaptive finite element method was shown recently for
an with modified marking strategy in
<cit.>. 
A different approach was, however,  taken in <cit.>, where 
convergence of the is proved, exploiting that the approximations
converge to a solution in the closure of the
adaptively created finite element spaces in the trial
space together with standard properties of the a posteriori bounds. 
The result covers a large class of inf-sup stable PDEs and all
practically relevant marking strategies without yielding convergence rates though.



Karakashian and Pascal <cit.> gave the first
proof of convergence for an adaptive based on a symmetric
interior penalty scheme () with Dörfler marking for Poisson's problem. Their
proof 
addresses the
challenge of negative power of h in the estimator, by showing that
the discontinuity-penalization term can be controlled by the element
and jump residuals only, provided that the discontinuity-penalisation parameter, henceforth denoted by σ,
is chosen to be sufficiently large; the element and jump
residuals involve only positive powers of h and, therefore, 
can be controlled similarly as for conforming methods. The optimality
of the adaptive was shown in <cit.>; see
also <cit.>. 

The standard error analysis of the requires
that σ is sufficiently large for the respective bilinear from
to be coercive with respect to an energy-like norm. It is not known in
general, however, whether the choice of σ required for
coercivity of the interior penalty bilinear form is large enough
to ensure that the discontinuity-penalization term can be controlled
by the element and jump residuals only. Therefore, the convergence of
is still open for values of σ large
enough for coercivity but, perhaps, not large enough for the
crucial result from <cit.> to hold. To the best
of our knowledge, the only result in this direction is the proof of
convergence of a weakly overpenalized for linear elements
<cit.>, utilizing the intimate relation between this
method and the lowest order Crouzeix-Raviart elements. 


This work is concerned with proving that the converges for
all values of σ for which the method is coercive,
thereby settling the above discrepancy between the magnitude of
σ required for coercivity and the, typically much larger,
values required for proof of convergence of . Apart from
settling this open problem theoretically, this new result has some
important consequences in practical computations: it is well known
that as σ grows, the condition number of the respective
stiffness matrix also grows. Therefore, the magnitude of the
discontinuity-penalization parameter σ affects the performance
of iterative linear solvers, whose complexity is also typically
included in algorithmic optimality discussions of adaptive finite
elements. In addition, the theory presented here includes a large
class of practically
relevant marking strategies and covers popular 
discontinuous Galerkin methods like the local discontinuous Galerkin
method (LDG) and even the nonsymmetric interior penalty method
(), which are coercive for any σ>0. Moreover, we
  expect that it can be generalised to non-conforming
  discretisations for a number of other problems
  like the Stokes equations or fourth order elliptic problems.
  However, as for the
  conforming counterpart <cit.>, no
  convergence rates are guaranteed.



The proof of convergence of the , discussed
below, is motivated by the basic convergence for the conforming adaptive
finite element framework of Morin, Siebert and Veeser
<cit.>. More specifically, we extend
considerably the ideas from  <cit.> and 
<cit.> to be able
to address the crucial challenge that the limits of solutions,
constructed by the adaptive algorithm, do not necessarily belong to
the energy space of the boundary value problem as well as to conclude
convergence from a perturbed best approximation result. 

To highlight the key
theoretical developments without the need to resort to complicated
notation, we prefer to focus on the simple setting of the Poisson
problem with essential homogeneous boundary conditions and conforming
shape regular triangulations. We believe,
however, that the results presented below are valid for general
elliptic PDEs including convection and reaction phenomena as well as
for some classes of non conforming meshes; compare with <cit.>.  



The remainder of this work is structured as follows. In
Section <ref> we shall introduce the framework for Poisson's equation and state the main result, which is
then proved in Section <ref> after 
some auxiliary results,
needed to generalise <cit.>,
are provided in Sections <ref> and <ref>. 
In particular, in Section <ref> a space is presented,
which is generated from limits of discrete discontinuous functions in
the sequence of discontinuous Galerkin spaces constructed by
. Section <ref> is then concerned with proving that  
the sequence of discontinuous Galerkin solutions produced by converges indeed to a generalised Galerkin solution in this limit space.
This follows from an (almost) best-approximation property,
generalising the ideas in <cit.>.


 






§ THE AND THE MAIN RESULT


Let a measurable set
ω and a m∈. We consider the Lebesgue
space L^2(ω;^m) of square integrable
functions over ω with values in ^m, with inner product  [ω]·· 
and associated norm [ω]·. We also set L^2(ω):=L^2(ω;).
The
Sobolev space H^1(ω) is the space of all functions in
L^2(ω) whose weak gradient is in L^2(ω;^d), for d∈. 




Thanks to the Poincaré-Friedrichs' inequality, the closure
H_0^1(ω) of C_0^∞(ω) in H^1(ω) 
is a Hilbert space with inner product
[ω]∇·∇· and norm
[ω]∇·.  
Also, we denote
the dual space H^-1(ω) of H_0^1(ω), with the norm
[H^-1(ω)]v:=sup_w∈
  H_0^1(ω)vw/[ω]∇ w, v∈ H^-1(ω), with
dual brackets defined by vw:=v(w), for
w∈ H^1_0(ω).



Let Ω⊂ℝ^d,
d=2,3, be a bounded polygonal (d=2) or polyhedral (d=3)
Lipschitz domain. We consider the Poisson problem

    -Δ u=f  in  Ω,
          
      u=0  on  ∂Ω,

with f∈ L^2(Ω). The weak formulation of (<ref>) reads:
find u∈ H^1_0(Ω), such that 

    [Ω]∇ u∇ v=[Ω]fv    for
        all v∈ H_0^1(Ω).

From the Riesz representation theorem, it follows that the solution
u exists and is unique.



 §.§ Discontinuous Galerkin method

  

Let  be a conforming (that is, not containing any hanging nodes) 
subdivision of Ω into disjoint
closed simplicial 
elements   so that Ω̅=⋃{:∈}
and set h_:=||^1/d.  





Let =() be the set of (d-1)-dimensional element faces
 associated with the subdivision  including ∂Ω, and let
=()⊂ by the subset of 
interior faces only. 
We also introduce the
mesh size function :Ω→ℝ, defined by (x):=
h_, if x∈\∂ and (x)=h_:=
|S|^1/(d-1), if x∈ S∈
and set
  Γ=Γ()=⋃{S:S∈} and Γ=Γ()=⋃{S:S∈}.
We assume that  is derived by iterative or recursive newest vertex bisection of
an initial conforming mesh _0; see <cit.>.
We denote  by  the family of shape regular triangulations consisting of such
subdivisions of _0.






Let
𝒫_r() denote the the space of all polynomials on 
of  degree at most r∈, we define the discontinuous finite element space

    :=∏_∈_r()⊂∏_∈W^1,p()=:
      W^1,p(),    1≤ p≤∞,

and H^1():=W^1,2().
Let =() be the set of Lagrange nodes of  and
define the neighbourhood of a node z∈() by
(z):={'∈:z∈'}, 
and the union of its elements by
(z)=⋃{'∈:z∈'}. We also define the corresponding neighbourhoods for all elements ∈ by
():={'∈:∩'≠∅}
and
()=⋃{'∈:'∩≠∅}=⋃{(z):z∈()∩}, respectively, and set  (S):=⋃{∈:S⊂}; compare with Figure <ref>.
 The numbers of neighbours
#(z) and #() are uniformly bounded for all
z∈, respectively
∈, depending on the shape regularity of  and, thus,
on _0.




Let ^+, ^- be two generic elements sharing a face
S:=^+∩^-∈ and let ^+ and
^- the outward normal vectors of ^+ respectively
^- on S. For
q:Ω→ℝ and ϕ⃗:Ω→ℝ^d, let
q^±:=q|_S∩∂^± and
ϕ⃗^±:=ϕ⃗|_S∩∂^±, and set

q|_   :=1/2(q^+ + q^-),        ϕ⃗|_   :=1/2(ϕ⃗^+ + ϕ⃗^-),


q|_   :=q^+^++q^-^-,        ϕ⃗|_   :=ϕ⃗^+·^++ϕ⃗^-·^-;

if S⊂∂∩∂Ω, we set
ϕ⃗|_ :=ϕ⃗^+  and
q|_ :=q^+^+. 



In order to define the discontinuous Galerkin schemes, we
  introduce the following local lifting operators. For S∈,
  we define  :L^2()^d→∏_∈_ℓ()^d and :L^2()→∏_∈_ℓ()^d by 
  
    
    ∫_Ω(ϕ⃗)·τ⃗   =∫_ϕ⃗·τ⃗    ∀τ⃗∈∏_∈_ℓ()^d
                         and∫_Ω(q)·τ⃗   =∫_ qτ⃗    ∀τ⃗∈∏_∈_ℓ()^d,

    with ℓ∈{r,r+1}.
    Note that (q) and (ϕ⃗) vanish outside 
    (S). Moreover, using the local definition and the
    boundedness of the lifting operators in a reference situation
    together with standard scaling arguments, we have 
    for ϕ⃗∈_r()^d and
    q∈_r() that
    
    (ϕ⃗)[]^-1/2ϕ⃗  and  (q)[]^-1/2q;

    
    compare with <cit.>. 
    Also, here and below we write a ≲ b
    when a ≤ Cb for a constant C not depending on the local mesh size of  or other essential quantities
    for the arguments presented below. 
    Observing that the sets (S), ∈ do
    overlap  at most d+1 times, we have for the 
    global lifting operators :L^2(Γ)^d→^d and
    :L^2(Γ)→^d defined by 
    
    (ϕ⃗):=∑_∈(ϕ⃗)    and    (q):=∑_∈(q),

  that 
    
    (v)[Γ]^-1/2 v  and  (β⃗·v)β⃗[Γ]^-1/2 v

    for all v∈ and β⃗∈^d.
  


We define the bilinear form ··:×→ℝ by

    wv   :=∫_∇ w·∇ v  x
           -∫_(∇ w·v+θ∇ v·w)
          +∫_(β⃗·w∇ v+∇
             wβ⃗·v)
          + ∫_Ωγ((w)+(β⃗·w))·((v)+(β⃗·v))
         +∫_σ/w·v  s;

for θ∈{±1}, γ∈{0,1}, β⃗∈^d and σ≥0.
Here we have used the short-hand notation

    ∫_·:=∑_∈∫_·    and    ∫_·:=∑_S∈∫_·.

We consider the choices θ=1, β⃗=0⃗, and γ=0 yielding the
symmetric interior penalty method () <cit.>, θ=-1,
β⃗=0⃗, and γ=0 which gives the nonsymmetric interior
penalty methods () <cit.>, and θ=1, β⃗∈^d, and
γ=1 which yields the local discontinuous Galerkin method () <cit.>;
compare also with <cit.> and <cit.>.  


In all three cases, the corresponding discontinuous
Galerkin finite element method () then reads: find u_∈ such that

    u_v_=∫_Ω f v_  x =:l(v_),  for all  v_∈.

Upon denoting by v the piecewise
gradient v|_=∇ v|_ for all ∈, the corresponding energy norm · is defined by

    w   :=( w^2
        +σ̅[Γ]^-1/2w^2)^1/2,

for w|_∈ H^1(), ∈. Here σ̅:=max{1,σ}.
Also, for some
subset ℳ⊂ with ω=⋃{|∈ℳ}, we define 

    [ℳ]w   :=([ω] w^2
        +σ̅[Γ(ℳ)]^-1/2w^2)^1/2.



If for we have σ:=C_σ r^2 for some
constant C_σ>0 sufficiently large, σ>0 for and
for  σ>0 when ℓ=r and σ=0 when ℓ=r+1
  (<cit.>), 
then there exists α=α(σ)>0, such that 

    αw^2≤ww    ∀ w∈ H^1(),

i.e. all three [s] are
coercive in . Note that coercivity (<ref>) holds true also for functions in H^1() after extending the discrete bilinear form using some liftings; see, e.g.,
<cit.> for details. 
The choice σ̅=max{1,σ} accounts
for the fact that we can have σ=0 for the LDG
in <cit.>.

From standard scaling arguments, we conclude
 the following local Poincaré-Friedrichs inequality
 from <cit.>. 


  Let  be a triangulation of Ω and _⋆ some
  refinement of . Then, for v∈(_⋆),
  ∈ and v_:=|()|^-1∫_()
  v, we have 
  
    [()]v-v_^2∫_()^2|
        v|^2+∫_S∈_⋆, S⊂()^2[_⋆]^-1v^2,

  where _⋆=(_⋆) and the hidden constant
  depends on d and on the shape regularity of ().




The next important result from <cit.> (compare also
with  <cit.> and <cit.>) quantifies the local distance
of a discrete non-conforming function to the conforming subspace with
the help of the of the scaled jump terms. 

  For ∈, there exists an interpolation operator
  ℐ_:H^1()→∩ H_0^1(Ω), such that we have 
  
    []^-1/2(v-ℐ_
        v)^2+[]∇(v-ℐ_ v)^2
        ∫_∂^-1v^2,

  for
  all ∈ and v∈.   
  
  
 
From this, we can easily deduce the following broken Friedrichs type
inequality; compare also with <cit.>.

  Let ∈, then
  
    [Ω]v[]v  for all v∈.

  
  
  



Let
BV(Ω) denote the Banach space of functions with bounded
variation equiped with the norm 

    [BV(Ω)]v=[L^1(Ω)]v+|Dv|(Ω),

where Dv is the measure representing the distributional derivative
of v with total variation 

    |Dv|(Ω)=sup_ϕ∈ C_0^1(Ω)^d, [L^∞(Ω)≤
      1]ϕ∫_Ω vϕ.

Here the supremum is taken over the space C_0^1(Ω)^d of all vector
valued continuously differentiable functions with compact support in Ω.

Another crucial result <cit.> states then that the 
total variation of the distributional derivative of broken Sobolev functions 
is bounded by the discontiuous Galerkin norm.
 
  For ∈ we have that
  
    |Dv|(Ω)[L^1(Ω)]
        v+∫_v[]v  for all v∈ H^1().

  
  
  





 §.§ A posteriori error bound



We recall the a posteriori results from
<cit.>;
compare also with <cit.>. 

For v∈(), we define the local error indicators for
∈ by 

    _(v,):=(∫_^2|f+Δ
          v|^2+∫_∂∩Ω∇
      v^2+σ∫_∂^-1v^2)^1/2;

when v=, we shall write _():=_(,).
Also, for ℳ⊂, we set 

    _(v,ℳ):=(∑_∈ℳ(v,)^2)^1/2.




  
  
  Let u∈ H_0^1(Ω) be the solution of (<ref>) and ∈ its respective approximation (<ref>) on the grid ∈. Then,
  
    []u- ^2∑_∈_()^2.

  
  


The efficiency of the estimator follows with the standard bubble
function technique of Verfürth <cit.>;
compare also with <cit.>,
<cit.> and Proposition <ref> below.

  Let u∈ H_0^1(Ω) be the solution of (<ref>)
  and let ∈. Then, for all v∈ and ∈, we have 
  
    ∫_^2|f+Δ
          v|^2+∫_∂∩Ω∇
      v^2
    [()]u-v^2+[()] (u-v)^2+((),f)^2,

with data-oscillation defined by
  
    (ℳ,f):=(∑_'∈ℳ(,f)^2)^1/2,  where  (,f):=inf_f_∈_r-1[] (f-f_),

  for all ℳ⊂. In particular, this implies  
  
    _(v,)[()]v-u+((),f).

  
  














    Note that the presented theory obviously applies to all locally
    equivalent estimators as well; compare e.g. with
    <cit.>. For
    the sake of a unified 
    presentation, we restrict ourselves to the above representation. 




 §.§ Adaptive discontinuous Galerin finite element method ()





The adaptive algorithm, whose convergence will be shown below, reads as follows.
[] Starting from an initial triangulation _0,
  the adaptive algorithm is an iteration of the following form 

  0.6
    
    
  * =
      ([_k]);
    
  * {_k()}_∈_k=(,);
    
  * =({_k()}_∈_k,);
    
  * [k+1]=(,); increment k.
    
  

Here we have used the notation _k():=_(), for brevity.



  
 We assume that the output 

    =()

is the approximation (<ref>) of u with respect to . 



  
 We suppose that 

    {_()}_∈:=(,)

computes the error indicators from Section <ref>.



  

We assume that the output 

    ℳ:=({_()}_∈,)

of marked elements satisfies 

    _()≤ g(_(ℳ)),     for
      all ∈∖ℳ.

Here g:_0^+→_0^+ is a fixed function, which is continuous in
0 with g(0)=0, i.e. lim_ϵ→ 0g(ϵ)=0.



  

We assume for ℳ⊂∈, that for the refined grid

    :=(,ℳ)

we have 

    ∈ℳ  ⇒  ∈∖,

i.e., each marked element is refined at least once. 



 §.§ The main result



The main result of this work states that the sequence of
discontinuous Galerkin approxiations, produced by , converges to
the exact solution of (<ref>).

  We have that 
  
    _k()→ 0   as k→∞.

  In particular, this implies that 
  
    [k]u-→ 0  as k→∞.









§ A LIMIT SPACE AND QUASI-INTERPOLATION


In this section we shall first introduce a new limit space _∞ of the sequence of
adaptively constructed discontinuous finite element spaces
{()}_k∈. A new
quasi-interpolation operator is then introduced in
Section <ref> in order to 
to prove that there exists a unique Galerkin solution u_∞ of a
generalised discontinuous Galerkin problem in _∞.




 §.§ Sequence of partitions


The produces a sequence {}_k∈_0 of nested
admissible partitions of Ω. Following
<cit.>, we define

    ^+:=⋃_k≥0⋂_j≥
      k[j],    and    Ω^+:=Ω(^+)

to be the set and domain of all elements, respectively, which eventually will not be refined any
more; here
Ω(X):=interior(⋃{:∈
  X}) for a collection of elements X. 
We also define the complementary domain
Ω^-Ω∖Ω^+. 
For the ease of presentation, in what follows, we shall
replace  subscripts  by k to indicate the underlying
triangulation, e.g. we write () instead of
[]().




The following result states that neighbours of elements in ^+
are eventually also elements of ^+; cf.,
<cit.>. 

  For ∈^+ there exists a constant K=K()∈_0, such that 
  
    ()=[K]()    for all  k≥ K,

  i.e., we have ()⊂^+ for all k≥ K.



 
Next, for a fixed k∈_0, we set 
  
    
    
    
    
    ^+   :=∩^+,       Ω_k^+   :=Ω(^+),
    

    ^++   :={∈()⊂^+},       Ω_k^++   :=Ω(^++),
    

    ^-   :=∖
    ^++
           Ω_k^-   :=Ω(^-);
  
  compare also with Figure <ref>.
  This notation is also adopted for the corresponding faces, e.g., we
  denote _k^+:=(^+) and
  _k^+:=(^+) and correspondingly
  for all other above sub-triangulations of . 
  

  The next lemma states that the meshsize of elements in
   ^- converges uniformly to zero; compare also
   with <cit.> and  <cit.>.
  
  
  
  
  
      We have that 
       lim_k→∞h_kχ_Ω_k^-_L^∞(Ω)
      =0, with χ_Ω_k^- denoting the characteristic
      function of Ω_k^-.

      In particular, this implies that lim_k→∞|Ω((^-_k))∖Ω^-|=
       lim_k→∞|Ω_k^-∖Ω^-|=0. 
    

    
      The first claim is proved in <cit.>.

      In order to prove the second claim, we first observe that
      
    |Ω_k^-∖Ω^-|=|Ω(^+∖^++)|.

      For ℓ∈, it 
      follows from Lemma <ref> and #[ℓ]^+<∞, 
      that there exists K(ℓ), such that
      [ℓ]^+⊂[K(ℓ)]^++ and thus
      
    |Ω(^+∖_K(ℓ)^++)|≤ |
           Ω(^+∖_ℓ^+)|→ 0    as ℓ→∞,

      i.e. a subsequence of
      {|Ω(^+∖^++)|}_k vanishes. However,
      since the sequence is monotone decreasing, it must vanish as a whole. To conclude, we first realise that
      |Ω((^-_k))∖Ω^-|≤
     |Ω_k^-∖Ω^-|+
     |Ω((^-_k)∖^-)|; it remains to
     prove that the latter term vanishes as k→∞. To this end, we observe that
      
    (^-_k)∖^-⊂^++∖^3+,

      with
      ^3+={∈()∈^++}. Indeed,
      assume that ∈^3+∩(^-), then
      there exists '∈() with
      '∈^++∩^-; this is a
      contradiction. Consequently, we have
      
    |Ω((^-_k)∖^-)|≤
            |Ω(^++∖^3+)|≤
            |Ω(^+∖^3+)|→
            0  as k→∞,

      where the last limit follows by iterating the reasoning from (<ref>). 
    








 §.§ The limit space


In this section, we shall investigate the limit of the finite element
spaces _k:=(), k∈. To this end, we
define

    _∞:={v∈ BV(Ω)   :
                                        v|_Ω^-∈
                                        H^1_∂Ω∩∂Ω^-(Ω^-) and v|_∈_r ∀∈^+
      
         such that ∃{v_k}_k∈, v_k∈_k
                                         with lim sup_k→∞[k]v_k<∞
         and lim_k→∞[k]v-v_k+[Ω]v-v_k=0
                                        };

here H^1_∂Ω∩∂Ω^-(Ω^-) denotes the
space of functions from H_0^1(Ω) restricted to
Ω^-. Moreover, we have extended the definition of the
piece-wise gradient to

    v∈ L^2(Ω)^d:   v|_Ω^-=∇
      v|_Ω^-  and   v|_=∇
      v|_ ∀∈^+.

Note that for v∈ BV(Ω) there exists the L^1-trace of v on
Γ_k=⋃{S:S∈_k}; compare e.g. with the trace
theorem <cit.>. In other words, v is
measurable with respect to the (d-1)-dimensional Hausdorff measure on
_k and, therefore, the term [k]v, v∈_∞, makes sense. 
Obviously, we have _k∩ C(Ω)⊂_∞ for all k∈ and, thus,  _∞ is not empty.




Setting h_+:=h_^+ and ^+:=(^+), we define

    [∞]vw   :=∫_Ω v· w
      
      
      + σ̅∫_^+[+]^-1vw,

and 
[∞]v:=[∞]vv^1/2,
for all v,w∈_∞. 








































We shall next list some basic properties of the space _∞. 




  For v∈_∞, we have 
  
    [k]v↗[∞]v<∞
        
           as 
        k→∞.

  In particular, for fixed ℓ∈, let
  ∈[ℓ]; then, we have
  
    ∫_{S∈_k:S⊂}h_k^-1v^2↗∫_{S∈^+:S⊂}h_+^-1v^2,  as k→∞.





  
  
  
  
  
  
  
  
  
  
  
  
  Since v∈_∞, there exists {v_k}_k∈, v_k∈_k
  with lim_k→∞[k]v-v_k=0 and 
  lim sup_k→∞[k]v_k<∞. 
  We first observe that 
  
    [k]v≤[k]v-v_k+[k]v_k<∞

  uniformly in k. 
  Thanks to the mesh-size reduction, i.e. h_m≤ h_k for all m≥
  k, we conclude that
  
    ∫__kh_k^-1v^2≤∫__kh_m^-1v^2≤∫__mh_m^-1v^2,

    thanks to the inclusion 
    ⋃_∈_k⊂⋃_∈_m. 
  
  Therefore, we have 
  [k]v≤[m]v for all
  m≥ k and, thus, {[k]v}_k∈ converges.
  Consequently, for ϵ>0 there exists K=K(ϵ), such
  that for all k≥ K and m>k large enough, we have
  
    ϵ>|[m]v^2-[k]v^2|   =σ̅∫__m∖(_m∩_k)[m]^-1v^2-σ̅∫__k∖(_m∩_k)[k]^-1v^2
       ≥ (2^1/(d-1)-1)  σ̅∫__k∖(_m∩_k)[k]^-1v^2
       ≥ (2^1/(d-1)-1)  σ̅∫__k∖_k^+[k]^-1v^2.

  This follows from the fact
  that [m]|_≤ 2^-1/(d-1)[k]|_ for all
  S∈_k∖(_m∩_k)
   together with 
  _k^+=_m∩_k for sufficiently large m>k. 
  
  Therefore, we have 
  ∫__k∖_k^+[k]^-1v^2→ 0
  as k→∞ and, thus, 
  
    [k]v^2   = 
                 
    
    
                         ∫_Ω | v|^2 
                        + σ̅∫__k^+h_k^-1v^2
                        +σ̅∫__k∖_k^+h_k^-1v^2→[∞]v^2+0.

  This proves the first claim. The second claim is a localised version
  and follows completely analogously. 



As a consequence, we have that Friedrichs and Poincaré inequalities
are inherited to _∞.

  We have
  
    [Ω]v[∞]v  for all v∈_∞.



Since v∈_∞, there exists {v_k}_k∈, v_k∈_k
  with lim sup_k→∞[k]v_k<∞ and
  lim_k→∞[k]v-v_k+[Ω]v-v_k=0. It thus
  follows from Corollary <ref> and
  Proposition <ref> that
  
    [Ω]v=lim_k→∞[Ω]v_klim_k→∞[k]v_k=[∞]v.

  This is the assertion.




  Fix k∈ and let ∈. Then for v∈_∞ and
  v_:=1/|()|∫_()v, we have 
  
    [()]v-v_^2[()]h_k
        v^2+∫_{S∈^+:S⊂()}h_k^2h_+^-1v^2.





  By the definition of _∞, there exists 
  v_ℓ∈_ℓ, ℓ∈_0, with
  lim_ℓ→∞[ℓ]v-v_ℓ+[Ω]v-v_ℓ=0 and
  lim sup_ℓ→∞[ℓ]v_ℓ<∞.  
  Therefore, we have 
  
    [()]
        v_ℓ^2+∫_{S∈_ℓ:S⊂()}h_ℓ^-1v_ℓ^2
    →[()]
        v^2+∫_{S∈^+:S⊂()}h_+^-1v^2  as ℓ→∞;

  see Proposition <ref>. Moreover, we have 
  
    [ω_k()]v_-v_ℓ,≤[ω_k()]v-v_ℓ≤[Ω]v-v_ℓ→ 0  as ℓ→∞,

  where
  v_ℓ,:=1/|()|∫_()v_ℓ. 
  We conclude with
  Proposition <ref> that
  
    [()]v-v_^2   ←[()]v_ℓ-v_ℓ,^2
        
       [()]h_k
        v_ℓ^2+∫_{S∈_ℓ:S⊂()}h_k^2h_ℓ^-1v_ℓ^2
       →[()]h_k
        v^2+∫_{S∈^+:S⊂()}h_k^2h_+^-1v^2,

  as ℓ→∞.




In order to extend the dG bilinear form (<ref>) to _∞, we need
  to define appropriate lifting operators.  For each ∈^+,
  there exists ℓ=ℓ()∈, such that ∈_ℓ^++
  
  .
  We define
  the local lifting operators
  [∞]:L^2(S)^d→ L^2(Ω)^d and 
  [∞]:L^2(S)→ L^2(Ω)^d by
  
    [∞]=[ℓ]:=[[ℓ]]    and    [∞]=[ℓ]:=[[ℓ]].

    From (<ref>) it is easy to see, that 
    [ℓ] and [ℓ] depend only on S and the
    at most two adjacent elements ,'∈[ℓ]^+ with ⊂∩'.
    Therefore, and thanks to the fact that 
    the ^+ are nested, we have that [ℓ]=[k]
    for all k≥ℓ and, thus,
    the definition is unique.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    We formally define the global lifting operators 
    
    by
    
    
    
    
    [∞]:=∑_∈^+[∞]    and    [∞]:=∑_∈^+[∞];

    here
    ^+:={∈^+:∉∂Ω}.


Moreover, from the local estimates 
  (<ref>), it is easy to see that for v∈_∞ and
  β⃗∈^d, we have that 
  ∑_∈_k^+[∞](v) and
  ∑_∈_k^+[∞](β⃗·v)
  are Cauchy sequences in L^2(Ω)^d. 
  Consequently,
  [∞](v),[∞](β⃗·v)∈
  L^2(Ω) are well posed 
  and we have 
  
    [∞](v)[Γ^+][+]^-1/2 v  and  [∞](β⃗·v)β⃗[Γ^+][+]^-1/2 v,

    where
    Γ^+=⋃{:∈^+} and
    Γ^+=⋃{:∈^+}. 
    This enables us to generalise the discontinuous Galerkin bilinear form   
    to _∞ setting
    
    [∞]wv   :=∫_Ω w· v  x -∫_^+(∇ w·v+θ∇ v·w)
          +∫_^+(β⃗·w∇ v+∇
             wβ⃗·v)
          + ∫_Ωγ([∞](w)+[∞](β⃗·w))·([∞](v)+[∞](β⃗·v))
         +∫_^+σ/h_+w·v  s,

 for v,w∈_∞.






  The space (_∞,[∞]··) is a
  Hilbert space.





  There exists a unique u_∞∈_∞, such that 
  
    [∞]u_∞v=∫_Ω fv    for
        all v∈_∞.

 


In order to prove the last two statements, we introduce
a new quasi-interpolation, which is
designed  in due consideration of the future 
refinements. The proofs of Lemma <ref> and
Corollary <ref> are postponed to the end of
Section <ref>.



 §.§ Quasi-interpolation



We shall now define a quasi-interpolation
operator Π_k, which maps into _∞∩_k; this will be a
key technical tool in the analysis. On the one 
hand, membership in _∞∩_k suggests to use some Clément
type interpolation since the mapped
functions need to be continuous in Ω^-.
On the other hand, the fact that the
may leave some elements (namely ^+⊃^++) unrefined, suggests
to define Π_k to be the identity on these elements. Note that the
quasi-interpolation operator
from <cit.> 
is motivated by a similar idea in order to map from one Crouzeix-Raviart
space into its intersection with a finer one. 



For fixed k∈, let 
{Φ_z^:∈, z∈_k()}
be the Lagrange basis of :=[], i.e., Φ_z^ is a piecewise polynomial
of degree r with 
(Φ_z^) = and 

    Φ_z^(y)=δ_zy    for
      all z,y∈_k.

Its dual basis is then the set {Ψ_z^:∈, z∈_k()} of piecewise polynomials
of degree r, such that 
(Ψ_z^) = and

    Ψ_y^Φ_z^=δ_zy    for
      all z,y∈_k().

For all ℓ≥ k, we define Π_k:L^1(Ω)→ L^1(Ω) by 

    Π_kv:=∑_∈∑_z∈_k()
      (Π_kv)|_(z)  Φ_z^,

where for z∈_k() we have that 

    (Π_kv)|_(z):=
      ∫_
        vΨ_z^,     if 
        (z)∩^++≠∅
    
        0,     else if z∈∂Ω
    ∑_'∈(z)|'|/|(z)|∫_'vΨ_z^',     else.

Beyond standard stability and interpolation estimates for
H_0^1(Ω) functions <cit.>, we
list the following properties related to our setting.








  The operator Π_k:L^1(Ω)→ L^1(Ω) defined
  in (<ref>) has the following properties:
  
  
  *  Π_k:L^p(Ω)→ L^p(Ω) is a linear and
    bounded projection for all 1≤ p≤∞. In particular, we have that 
    
    [L^p()]Π_kv[L^p(())]v,

    where the constant solely depends on p, r, d, and  the shape regularity
    of _0. 
  
  *  Π_kv∈ for all v∈ L^1(Ω);
     
  *  Π_kv|_=v|_, if ∈ and
    v|_()∈_r(ω_k());
    
    
    
    
    
    
    
    
    
    
    
  
  *  Π_kv|_=v|_, if ∈^++ and
    v|_∈_r(); if moreover v∈, then also
    v-Π_kv|_S≡0 for all ∈_k^++. 
  
  *  Π_kv|_Ω∖Ω_k^+∈
    C(Ω∖Ω_k^+) and Π_kv
      =0 on ∂(Ω∖Ω^+_k);
  
  *  Π_k v=v, for all v∈_k with v|_Ω∖Ω_k^++∈
    C(Ω∖Ω_k^++);
  
  *  Π_kv∈_∞, and we have [k]Π_kv=[∞]Π_kv.
  



  Claims (<ref>)–(<ref>) follow by standard estimates for
  the Scott-Zhang operator <cit.>. 

  Assertion (<ref>) is a consequence of the
  definition (<ref>) of Π_k since ∈^++
  implies that ()∩^++=().
  Note that v∈ implies v|_∈_r() for all
  ∈ and thus (Π_k v)|_(z)=v|_(z) for all
  ∈(z) if
  (z)∩^++≠∅. This is in particular the
  case when z∈∩_k with ∈_k^++.

  For ∈∖^+, we have that
  (z)∩^++=∅ since otherwise there exists
  '∈()∩^++ and thus ∈('), which implies ∈^+, thanks to the
  definition of ^++. Therefore, (<ref>) implies    
  that Π_kv is continuous on 
  Ω∖Ω_k^+. Moreover, for z∈_k()∩Ω∖Ω^+_k, definition (<ref>) is
    independent of  and thus Π_kv does not jump across the
    boundary Ω∖Ω_k^+.  
    This completes the proof of (<ref>).

  On the one hand, if v∈_k with v|_Ω∖Ω_k^+∈
  C(Ω∖Ω_k^++) then we have clearly
  Π_kv|_Ω∖Ω_k^+=v|_Ω∖Ω_k^+.
  On the other hand, we can
  conclude Π_kv|_Ω_k^++=v|_Ω_k^++
  from (<ref>). This yields (<ref>).

  The claim (<ref>) is
  an immediate consequence of (<ref>).




    Let v∈[ℓ] for some k≤ℓ∈_0∪{∞}. Then for all ∈, we have 
    
    ∫_∇Π_kv^2+∫_∂h_k^-1Π_kv^2
    ∫_() v^2+∑_'∈[ℓ], '⊂()∫_∂'h_ℓ^-1v^2,

 setting _ℓ:=^+ and h_ℓ:=h_+, when ℓ=∞.
 In particular, we  have 
    [k]Π_k v[ℓ]v.
  
  
  


  We begin by noting that, summing over all elements in  and
  accounting for the finite overlap of the domains (),
  ∈,  the global stability estimate is an immediate
  consequence of the corresponding local one.

  We first assume ℓ<∞. Let
  ∈^++⊂[ℓ]^++. Then, thanks to
  Lemma <ref>(<ref>), we have  
  Π_kv|_=v|_. Moreover, let '∈ such that
  ∩'∈_k; then (z)∋∈^++
  and thus (Π_kv)|_'(z)=v|_'(z), for all
  z∈_k()∩_k(').  Consequently, we have
  Π_kv=v
  on ∂, in other words
  
    ∫_∇Π_kv^2+∫_∂h_k^-1Π_kv^2 
        = ∫_∇ v^2+∫_∂h_k^-1v^2 .


  Let now ∈_k be arbitrary. Then, an inverse estimate and the local stability
  (Lemma <ref> (<ref>) and (<ref>)) for
  v_:=1/|()|∫_() v∈, imply 
  
    ∫_∇Π_kv^2   ∫_ h_k^-2Π_k(v-v_)^2∫_()h_k^-2v-v_^2
       ∑_'⊂(),'∈[ℓ]∫_'∇ v^2+
          ∫_∂'h_ℓ^-1v^2;

  here the last estimate follows from the broken Poincaré inequality,
  Proposition <ref>. 

  If now for all '∈, with '⊂(), we have
  '∉^++, which
  implies ∈∖^++. Then,
  thanks to Lemma <ref>(<ref>), we
  have that Π_kv is continuous across ∂, i.e.,
  Π_kv|_∂=0. 
  On the contrary, assuming that there exists '∈^++,
  with '∈(), we conclude that
  ∈(') and thus ∈^+.
  From the local quasi
  uniformity, we thus have for all ”∈_ℓ with  ”∩≠∅
  that |”| ||.
  Let z∈_k(); then, according to (<ref>), we have that
  
    Π_kv|_∂(z)=
        v|_∂(z),     if ∃'∈(z)∩^++;
          
    
          0,   else.




































  Using standard scaling arguments, this implies
  
    ∫_∂Π_kv^2   
        |∂|∑_z∈_k∩∂(Π_kv|_∂(z))^2
                                               = 
        |∂|∑_z∈_k∩∂(v|_∂(z))^2
        
       ≤
         |∂|∑_z∈_ℓ∩∂(v|_∂(z))^2
          ∫_∂v^2.

  Combining this with (<ref>) proves the local
 bound in the case ℓ<∞.

  
  For ℓ=∞, we observe
  that a bound similar to (<ref>) can be obtained with
  Lemma <ref> instead of
  Proposition <ref>. The local bound follows then by
  arguing as in the case ℓ<∞.




  For v∈_ℓ, k≤ℓ∈∪{∞}, we have that 
  
    ∫_| v-Π_k v|^2 + ∫_^-2
        |v-Π_k v|^2+∫_∂^-1v-Π_k v^2
        
    ∫_()| v|^2 +∑_S∈_ℓ,S⊂()∫_S^-1v^2,

  where we set _ℓ:=^+ and h_ℓ:=h_+, when ℓ=∞.
  The constant depends only on d, r and the shape
    regularity of [0].


  The claim follows from Lemma <ref>(<ref>), together
  with the stability Lemma <ref> and the local Poincaré inequality
  from Proposition <ref>, respectively,
  Lemma <ref>. 





The next result concerns the convergence of the quasi-interpolation.

  Let v∈_∞; then,
  
    [k]v-Π_k v→ 0     and    [∞]v-Π_k v→ 0

  as k→∞.



 For brevity, set
  v_k:=Π_k v∈. Thanks to Lemma <ref> and Lemma <ref>(<ref>) and
  (<ref>),
  we have that
  
    [k]v-v_k^2   ∫_∖^++| v-
        v_k|^2+
        ∫__k∖_k^++[k]^-1v-v_k^2
       ≤∫_^-| v-
        v_k|^2+
        ∫__k^-[k]^-1v-v_k^2
       =I^-_k +II^-_k.

  We conclude from  Lemma <ref> that
  
    II^-_k   =
                   ∫__k^-[k]^-1v-v_k^2∑_∈^-∑_'∈^+,
        '⊂() ∫_∂'[+]^-1v^2
       ∫_^+∖_k^++[+]^-1v^2.

  The term on the right hand side is the tail of a convergent series, since it is
  bounded thanks to 
  [∞]v<∞ and all of its summands are positive. Therefore,  II_k^-→0 as
  k→∞. 

  Thus, it remains to prove that I_k^-→ 0 as k→∞. To this
  end, we recall that thanks to the  definition of
  H^1_∂Ω∩∂Ω^-(Ω^-) we have that
  v|_Ω^-=ṽ|_Ω^- for some function ṽ∈ H_0^1(Ω). Since H_0^2(Ω) is dense in H_0^1(Ω), for ϵ>0, there exists v_ϵ∈ H^2_0(Ω) 
  such
  that ṽ-v_ϵ_H^1(Ω^-)≤ṽ-v_ϵ_H^1(Ω)<ϵ.
  Combining
  Lemma <ref>(<ref>) and (<ref>) with
  standard estimates
  <cit.> for H_0^1(Ω)
  functions, with
  the  Bramble-Hilbert Lemma (see, e.g., <cit.>),
  we 
  obtain 
  
    ∫_^-| v-∇
        v_k|^2
       ∫_^-|∇ v_ϵ-∇Π_kv_ϵ|^2+| (v- v_ϵ)-∇Π_k(v-v_ϵ)|^2
       ∫_(^-)[k]^2∑_|α|=2|D^α
          v_ϵ|^2+∫_(^-)|
          (v-v_ϵ)|^2
       h_kχ_Ω_k^-_L^∞(Ω)^2
          ∫_Ω∑_|α|=2|D^α
          v_ϵ|^2+∫_(^-)|
          (v-v_ϵ)|^2,

  where we have used that
  h_k_L^∞(Ω((^-)))h_kχ_Ω_k^-_L^∞(Ω)→ 0 as k→∞, thanks to the local quasi-uniformity
  of  and 
  Lemma <ref>. Moreover, we conclude
  ∫_(^-)| (v-v_ϵ)|^2→∫_Ω^-|∇ (v-v_ϵ)|^2<ϵ from
  Lemma <ref> and the absolute continuity of the
  Lebesgue integral. Consequently, 
  lim_k→∞I^-_kϵ, which completes the proof
  of the first claim, since ϵ>0 is arbitrary. 

  The second claim follows similarly by replacing _k by
  ^+ and noting that [k]Π_kv=[∞]Π_k v,
  since Π_k v is continuous in Ω∖Ω^+.






  The positivity of [∞]· on _∞
    follows from Lemma <ref> together with 
    Corollary <ref>.  
  
   
  In order to prove that _∞ is complete with respect to
  [∞]·, let 
  {v^ℓ}_ℓ∈⊂_∞ be a Cauchy sequence with
  respect to [∞]·. Note that thanks to the Friedrichs
  inequality (Corollary <ref>),
  there exists the limit v^ℓ→ v∈ L^2(Ω); this is the
  candidate for the limit of v^ℓ in _∞. 

  We first observe that, 
  since 
  v^ℓ|_∈_r for all ∈^+, it follows
  from the definition of [∞]· that 
  v|_∈_r for all ∈^+.
  Moreover, 
  Propositions <ref> and <ref> imply
  that {v^ℓ}_ℓ∈ is also a Cauchy Sequence in
  BV(Ω) and thus v∈ BV(Ω).
  Therefore, v has L^1-traces on each ∂,
  ∈, k∈.

  Next, we deal with the jump terms. To this end, we first
  observe that, for k∈, {v^ℓ}_ℓ∈ is also a Cauchy sequence with
  respect to the [k]·-norm and uniqueness of
  limits imply  v^ℓ|_→v|_ in L^2() as
  ℓ→∞, for all
  ∈_k, in the sense of traces.
  Let ϵ>0 arbitrary fixed, then there exists L=L(ϵ),
  such that [k]v^j-v^ℓ≤[∞]v^j-v^ℓ≤ϵ for all j≥ℓ≥ L. Fix ℓ≥ L, then thanks to
  Proposition <ref>, there exists K≡ K(ϵ,L), such that for all k≥ K, we have
  
    ∫__k∖_k^+h_+^-1v^L^2≤ϵ^2.

  Consequently, we have

    ∫__k[k]^-1v^2    =
        ∫__k∖_k^+[k]^-1v^2
        +∫__k^+[k]^-1v^2 
       =lim_ℓ→∞∫__k∖_k^+[k]^-1v^ℓ^2 +∫__k^+[k]^-1v^2
        .

Thus,  it follows from

    ∫__k∖_k^+[k]^-1v^ℓ^2 ≤ 2[k]v^ℓ-v^L^2+2∫__k∖_k^+[k]^-1v^L^2 ≤ 4ϵ^2,

for ℓ≥ L, that

    ∫__k[k]^-1v^2 →∫_^+[+]^-1v^2     as  k→∞,

since ϵ>0 is arbitrary.

We next verify that v|_Ω^-∈
    H^1_∂Ω∩∂Ω^-(Ω^-), i.e., that v is a
  restriction of a function from H_0^1(Ω) to Ω^-. To
  this end, for each ℓ,m∈, we
  define v^ℓ_m:=Π_mv^ℓ∈_m for ℓ≥ m∈ and since v^ℓ_m∈
  C(Ω∖Ω^+_k)⊂
  C(Ω∖Ω^+_m) (see
  Lemma <ref>(<ref>)) for k≥ m,  we have that
  [m]v_m^ℓ=[k]v_m^ℓ=[∞]v_m^ℓ. 
  
  
  
  Thanks to Lemma <ref>, for each ℓ∈, there exists a
  monotone sequence 
  {m_ℓ}_ℓ∈, such that [∞]v^ℓ-v^ℓ_m_ℓ≤1/ℓ and
  thus
   
    [m_ℓ]v_m_ℓ^ℓ=[∞]v_m_ℓ^ℓ≤[∞]v^ℓ-v_m_ℓ^ℓ
    +[∞]v^ℓ<1/ℓ+[∞]v^ℓ<∞

   Consequently, the conforming interpolation
   ℐ_m_ℓv_m_ℓ^ℓ∈_m_ℓ∩ H_0^1(Ω) 
  from Proposition <ref> is bounded uniformly in
  H_0^1(Ω) and thus there exists a weak limit ṽ∈
  H_0^1(Ω) of a subsequence, which for convenience we denote with the same label. Moreover, again from
  Proposition <ref>, we have
  
    [Ω^-]v-ℐ_m_ℓv^ℓ_m_ℓ   ≤[Ω]v-v^ℓ+
          [Ω]v^ℓ-v^ℓ_m_ℓ+
          [Ω^-]v^ℓ_m_ℓ-ℐ_m_ℓv^ℓ_m_ℓ
       [Ω]v-v^ℓ+1/ℓ+
          ∫_([m_ℓ]∖[m_ℓ]^+)v^ℓ_m_ℓ^2
       ≤[Ω]v-v^ℓ+2/ℓ+ [L^∞(Ω)]χ_Ω_m_ℓ^-h_m_ℓ[m_ℓ]v^ℓ^2,

  which vanishes as ℓ→∞ thanks to the Friedrichs
  inequality (Corollary <ref>) and
  Lemma <ref>. Therefore, v|_Ω^-=ṽ|_Ω^- and we can define the piecewise gradient of v as
  in (<ref>).


  We shall next show that [∞]v-v^ℓ→0 as ℓ→∞. Arguing similar as for (<ref>), we have 
  
    ∫_^+h_+^-1v-v^ℓ^2→
        0  as ℓ→∞.

  Consequently, it remains to show that v -
    v^ℓ→ 0 as ℓ→∞. To this end, we observe that
  { v^ℓ}_ℓ is a Cauchy Sequence in L^2(Ω)^d
  and thus there exists d⃗∈ L^2(Ω)^d with v^ℓ-d⃗→ 0 as ℓ→∞ and it thus suffices to prove
  d⃗= v. Let ϕ∈ C_0^∞(Ω), then we have
  from Lemma <ref>
  for the distributional derivative on the one hand, that
  
    ⟨ Dv^ℓ,ϕ⟩   =∫_Ω v^ℓ·ϕ -∫_^+v^ℓ·ϕ→∫_Ωd⃗·ϕ -∫_^+v·ϕ

  as ℓ→∞.
  On the other hand, 
  
    ⟨ Dv^ℓ,ϕ⟩   =∫_Ω∖Ω_k^+ v^ℓ·ϕ +∫_Ω_k^+ v^ℓ·ϕ -∫_^+v^ℓ·ϕ
       =∫_Ω∖Ω_k^+∇ℐ_m_ℓv^ℓ_m_ℓ·ϕ
          +∫_Ω∖Ω_k^+
          (v^ℓ-ℐ_m_ℓv^ℓ_m_ℓ)
          ·ϕ
         +∫_Ω_k^+ v^ℓ·ϕ
          -∫_^+v^ℓ·ϕ.

  In order to estimate the second term, we employ
  Proposition <ref>, and obtain for some arbitrary given
  ϵ>0 that 
  
    ∫_Ω∖Ω_k^+
        (v^ℓ-ℐ_m_ℓv^ℓ_m_ℓ)^2   ∫_Ω
                                                            (v^ℓ-v^ℓ_m_ℓ)^2
         +∫_Ω∖Ω_k^+
                                                            (v^ℓ_m_ℓ-ℐ_m_ℓv^ℓ_m_ℓ)^2
       1/ℓ+∫__k^-h_m_ℓ^-1v^ℓ^2≤1/ℓ+ϵ^2

  for all ℓ≥ L(ϵ) and k≥ K(ϵ,L) similarly as in (<ref>).
  Recalling that ṽ is the weak limit of
  {ℐ_m_ℓv^ℓ_m_ℓ}_ℓ in H_0^1(Ω)
  and v^ℓ converges strongly in _r(_k^+),
  we thus conclude that
  
    ∫_Ω(χ_Ω∖Ω_k^+∇ṽ+χ_Ω_k^+ v-d⃗)·ϕ≤ϵ[Ω]ϕ.

  Recalling (<ref>) the assertion follows by letting k→∞ from the uniform integrability
  of ∇ṽ and v|_Ω^+.


  Finally note that _k⊂_j for j≥ k and thus defining 
  w_k:=v_m_ℓ^ℓ for  k∈{m_ℓ,…,m_ℓ+1-1}
  yields w_k∈_k. Consequently, 
  
    v-w_k+[k]v-w_k[∞]v-w_k=[∞]v-v_m_ℓ^ℓ≤[∞]v-v^ℓ+1/ℓ,

  where we have used that the
  Friedrichs inequality (Corollary <ref>) is
  inherited since
  [∞]v-w_k=lim_ℓ→∞[∞]v_ℓ-w_k.
The right-hand side
  vanishes because ℓ→∞ as
  k→∞. 





  The assertion follows from Lemma <ref> and the observation that 
  
    [∞]v^2[∞]vv    and    [∞]vw[∞]v[∞]w

  for all v,w∈_∞. Indeed, the continuity follows with standard
  techniques using (<ref>) and the coercivity is a consequence of 
  
    [∞]Π_kv^2=
        [k]Π_kv^2[k]Π_kvΠ_kv=[∞]Π_kvΠ_kv

  and Lemma <ref>.








§ (ALMOST) BEST APPROXIMATION PROPERTY



In this section we shall prove that the solution
u_∞∈_∞ of (<ref>) is indeed
the limit of the discontinuous Galerkin solutions produced by
. 
This is a consequence of the density of spaces
{_k}_k∈_0 in _∞ and the (almost) best approximation
property of discontinuous Galerkin solutions; the latter generalises <cit.>. 




  Let u_∞∈_∞ be the solution of (<ref>) and u_k∈ be the
  approximation from (<ref>) on  for some k∈
  and u_∞ the unique solution of the limit problem from
  Corollary <ref>.
  Then, we have
  
    [k]u_∞-u_k[∞]u_∞-Π_ku_∞+fu_k-Π_ku_k-[k]Π_ku_∞u_k-Π_ku_k/[k]u_k-Π_ku_∞.




  Assume that u_k≠Π_ku_∞∈∩_∞ and set ψ=u_k-Π_ku_∞. Then, we have from (<ref>) that
  
    α[k]u_k-Π_ku_∞^2   ≤[k]u_k-Π_ku_∞ψ=fψ-[k]Π_ku_∞ψ
       =fΠ_kψ+fψ-Π_kψ-[k]Π_ku_∞ψ
       =([∞]u_∞Π_kψ-[k]Π_ku_∞Π_kψ)
          
         
          +(fψ-Π_kψ-[k]Π_ku_∞ψ-Π_kψ)≡ (I)+(II),

 using that Π_kψ∈_k∩_∞ from Lemma <ref>(<ref>). 







  For (I), we have, respectively, 
  
    (I)    =∫_Ω u_∞·Π_kψ  x
          -∫_^+(∇
            u_∞·Π_kψ+θ∇Π_kψ·u_∞)
         
          +∫_^+(β⃗·u_∞∇Π_kψ+∇
             u_∞β⃗·Π_kψ)
          + ∫_Ωγ([∞](u_∞)+[∞](β⃗·u_∞))·([∞](Π_kψ)+[∞](β⃗·Π_kψ))
         +∫_^+σ/[+]u_∞·Π_kψ 
          s
          
         -∫_ΩΠ_ku_∞·Π_kψ  x
          +∫__k(∇Π_ku_∞·Π_kψ+θ∇Π_kψ·Π_ku_∞)
          -∫_^+(β⃗·Π_ku_∞∇Π_kψ+∇Π_ku_∞β⃗·Π_kψ)
          - ∫_Ωγ([k](Π_ku_∞)+[k](β⃗·Π_ku_∞))·([k](Π_kψ)+[k](β⃗·Π_kψ))
         -∫_^+σ/[k]Π_ku_∞·Π_kψ 
          s
          
    
          
          
          
          
          
          
          
          
          
          
          
             =∫_Ω (u_∞-Π_ku_∞)·Π_kψ  x
          
          -∫_^+_k∇
            (u_∞-Π_ku_∞)·Π_kψ-θ∫_^+∇Π_kψ·u_∞-Π_ku_∞
          +∫_^+(β⃗·u_∞-Π_ku_∞∇Π_kψ+∇ u_∞-∇Π_ku_∞β⃗·Π_kψ)
         + ∫_Ωγ([∞](u_∞-Π_ku_∞)+[∞](β⃗·u_∞-Π_ku_∞))
            
           ·([∞](Π_kψ)+[∞](β⃗·Π_kψ))
         +∫_^+_kσ/[k]u_∞-Π_ku_∞·Π_kψ 
          s
          
       [∞]u_∞
            -Π_ku_∞[∞]Π_kψ=[∞]u_∞
            -Π_ku_∞[k]Π_kψ
       [∞]u_∞
            -Π_ku_∞[k]u_k-Π_ku_∞;

  here we used that Π_ku_∞,Π_kψ∈∩_∞, h_∞=h_k
  on _k^+ and that Π_ku_∞
  and Π_kψ are continuous on Ω∖Ω^+_k, i.e.,
  Π_ku_∞
  =Π_kψ=0 on ^+∖_k^+, 
  which follows from
  Lemma <ref>. 
  Note that this and Π_ku_∞
  =Π_kψ=0 on 
  ∂(Ω∖Ω_k^+) from Lemma <ref>
  also implies that
  [k](Π_kψ)=[∞](Π_kψ) and
  [k](Π_ku_∞)=[∞](Π_ku_∞) as well as
  the corresponding relations between [k] and [∞];
  compare with (<ref>). Thus, the above estimate follows
  from the Cauchy-Schwarz inequality, application of inverse inequalities in
  conjunction with the stability of the lifting
  operators (<ref>),
  and Lemma <ref>.  


  Consequently, triangle
  inequality and the above imply
  
    [k]u_∞-   ≤[k]u_∞-Π_ku_∞+[k]-Π_ku_∞
       [k]u_∞-Π_ku_∞+[∞]u_∞-Π_ku_∞
           +
        fψ-Π_kψ-[k]Π_ku_∞ψ-Π_kψ/[k]u_k-Π_ku_∞.

  Thanks to [k]u_∞-Π_ku_∞≤[∞]u_∞-Π_ku_∞,
  this proves the assertion.


The properties of  the quasi-interpolation (<ref>) allow for the
consistency term in Lemma <ref> to be
bounded
by the a posteriori indicators of essentially the elements, which 
will experience further refinements. 


  Let u_∞∈_∞ be the solution of (<ref>) and u_k∈ be the
  approximation from (<ref>) on  for some k∈. Then, we have 
  
    fu_k-Π_ku_k-[k]Π_ku_∞u_k-Π_ku_k/[k]u_k-Π_ku_∞(∑_∈_k∖_k^3+_k(Π_ku_∞,)^2)^1/2,

  where ^3+:={∈:()⊂^++}.



  Let v_k:=Π_ku_∞ and ϕ:=u_k-Π_k
  u_k=u_k-Π_ku_∞-Π_k(u_k-Π_ku_∞). Then, using
  integration by parts, we have
  
    fϕ   -[k]v_kϕ
       =∫_(f+Δ
                                            v_k)ϕ
                                            -∫__k∇
                                            v_kϕ
          +∫__kθ∇ϕv_k
          -∫__k(β⃗·v_k∇ϕ+∇
             v_kβ⃗·ϕ)
          - ∫_Ωγ([k](v_k)+[k](β⃗·v_k))·([k](ϕ)+[k](β⃗·ϕ))
         -σ∫__kh_k^-1v_kϕ.

  Thanks to properties of Π_k (see Lemma <ref>), we have
  that v_k|_≡0 for
  S∈_k∖_k^+, v_k|_Ω∖Ω_k^+≡0,
  ϕ|_≡0 for ∈^++, and ϕ|_S≡0 for S∈_k^++. Therefore, we have 
  
    fϕ   -[k]v_kϕ
       =∫_∖^++(f+Δ v_k)ϕ
          -∫__k∖_k^++∇
            v_kϕ
         +θ∫__k^+∇ϕv_k
          -∫__k^+β⃗·v_k∇ϕ-∫__k∖_k^++∇ v_kβ⃗·ϕ
          - ∫_Ωγ([k](v_k)+[k](β⃗·v_k))·([k](ϕ)+[k](β⃗·ϕ))
         -σ∫__k^+∖_k^++h_k^-1v_kϕ.
    
The last term on the right-hand side of (<ref>) can be estimated using Cauchy-Schwarz' inequality;
  for the first two terms we use the interpolation estimates from
  Corollary <ref> for 
    ϕ=ψ-Π_kψ with ψ=u_k-Π_ku_∞∈_k as to
    obtain 
    
    ∫_∖^++(f+Δ
                                            v_k)ϕ
                                            -∫__k∖_k^++∇
                                            v_kϕ
    [(∫_∖^++h_k^2|f+Δ
          v_k|^2)^1/2
          + (∫__k∖_k^++ h_k∇
          v_k^2)^1/2][k]u_k-Π_ku_∞.

    Moreover, from ϕ|_≡0, ∈^++, we have
    that ϕ|_(S)≡0 and thus
      ∇ϕ|_≡0 for all 
    ∈_k^3+=(^3+). Therefore, by
    standard trace inequalities, inverse estimates and
    Corollary <ref>, we have  that
    
    ∫__k^+∇ϕv_k   =
                                                            ∫__k^+∖_k^3+∇ϕv_k(∫__k^+∖_k^3+
                                                            h_k^-1v_k^2)^1/2[k]ϕ.

     A similar argument yields 
     
    ∫__k^+β⃗·v_k∇ϕ   =∫__k^+∖_k^3+β⃗·v_k∇ϕ
       β⃗(∫__k^+∖_k^3+
                                                            h_k^-1v_k^2)^1/2[k]ϕ.

     Finally we have with (<ref>) and the local support
     of the local liftings, that
     
    ∫_Ω[k](v_k)·[k](ϕ)   =∫_Ω(∑_∈_k^+
           [k](v_k))
           ·(∑_∈_k∖_k^++[k](ϕ))
       = ∫_^+∖^++[k](v_k)·[k](ϕ)
       (∫__k^+∖_k^3+^-1v_k^2)^1/2[k]ϕ.

     Similar bounds hold for the remaining terms in (<ref>).
     Combining the above observations proves the desired assertion.





In order to conclude convergence of the sequence of discrete discontinuous
Galerkin approximations from Lemma <ref>, we need to control
the error estimator. To this end, we shall use Verfürth's bubble
function technique. 

Let n∈, such that 
n uniform refinements of an element ensure that the element as well
as  each of its sides have at least one interior node.
We specify the elements in , which neighbourhood is eventually uniformly
refined n times by

    _k^0{∈   ∃ℓ=ℓ()≥ k+n such that
       all '∈() are n 
    times uniformly refined in _ℓ}

This
guarantees that suitable discrete interior and side bubble
functions are available in _∞ for all ∈^0;
(compare also with <cit.>, <cit.> and
<cit.>). 
We define Ω_k^0:=Ω(^0)
⊂Ω_k^-. Introducing Ω_k^⋆=Ω(^⋆) with
^⋆∖(^++∪^0),
we have from 
<cit.> that

    |Ω(^⋆)|
      → 0     as k→∞.





  Let u_∞ be the solution of (<ref>). 
  Then, for every ∈^0 and v∈, k∈, we have
  
    ∫_h_k^2|f+Δ
          v|^2+∫_∂∩Ω h_k
            v^2
       [()] (u_∞-v)^2
            +∫_{S∈^+:S⊂()}h_+^-1u_∞-v^2
           +((),f)^2;

  in particular, we also have 
  
    ∑_∈_k^0∫_h_k^2|f+Δ
          v|^2+∫_∂∩Ω h_k
            v^2
        
    [∞]u_∞-v^2+∑_∈^0∑_'∈ω_k()(',f)^2.

  Note that since v∈⊄_∞ in general, the above terms may be equal to infinity.






  The proof follows from standard techniques; compare
  e.g. <cit.> by replacing
  Verfürth's bubble functions by their discrete counterparts. 
  However, in order to keep the presentation self-contained, we
  provide a sketch of the proof. Let ∈^0, then, thanks
  to the  definition of ^0, there exists some ℓ>k such that there exists a
  piecewise affine discrete bubble function
  ϕ_∈_ℓ∩ C(Ω) 
  satisfying ϕ_∈ H_0^1(), and
  
    h_^d[L^∞()]∇ qϕ_^2[]∇ qϕ_^2
        h_^-2[]q^2  for all q∈_r-1 (E);

  compare <cit.>.
  Let f_∈_r-1(E) an arbitrary polynomial. Observing that
  (f_+Δ v)ϕ_∈ C(Ω) and thus does not jump across
  sides, we have by
  equivalence of norms on finite dimensional spaces and a scaled trace inequality, that 
  
    ∫_ |f_+Δ v|^2
       ∫_ (f_+Δ v) (f_+Δ v)ϕ_
       = [∞]u_∞-v(f_+Δ v)ϕ_-∫_
          (f-f_) (f_+Δ v)ϕ_
       [] (u_∞ - v)[]∇(f_+Δ v)ϕ_
          - ∫_^+u_∞-v∇(f_+Δ v)ϕ_
         +[]f-f_[](f_+Δ v)ϕ_.

  From (<ref>) and standard inverse estimates,  we conclude that 
  
    ∫_^+u_∞-v∇(f_+Δ
        v)ϕ_
       ≤∑_S∈^+,S⊂∫_Su_∞-v^2[L^∞()]∇(f_+Δ
        v)ϕ_
       (∫_^+h_+^d-1u_∞-v^2)^1/2
          h_^-1-d/2[]f_+Δ
        v
       (∫_^+h_+^-1u_∞-v^2)^1/2
      h_^-1[]f_+Δ
        v,

  since h_+≤ h_ on .
  Therefore, we arrive at 
  
    ∫_ h_k^2|f_+Δ v|^2   []
            (u_∞ - v)^2+ ∑_S∈^+,S⊂∫_ h_+^-1u_∞-v^2
         +h_^2[]f-f_^2.

  Thanks to the definition of ^0, the same bound applies for
  all '∈().
  
  
  We now turn to investigate the jump terms. To this end, we fix
  one ∈_k, ⊂. Thanks to the definition of
  ^0, there exists 
  
  a hat function ϕ_∈_ℓ∩ C(Ω)∩ H_0^1((S)),
  and for 
  q∈_r-1(), let q̃∈_r-1((S)) be some extension, such that
  
    h_^d [L^∞((S))]∇q̃ϕ_[(S)]q̃ϕ_^2 h_∫_  |q|^2.

  Noting that ∇ v∈_r-1(S), we
  have, by the equivalence of norms on finite dimensional spaces, that 
  
    ∫_S∇
            v^2   ∫_S∇ v^2ϕ_
       = 
          [∞]u_∞-v∇
                vϕ_ - ∫_ω_k(S)
          (f+Δ v)∇ vϕ_
       [(S)](u_∞-v)[(S)]∇∇
                vϕ_
          + ∫_^+u_∞-v∇∇
                vϕ_
         +([]f+Δ v^2+[']f+Δ v^2)^1/2[(S)]∇
                vϕ_.

   Similarly, as for the element residual, we have that 
  
    ∫_^+u_∞-v∇∇
                vϕ_
    (∑_'∈^+,
        '⊂(S)h_+^-1u_∞-v^2)^1/2(∫_ h_∇ v^2)^1/2,

using (<ref>).  Combining this with (<ref>), we obtain
  
    ∫_Sh_∇
            v^2   [(S)](u_∞-v)^2+ ∑_'∈^+,
          '⊂(S)∫_ h_+^-1u_∞-v^2 
          + h_^2[]f+Δ v^2+h_'^2[']f+Δ
          v^2.

  Finally applying the bound (<ref>) to
  ,'∈(), we have proved the first assertion. 

  The second assertion follows, then, by summing over all
  ∈^0 together with an observation from
  <cit.>, which we sketch here in order to keep the
  this work self-contained. Let
  M:=max{#():∈^0} 
  be the maximal number of neighbours, then ^0 can be split
  into M^2+1 subsets _k,0^0,…, _k,M^2^0 such
  that for each j, we have that ',∈_k,j^0 with
  ≠' implies that ()∩(')=∅.
  Consequently, we have 
  
    ∑_∈^0[()]
          (u_∞-v)^2 
           ≤∑_j=0^M^2∑_∈_k,j^0[()]
          (u_∞-v)^2
        
       ≤ (M^2+1) [Ω_k^0]
          (u_∞-v)^2.
  
  Together with similar estimates for the jump terms and the
  oscillations  the second assertion follows from the first one.




  Let u_∞ the solution of (<ref>) and u_k∈ be the
approximation from (<ref>) on  for some k∈.
  Then, 
  
    [k]u_∞-u_k→ 0   as k→∞

  and in particular [Ω]u_∞-u_k→ 0 as k→∞.




  Thanks to Lemma <ref>, Lemma <ref> and
  Lemma <ref>, we have that 
  
    lim_k→∞[k]u_∞-u_k^2   lim_k→∞[∞]u_∞-v_k^2+∑_∈_k∖_k^3+_k(v_k,)^2
        
       =lim_k→∞∑_∈_k∖_k^3+_k(v_k,)^2,

  where v_k:=Π_ku_∞.
  
  
  
  
  
  
  We conclude
  from (<ref>) that 
  
    |Ω∖(Ω_k^0∪Ω_k^3+)|   ≤|Ω∖(Ω_k^0∪Ω_k^++)|+|Ω_k^++∖Ω^3+_k| 
        
        
        
          → 0 ,

  as k→∞. Indeed, for k∈, it
  follows from Lemma <ref> and 
  #[k]^+<∞, that there exists K=K(k), such that 
  [k]^+⊂_K^3+,
  i.e. |Ω^+∖Ω^3+_K|≤
  |Ω^+∖Ω^+_k|→0 as k→∞.
  Thanks to monotonicity we conclude
  that |Ω^++_k∖Ω^3+_k|≤|Ω^+∖Ω^3+_k| → 0 as k→∞.
  We next show that this implies
  
    ∑_∈_k∖(_k^0∪_k^3+)_k(v_k,)^2→
        0.

  Lemma <ref> implies that
  [∞]u_∞-v_k→0 and, thus, the interior
  residual and the gradient
  jumps part of the estimator vanish due to uniform
  integrability. Moreover, it follows from Proposition <ref> that
  
    ∫_(∖(_k^0∪_k^3+))
        h_k^-1v_k^2   ∫_(∖^3+)
        h_k^-1u_∞^2+[k]u_∞-v_k^2
          
       ≤∫_(^+∖^3+)
        h_+^-1u_∞^2+[k]u_∞-v_k^2.

  
  The last term on the right-hand side of the above estimate vanishes thanks to Lemma <ref>. 
  Again, letting K=K(k), such that 
  ^+⊂_K^3+, we have
  
    ∫_(^+∖[K(k)]^3+)
        h_+^-1u_∞^2≤∫_(^+∖^+)
        h_+^-1u_∞^2→ 0,  as k→∞.

  Thanks to monotonicity, we thus conclude ∫_(^+∖^3+)
    h_+^-1u_∞^2→0, as k→∞.


  On the remaining elements ^-, it follows from Proposition <ref> 
  that 
  
    ∑_∈_k^0_k(v_k,)^2[∞]u_∞-v_k^2+
        ∑_∈_k^0((),f)^2.

  The first term on the right-hand side vanishes due to Lemma <ref>. For the
  second term we observe that |⋃{():∈^0}|| Ω_k^0|, depending on the shape regularity of
  _0 and, therefore, it vanishes since 
  
    [L^∞(Ω)]h_kχ_Ω_k^0≤[L^∞(Ω)]h_kχ_Ω_k^-→ 0   as k→∞,

  thanks to Lemma <ref>.

  The second limit follows then from
    
    [Ω]u_∞-u_k   ≤[Ω]u_∞-Π_k
                                       u_∞+[Ω]Π_ku_∞-u_k
       [∞]u_∞-Π_k
          u_∞+[k]Π_ku_∞-u_k,

  which vanishes due to the above observations.
  
  




§ PROOF OF THE MAIN RESULT



We are now in the position to prove that the error estimator
vanishes by splitting the estimator according to 

    =^0∪^++∪^⋆

and consider each part separately following the ideas of <cit.>.
This in turn implies that the sequence of discontinuous Galerkin
  approximations produced by indeed converges to the exact
  solution of (<ref>).


  We have that 
  
    _k(^0)→ 0 ,  as k→∞.



  Thanks to Proposition <ref>, we have 
  
    ∑_∈_k^0∫_h_k^2|f+Δ|^2+∫_∂∩Ω h_k∇^2
        
    [∞]u_∞-^2+∑_∈^0((),f)^2.

  The right-hand side vanishes thanks to Theorem <ref>
  and (<ref>). 
  
  
  
  
  
  

  It remains to prove that 
  
    ∫_(^0)h_k^-1^2→0,  as k→∞.

  By definition,
  Ω_k^0⊂Ω∖Ω_k^+ and, thanks to
  Lemma <ref>(<ref>), we have that Π_k u_∞∈ C(Ω∖Ω_k^+). 
  Therefore, we conclude
  
    ∫_(^0)h_k^-1^2   =
        ∫_(^0)h_k^-1-Π_ku_∞^2≤[k]-Π_ku_∞→0

  as k→∞; see Lemma <ref> and Theorem <ref>.



  We have that 
  
    lim_k→∞_k(^⋆)=0.



  We conclude from the lower bound (Proposition <ref>) that 
  
    ∑_∈^⋆∫_h_k^2|f+Δ|^2+∫_∂ h_k∇^2
       ∑_∈^⋆[ω_k()]u-^2+[ω_k()]∇
              u-^2+((),f)^2
            
    
            
               ∑_∈^⋆{[ω_k()]u^2+[ω_k()]u_∞-^2+[ω_k()]u_∞^2
            
           +[ω_k()]∇
              u^2+[ω_k()]
              u_∞-^2+[ω_k()]
              u_∞^2 
           +((),f)^2}.

  This vanishes as k→∞ thanks to Theorem <ref> and (<ref>),
  together with the uniform integrability of the terms involving u
  and u_∞. Note that
  |⋃{():∈^⋆}|
  |Ω_k^⋆|, with the constant depending on the shape regularity of _0.

  It remains to prove
  
    ∫_(^⋆)h_k^-1^2→0,  as k→∞.

  To this end, we observe that
  
    ∫_(^⋆)h_k^-1^2   =
        ∫_(^⋆)h_k^-1-Π_ku_∞^2+∫_(^⋆)h_k^-1Π_ku_∞^2
       ≤1/σ̅[k]-Π_ku_∞^2+∫_(^⋆)h_k^-1Π_ku_∞^2.

  As in the proof of Lemma <ref>, we have that the first term
  vanishes as k→∞. Thanks to Lemma <ref>, there
  exists ℓ(k)≥ K(k)≥ k such that
  ^+⊂[K(k)]^++ and
  [K(k)]^+⊂[ℓ(k)]^++.
  Consequently, we have that Π_ℓ u_∞|_ =0 for all
  S∈; see Lemma <ref>(<ref>).  
  Therefore, we conclude from Lemma <ref> that
  
    σ∫_(^⋆)h_k^-1Π_ku_∞^2   =σ∫_(^⋆)h_k^-1Π_ku_∞-Π_ℓ
        u_∞^2
       [k]Π_ku_∞-
        u_∞^2+[ℓ]u_∞-Π_ℓ
        u_∞^2→0,

  as k→∞.




  We have 
  
    _k(^++)→0  as k→∞.


Step 1: By definition, elements in ^++ will not be
  subdivided, i.e. we have that
  ℳ_k⊂∖^++; compare
  with (<ref>). As a consequence of Lemmas <ref>
  and <ref>,
  we conclude from (<ref>) for all
  ∈^++
  that 
  
    _k()≤lim_k→∞g(_k(ℳ_k))=
        lim_k→∞ g(_k(^-∪_k^⋆))→0,

  as k→∞. We shall reformulate the above element-wise convergence
  in an integral framework, in order to conclude _k(^++)→ 0
  as k→∞ via a generalised version of the dominated
  convergence theorem. To this end, we shall consider some
  properties of the error indicators.
  
  Step 2:
  Thanks to the definition of ^++, we have for all
  ∈^++, that
  ω_k()=ω_ℓ()=:ω() and
  ()=[ℓ]()=() for all ℓ≥ k.
  Therefore, we obtain by the lower bound,
  Proposition <ref>, that 
  
    _k()^2   [()]-u^2+((),f)^2
          
       [()]-u_∞^2+[()]u_∞^2+[H^1(ω())]u^2
          +[ω()]f^2
          
       =:[()]-u_∞^2+C_^2.

  Arguing as in the proof of Proposition <ref>, 
  we can conclude from the local estimate that
  
    ∑_∈^++C_^2 [∞]u_∞^2+[H^1(Ω)]u^2+f_Ω^2<∞

  independently of k.
  
  Step 3: We shall now reformulate _k(^++) in
  integral form. Note that thanks to Lemma <ref>, we have that
  ^+=⋃_k∈_0^+=⋃_k∈_0^++,
  and also that the sequence {^++}_k∈_0 is nested. For
  x∈Ω^+, let 
  
    ℓ=ℓ(x):=min{k∈_0:  there
        exists ∈_k^++ such that x∈}.

  Then, we define 
  
    ϵ_k(x):=M_k(x):=0  for k<ℓ,
        andϵ_k(x):=1/||_k^2(),     M_k:=1/||([()]-u_∞^2+C_^2)  for k≥ℓ.

  Consequently, for any k∈_0, we have 
  
    _k(^++)^2=∫_Ω^+ϵ_k(x).

  Moreover, thanks to the fact that the sequence
  {^++}_k∈_0 is nested, we conclude from (<ref>) that 
  
    lim_k→∞ϵ_k(x)=lim_k→∞1/||_k^2()=0.

  It follows from (<ref>) and (<ref>)
  that M_k is an integrable majorant for ϵ_k.

Step 4: We shall show that the majorants {M_k}_k∈_0
converge in L^1(Ω^+) to 

    M(x):=1/||C_^2,   for x∈  and  ∈^+.

Then the assertion follows from a generalised majorised convergence
theorem; see <cit.>. 
In fact, by the definition of M_k, we have that 

    [L^1(Ω^+)]M_k-M=∑_∈^++[L^1()]M_k-M+∑_∈^+∖^++[L^1()]M.

The latter term vanishes since it is the tail of a converging series (compare
with (<ref>)) and for the former term, we have, thanks to
Theorem <ref>, that

    ∑_∈^++[L^1()]M_k-M=∑_∈^++[()]-u_∞^2[k]u_k-u_∞→ 0

as k→∞.






  The assertion follows from Proposition <ref> together
  with Lemmas <ref>, <ref>, and
  <ref> recalling
  the splitting (<ref>).




§ ACKNOWLEDGEMENT

We thank the anonymous referee of  the paper
<cit.> for finding a highly non-trivial
counterexample to the first statement in
<cit.>, which lead to this
corrected version of <cit.>.






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