HAL Id: hal-02941277
https://hal.archives-ouvertes.fr/hal-02941277
Submitted on 16 Sep 2020
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
flow on general meshes
Julien Coatléven
To cite this version:
https://doi.org/10.1051/m2an/2020025 www.esaim-m2an.org
SOME MULTIPLE FLOW DIRECTION ALGORITHMS FOR OVERLAND FLOW
ON GENERAL MESHES
Julien Coatl´
even
˚Abstract.After recalling the most classical multiple flow direction algorithms (MFD), we establish their equivalence with a well chosen discretization of Manning–Strickler models for water flow. From this analogy, we derive a new MFD algorithm that remains valid on general, possibly non conforming meshes. We also derive a convergence theory for MFD algorithms based on the Manning–Strickler models. Numerical experiments illustrate the good behavior of the method even on distorted meshes.
Mathematics Subject Classification. 65N08, 65N12, 65N15.
Received September 17, 2019. Accepted April 9, 2020.
1. Introduction
Overland water flow plays a major role in hydrogeology at many time scales, from million of years for stratigraphic studies of interest for the oil and gas industry to several days or weeks for predicting river flooding and landslides. Intermediate time scales are also increasingly studied as they are of crucial interest for climatic forecasts, from glacier withdrawal to desert expansion. Of course, a huge literature exists in the corresponding fields, describing the physical phenomenons involved with an equally huge model diversity (Navier–Stokes, Stokes, shallow water, etc...). Due to the complexity underlying flow models, approximate, phenomenologically based models have been developed to increase computational speed. Among those models, a common approach that has given satisfactory results is the one based on the so-called multiple flow directions algorithms (MFD). The idea underlying all MFD algorithms is that water routing at large space scales will be mainly governed by the topographic slope. Using digital elevation models that provide a meshed representation of the topography, those algorithms compute water flow by distributing water from the topographically higher discrete cells to the topographically lower ones, each distribution formula corresponding to a specific MFD algorithm. In most cases, the MFD algorithms are developed assuming that the mesh is a uniform cartesian grid, with the same space step in each direction. Historically, the first MFD algorithms were in fact single flow direction algorithms, where the distribution formula selected only one neighbour, generally the one with the steepest slope. The deficiencies of such simple algorithms are the origin of the development of MFD algorithms (see [26]). The most classical MFD algorithm [16,22] uses directly the slope to distribute the flow, while models using powers of the slope were developed to concentrate the flow and limit diffusion effects due to the use of coarse meshes Keywords and phrases. Multiple flow direction algorithm, overland flow, virtual element method, hybrid finite volume, general meshes.
IFP ´Energies nouvelles, 1 et 4 avenue de Bois-Pr´eau, Rueil-Malmaison 92852, France.
˚Corresponding author:julien.coatleven@ifpen.fr
(see [18,21,25]). Some attempts have been made to apply MFD algorithms on more general meshes, with an emphasis on triangular ones (see [24,27]). We refer the reader to [9] for a comparison of some of those methods, and to the references in the aforementioned papers for a more complete overview of the tremendous literature on MFD algorithms, on which we will not try to be exhaustive.
All those algorithms proceed sequentially from the higher cells to the lower ones, however as was already noticed in [23], using an MFD algorithm is in fact equivalent to solving a linear system using a specific cell ordering. The key idea underlying the present paper comes from a deeper study of the linear system associated with one of the earliest MFD algorithm. Indeed, we will explain that this linear system is completely equivalent to a two point flux finite volume scheme (see [10]) applied to a stationary Manning–Strickler model for water flow. It was then clear that replacing the TPFA scheme that requires a strong orthogonality hypothesis on meshes to remain valid by more advanced flux approximation schemes would allow us to derive MFD algorithms adapted to general meshes. Moreover, this equivalence will also allow us to derive a theoretical framework within which we will be able to study the convergence properties of MFD algorithms. In our numerical experiments, we have considered two flux reconstructions inspired by two finite volume schemes: the hybrid finite volume (see [11,12]) and the virtual volume method (or conservative first order virtual element method, see [5]). Of course, other choices could have been made (for instance VAG finite volumes [13,14] or discontinuous Galerkin methods [7]), however those two schemes will be sufficient to illustrate our approach.
The paper will be organized as follows: after describing the data and meshes, we recall the most classical mul-tiple flow direction algorithms, and reformulate them in a more algebraic fashion. Next, using this reformulation we explain how they are linked with the TPFA scheme for a family of Manning–Strickler models. We elaborate on this basis to overcome the mesh limitations induced by the TPFA scheme, introducing a new family of MFD algorithms that will remain valid on a huge class of meshes, including those with hanging nodes. We then study the convergence properties of all methods and conclude by some numerical illustrations.
2. Classical multiple flow direction algorithms
2.1. Mesh and data description
Let Ω be a bounded polyhedral connected domain of R2, whose boundary is denoted BΩ “ ΩzΩ. We recall the usual notations describing a mesh ℳ “ p𝒯 , ℱ q of Ω. 𝒯 is a finite family of connected open disjoint polygonal subsets of Ω (the cells of the mesh), such that Ω “ Y𝐾P𝒯𝐾. For any 𝐾 P 𝒯 , we denote by |𝐾| the measure of |𝐾|, by B𝐾 “ 𝐾z𝐾 the boundary of 𝐾, by ℎ𝐾 its diameter and by 𝑥𝐾 its barycenter. ℱ is a finite family of disjoint subsets of hyperplanes of R2included in Ω (the faces of the mesh) such that, for all 𝜎 P ℱ , its measure is denoted |𝜎|, its diameter ℎ𝜎 and its barycenter 𝑥𝜎. For any 𝐾 P 𝒯 , there exists a subset ℱ𝐾 of ℱ such that B𝐾 “ Y𝜎Pℱ𝐾𝜎. Then, for any 𝜎 P ℱ , we denote by 𝒯𝜎“ t𝐾 P 𝒯 | 𝜎 P ℱ𝐾u. Next, for all 𝐾 P 𝒯 and all 𝜎 P ℱ𝐾,
we denote by 𝑛𝐾,𝜎the unit normal vector to 𝜎 outward to 𝐾, and 𝑑𝐾,𝜎“ ||𝑥𝜎´ 𝑥𝐾||. The set of boundary faces is denoted ℱext, while interior faces are denoted ℱint. Finally for any 𝜎 P ℱint, we denote 𝑑𝐾𝐿 “ ||𝑥𝐾´ 𝑥𝐿|| where 𝒯𝜎 “ t𝐾, 𝐿u. We assume that there exists a subset ℱext,inof ℱext such that:
BΩin“ ď
𝜎Pℱext,in
𝜎 where BΩin“ t𝑥 P BΩ | ∇𝑏 ¨ 𝑛 ą 0u
and we denote of course ℱext,out “ ℱextzℱext,in. As usual, ℎ “ max𝐾P𝒯ℎ𝐾 will denote the mesh size. In what follows, we will assume that our mesh satisfies:
(A1) There exists a real number 𝜌 ą 0 and a matching simplicial submesh 𝒮𝒯 of ℳ such that for any 𝑇 P 𝒮𝒯 , 𝜌ℎ𝑇 ď 𝑟𝑇 where 𝑟𝑇 is the inradius of 𝑇 , and for any 𝑇 P 𝒯 and any 𝑇 P 𝒮𝒯 such that 𝑇 Ă 𝐾, 𝜌ℎ𝐾 ď ℎ𝑇. From [3,7], we know that assumption (A1) implies that for any 𝑘 P N, there exists 𝐶𝑡𝑟,𝑘ą 0 independent on ℎ such that for any 𝐾 P 𝒯 , any 𝜎 P ℱ𝐾 and any 𝑝 P P𝑘p𝐾q:
Figure 1. Example of real topography.
There also exists 𝐶𝑡𝑟ą 0 independent on ℎ such that for any 𝑣 P 𝐻1p𝐾q: ||𝑣||𝐿2p𝜎qď 𝐶𝑡𝑟 ´ ℎ´1 𝐾 ||𝑣|| 2 𝐿2p𝐾q` ℎ𝐾||∇𝑣||2𝐿2p𝐾q ¯1{2 . (2.2)
Finally, assumption (A1) implies that for any integer 𝑘, there exists 𝐶poly,𝑘ą 0 such that for any 𝐾 P 𝒯 and any 𝑣 P 𝐻𝑠
p𝐾q with 𝑠 P t1, . . . , 𝑘 ` 1u, there exists 𝑝 P P𝑘p𝐾q such that
|𝑣 ´ 𝑝|𝐻𝑚p𝐾q` ℎ1{2𝐾 |𝑣 ´ 𝑝|𝐻𝑚pB𝐾qď 𝐶poly,𝑘ℎ𝑠´𝑚𝐾 |𝑣|𝐻𝑠p𝐾q for 𝑚 P t0, . . . , 𝑠 ´ 1u . (2.3)
In the estimates that will follow, the constant 𝐶 ą 0 will always denote by convention a quantity independent on the mesh size ℎ, whose value can change from line to line. Also by convention for any 𝜎 P ℱint we will denote 𝒯𝜎“ t𝐾, 𝐿u. In the same way, when considering a cell 𝐾 and one of its interior faces 𝜎 P ℱ𝐾X ℱint, by convention the cell 𝐿 will denote the other element of 𝒯𝜎.
In order to be able to establish error estimates, we assume that the topography, often called the basement in the geological community and thus usually denoted 𝑏, satisfies 𝑏 P 𝑊2,8pΩq and that there exists 𝛼 ą 0 such that ´∆𝑏 ě 𝛼 for almost every 𝑥 P Ω. However, from the practical point of view 𝑏 is only known through some pointwise values, very often given as a digitized surface elevation, on which some interpolation and upscaling or downscaling processes have been applied (see Fig.1). Thus, its Laplacian cannot be expected to be practically computable, and the above assumption should really be considered as an abstract technical requirement for establishing convergence estimates. Thus, our data will more realistically consists in some discrete mesh-based p𝐵𝐾q𝐾P𝒯 representation of 𝑏, where each 𝐵𝐾 can represent several pointwise values of 𝑏, complemented by a water source term 𝑓 P 𝐿8pΩq, possibly also only known through a mesh-based representation.
2.2. The classical multiple flow direction algorithm
generally found in the geological community, that is to say a purely algorithmic point of view. One of our first tasks will precisely consists in abstracting ourselves from this algorithmic setting. For any 𝐾 P 𝒯 , let 𝑏𝐾 be a value for the topography associated with cell 𝐾. To fix notations, consider for the moment that
𝑏𝐾 “ 1 |𝐾| ż 𝐾 𝑏 @𝐾 P 𝒯
but other approximations can be considered, for instance 𝑏𝐾 “ 𝑏 p𝑥𝐾q for any 𝐾 P 𝒯 if 𝑏 is regular enough. Multiple flow direction algorithms are based on formulae to distribute the water flow from a cell to its neighbours that are topographically lower. The most classical distribution formula consists simply in distributing the flow proportionally to the ratio 𝑠𝐾𝐿{𝑠𝐾of the discrete slope 𝑠𝐾𝐿between the high cell 𝐾 and the low cell 𝐿 regarding the total positive slope 𝑠𝐾 of the high cell 𝐾, where the discrete slope 𝑠𝐾𝐿 is given by
𝑠𝐾𝐿“ |𝜎| 𝑑𝐾𝐿
p𝑏𝐾´ 𝑏𝐿q and the total positive slope 𝑠𝐾 of cell 𝐾 is given by
𝑠𝐾 “ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿 |𝜎| 𝑑𝐾𝐿 p𝑏𝐾´ 𝑏𝐿q .
Notice that in many cases of the literature, as the MFD algorithm is applied on a uniform cartesian mesh with the same space step in every direction, the face measure |𝜎| is simply omitted (see for instance [16,22]), with no impact on the ratio 𝑠𝐾𝐿{𝑠𝐾. To give a detailed description of the most classical multiple flow direction algorithms, we need to introduce some notations. Let 𝑏0 “ max t𝑏𝐾| 𝐾 P 𝒯 u, 𝒯0 “ t𝐾 P 𝒯 | 𝑏𝐾 “ 𝑏0u and
ˆ
𝒯´1 “ H. The set 𝒯0 thus denotes the set of cells with maximum topographic height. Then, for any 𝑛 P N we define the set ˆ𝒯𝑛 of elements of 𝒯 by setting:
ˆ 𝒯𝑛“
ď
0ď𝑖ď𝑛
𝒯𝑖 where 𝒯𝑖“ t𝐾 P 𝒯 | 𝑏𝐾 “ 𝑏𝑖u and 𝑏𝑖“ max !
𝑏𝐾 | 𝐾 P 𝒯 z ˆ𝒯𝑖´1 )
.
By construction, as 𝒯 is a finite set there exists 𝑁𝑏 ą 0 such that 𝒯𝑁𝑏´1 ‰ H and ˆ𝒯𝑁𝑏´1 “ 𝒯 . Thus, for any
𝑛 ě 𝑁𝑏, we have 𝒯𝑛“ H and ˆ𝒯𝑛“ ˆ𝒯𝑁𝑏´1.
To model water sources, we define a family p𝑓𝐾q𝐾P𝒯 by setting: 𝑓𝐾 “ 1 |𝐾| ż 𝐾 𝑓 @𝐾 P 𝒯 .
The classical MFD algorithm (reformulated from [16,22]) then reads as follows (see Fig.2for a visual illustration, where the width of the arrows is roughly following the amount of distributed water):
(i) For any 𝐾 P 𝒯 , the water influx𝑞r𝐾 is initialized at 0. (ii) For any 𝐾 P 𝒯0, the water influx𝑞r𝐾 is given by𝑞r𝐾 “ |𝐾|𝑓𝐾. (iii) Loop for 𝑛 “ 1 . . . 𝑁𝑏´ 1.
(iii-1) For any 𝐿 P ˆ𝒯𝑛´1, we distribute the entire water influx r𝑞𝐿 to the neighbours that belong to 𝒯𝑛 propor-tionally to the slope between the cell 𝐿 and its neighbour, i.e.:
Figure 2. Basic principle of MFD algorithms: water is distributed to lower neighbouring cells proportionally to the slope.
where 𝑠𝐿 is the total positive slope in cell 𝐿, and ÐÝ denotes the action of updating𝑞r𝐾. (iii-2) For any 𝐾 P 𝒯𝑛, the water influx is complemented by the local sources by setting
r
𝑞𝐾ÐÝ𝑞r𝐾` |𝐾|𝑓𝐾. (iv) End loop for 𝑛 “ 1 . . . 𝑁𝑏´ 1.
The MFD algorithm admits a reformulation as a linear system that will play a key role in the remaining of the paper. To our knowledge, only [23] mentioned this reformulation, although without exhibiting an explicit formula. This link seemed to have been most of the time simply overlooked by the geological community: Theorem 2.1. The MFD algorithm is equivalent to solving the following linear system for the unknown p𝑞r𝐾q𝐾P𝒯 r 𝑞𝐾´ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿 |𝜎|r𝑞𝐿 𝑑𝐾𝐿𝑠𝐿 p𝑏𝐿´ 𝑏𝐾q “ |𝐾|𝑓𝐾 @𝐾 P 𝒯 (2.4) where 𝑠𝐾“ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿 |𝜎| 𝑑𝐾𝐿 p𝑏𝐾´ 𝑏𝐿q
using an ordering for the cells of 𝒯 based on decreasing topography 𝑏𝐾 and a lower triangular solver.
Proof. First, notice that as cells 𝐾 P 𝒯𝑛 are updated only at step 𝑛, their expression is complete past this step and given by:
However, by construction of the sets ˆ𝒯𝑖, any 𝐿 P 𝒯 such that 𝑏𝐿 ą 𝑏𝐾 for 𝐾 P 𝒯𝑛, will satisfy 𝐿 P ˆ𝒯𝑛´1. Thus, the above expression can be simplified in:
r 𝑞𝐾 “ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿 |𝜎|𝑞r𝐿 𝑑𝐾𝐿𝑠𝐿 p𝑏𝐿´ 𝑏𝐾q ` |𝐾|𝑓𝐾 for all 𝐾 P 𝒯 z𝒯𝑛
where the set ˆ𝒯𝑛´1 no longer appears. As for any 𝐾 P 𝒯 , there exists 0 ď 𝑛 ď 𝑁𝑏´ 1 such that 𝐾 P 𝒯𝑛, and noticing that 𝑠𝐿 ą 0 as soon as there exists 𝜎 P ℱ𝐿X ℱint, 𝒯𝜎 “ t𝐾, 𝐿u such that 𝑏𝐿ą 𝑏𝐾, (2.4) follows by induction. Finally, starting from (2.4), if one chooses an ordering for the cells of 𝒯 based on decreasing topography 𝑏𝐾, it is clear that the above system becomes a lower triangular one for r𝑞𝐾. It should be obvi-ous at this point that the associated lower triangular solver then coincides exactly with the classical MFD
algorithm.
A great advantage of considering the above system rather than its algorithmic counterpart is that it makes clear that triangular solvers are not the only possible linear solvers, which was indeed the main point of [23]. A wide range of linear solvers can be used to solve (2.4), and most importantly parallel solvers than can considerably speed up the solving process on meshes with a huge cell number.
Remark 2.2. Many variants of this classical MFD algorithm exist, which at least to the authors knowledge mainly consist in modifying the way the influx is distributed from an upper cell to its lower neighbours: powers of the slope instead of the slope, probabilistic repartitions, repartitions using a specified number of neighbours for a given mesh structure, etc... With the exception of distribution formulae that use powers of the slope instead of the slope itself, proceeding as we have done for the most classical MFD algorithm it is not difficult to show that all those variants can be rewritten under the form:
r 𝑞𝐾´ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿 𝜆𝜎|𝜎|𝑞r𝐿 𝑑𝐾𝐿𝑠𝐿 p𝑏𝐿´ 𝑏𝐾q “ |𝐾|𝑓𝐾 @𝐾 P 𝒯
with this time
𝑠𝐾 “ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿 𝜆𝜎|𝜎| 𝑑𝐾𝐿 p𝑏𝐾´ 𝑏𝐿q .
The coefficient 𝜆𝜎 associated with each face will take different values depending on the way one want to modify the influx repartition. Notice that we have chosen to not consider distribution formulae based on powers of the slope as they only had some technical difficulties with no fundamental differences in the analysis.
3. On the link with Manning–Strickler models
Consider the following classical transient Manning–Strickler model: ˇ ˇ ˇ ˇ ˇ ˇ B𝑢 B𝑡 ´ div p𝜆𝑢 𝑚∇𝑏q “ 𝑓 in Ω ´𝑢𝑚∇𝑏 ¨ 𝑛 “ 0 on BΩin
where 𝑢 is the water height, 𝑚 a parameter and 𝑛 denotes the outward normal to Ω. The coefficient 𝜆 P 𝐿8pΩq is the inverse of the Manning–Strickler coefficient expressing soil roughness, and is assumed to satisfy 0 ă 𝜆´ď 𝜆 ď 𝜆` ă `8 almost everywhere in Ω. Formally, the steady state associated with the above system is the following stationary Manning–Strickler model for overland flow:
Denoting 𝑢𝜆,𝑚 “ 𝜆𝑢𝑚 , 𝛽 “ ´∇𝑏 P 𝑊1,8pΩq and 𝜇 “ ´∆𝑏 ě 𝛼 ą 0 P 𝐿8pΩq, remark that (3.1) can be rewritten: ˇ ˇ ˇ ˇ ˇ 𝛽 ¨ ∇𝑢𝜆,𝑚` 𝜇𝑢𝜆,𝑚 “ 𝑓 in Ω 𝑢𝜆,𝑚“ 0 on BΩin. (3.2) Stationary transport problems of the form (3.2) have of course received a lot of attention in the existing literature, in particular as they correspond to a popular model for neutron transport. Their well-posedness has been considered for instance in [1,2], or more recently in [8,15,17]. Thus, from the results of [8,15] and the regularity hypotheses on 𝑏, 𝑓 , and 𝜆, we know that there exists a unique 𝑢 P 𝐿8
pΩq solution of (3.2). As the water height 𝑢 only appears through its 𝑚-th power 𝑢𝑚, in the remaining of the paper we will with a slight abuse of notations directly use 𝑢 instead of 𝑢𝑚.
We are now going to explain that the classical MFD algorithm coincides with a well chosen discretization of the stationary Manning–Strickler model. Assume that the mesh is orthogonal, i.e. there exists a family of centroids p𝑥𝐾q𝐾P𝒯 such that:
𝑥𝐾P ˚𝐾 @𝐾 P 𝒯 and
𝑥𝐿´ 𝑥𝐾 ||𝑥𝐿´ 𝑥𝐾||
“ 𝑛𝐾,𝜎 for 𝜎 P ℱint, 𝜎 “ t𝐾, 𝐿u
and let us denote 𝑑𝐾,𝜎 the distance of 𝑥𝐾 to the hyperplane containing 𝜎 for any 𝜎 P ℱ𝐾 and any 𝐾 P 𝒯 . Then, one can use a two-point finite volume scheme to discretize the steady-state Manning model. Assume that 𝑏𝐾 “ 𝑏 p𝑥𝐾q or at least a second order approximation of it. Denoting 𝑢𝐾 for any 𝐾 P 𝒯 the discrete water height unknown, if one further assumes that 𝑏𝜎 “ 𝑏𝐾 for any 𝜎 P ℱext and 𝐾 P 𝒯𝜎 which is generally what is done in practical applications of the MFD algorithm, for any 𝐾 P 𝒯 we get:
ÿ 𝜎Pℱ𝐾Xℱint 𝜏𝐾𝐿r𝑢 up 𝜎 p𝑏𝐾´ 𝑏𝐿q “ |𝐾|𝑓𝐾 where r𝑢up𝜎 “ 𝑢𝐾 if 𝑏𝐾 ě 𝑏𝐿 and r𝑢 up
𝜎 “ 𝑢𝐿 if 𝑏𝐾 ă 𝑏𝐿 and the transmissivity 𝜏𝐾𝐿 is given for instance by the harmonic mean: 𝜏𝐾𝐿“ |𝜎|𝜆𝐾𝜆𝐿 𝜆𝐾𝑑𝐿,𝜎` 𝜆𝐿𝑑𝐾,𝜎 where 𝜆𝐾 “ 1 |𝐾| ż 𝐾 𝜆.
Immediately we deduce the following equivalence result, which despite its simplicity is the cornerstone of the present paper:
Theorem 3.1. The TPFA scheme for Manning Strickler’s model is equivalent to the MFD algorithm. Proof. Gathering the faces by upwinding kind, we get:
ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿 𝜏𝐾𝐿𝑢𝐾p𝑏𝐾´ 𝑏𝐿q ´ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿 𝜏𝐾𝐿𝑢𝐿p𝑏𝐿´ 𝑏𝐾q “ |𝐾|𝑓𝐾. (3.3) Setting 𝑠𝐾 “ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿 𝜏𝐾𝐿p𝑏𝐾´ 𝑏𝐿q
and noticing that 𝑠𝐿 ą 0 as soon as there exists 𝜎 P ℱ𝐿X ℱint such that 𝑏𝐿 ą 𝑏𝐾, we see that equation (3.3) can be rewritten:
𝑠𝐾𝑢𝐾´
ÿ
𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿
𝜏𝐾𝐿𝑢𝐿p𝑏𝐿´ 𝑏𝐾q “ |𝐾|𝑓𝐾. Defining the water influx by𝑞r𝐾 “ 𝑠𝐾𝑢𝐾, we thus obtain:
r 𝑞𝐾´ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿 𝜏𝐾𝐿r 𝑞𝐿 𝑠𝐿 p𝑏𝐿´ 𝑏𝐾q “ |𝐾|𝑓𝐾. (3.4)
Surprisingly, this result seems to be absent of the MFD literature. The main reason is probably that formu-lations of MFD algorithms as linear systems are equally difficult to find. From this equivalence between the classical MFD and the two-point flux approximation (TPFA) of the classical Manning–Strickler model, some useful observations can be made. Probably the most surprising one is that the MFD unknown p𝑞r𝐾q𝐾P𝒯 can be used instead of 𝑢𝐾 to solve the two-point approximation of the Manning–Strickler model. Moreover, existence and uniqueness of solutions of the MFD problem (3.4) are now an immediate consequence of its equivalence with the MFD algorithm without requiring any hypothesis on the topography. Next, let us mention that when modeling overland flow, the quantity of interest is not the water height but the water discharge, i.e. the norm ||𝜆ℎ𝑚∇𝑏|| of the water flux vector. This is the reason why no water height explicitly appears in MFD algorithms. However, a crucial consequence of our equivalence result is that the usual unknownr𝑞𝐾 of the MFD algorithm, while an excellent choice from the algebraic perspective, is probably not the good quantity to represent the water discharge. Indeed, usingr𝑞𝐾 “ 𝑠𝐾𝑢𝐾 and the consistency of the two-point formula we see that it approximates:
r 𝑞𝐾 « ÿ 𝜎Pℱ𝐾 ż 𝜎 𝑢 p´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎q`
where 𝑣` “ maxp0, 𝑣q. We cannot expect such a r
𝑞𝐾 to be an approximation of the norm of 𝜆𝑢∇𝑏, as it approximates the accumulated influx in a cell which is a mesh dependent quantity. Thus, no kind of convergence let alone approximation properties can be expected for such a quantity in general. To effectively compute a discrete water discharge 𝑞𝐾 for each cell 𝐾 P 𝒯 , we reconstruct cellwise the water flux vector by setting:
𝑄𝐾“ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ą𝑏𝐿 𝜏𝐾𝐿𝑞r𝐾 |𝐾|𝑠𝐾 p𝑏𝐾´ 𝑏𝐿q p𝑥𝜎´ 𝑥𝐾q ´ ÿ 𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿 𝜏𝐾𝐿𝑞r𝐿 |𝐾|𝑠𝐿 p𝑏𝐿´ 𝑏𝐾q p𝑥𝜎´ 𝑥𝐾q .
The consistent water discharge is immediately given by 𝑞𝐾 “ ||𝑄𝐾||. We will illustrate in the numerical section the convergence deficiency of𝑞r𝐾, and how 𝑞𝐾 has a much better behavior. Obviously, for any 𝐾 P 𝒯 such that 𝑠𝐾 ‰ 0, we can compute an equivalent positive water height by setting 𝑢𝐾 “r𝑞𝐾{𝑠𝐾. Cells where 𝑠𝐾 “ 0 are cells where all discrete fluxes are ingoing fluxes, thus it is clear that one should either set 𝑢𝐾 “ `8 or 𝑢𝐾 “ 0 depending if water effectively reaches the cell or not. From the definition of 𝑄𝐾, we see that the value of 𝑢𝐾 for such cells will have no influence on the water discharge and its asymptotic convergence. It is thus clear that one can always define 𝑢𝐾 by setting:
𝑢𝐾 “ ˇ ˇ ˇ ˇ ˇ ˇ r 𝑞𝐾 𝑠𝐾 if 𝑠𝐾 ą 0 0 otherwise.
Remark 3.2. Notice that using the harmonic mean is not mandatory if one can use an approximation of 𝜆 directly on the faces, leading to the transmissivity 𝜏𝐾𝐿“ |𝜎|𝜆𝜎{𝑑𝐾𝐿in which case (3.4) will correspond to one of the many variants of the classical MFD.
4. Some conservative and consistent flux reconstructions on general meshes
4.1. A conservative flux reconstruction formula
For any cell 𝐾 P 𝒯 , let us denote 𝐵𝐾 “ 𝒟𝐾p𝑏q the local values associated with the topography 𝑏, and 𝑋𝐾 “ R𝒩𝐾 the set of local values, with 𝒩𝐾 the number of local values. The operator 𝒟𝐾 : 𝐻1p𝐾q ÞÝÑ 𝑋𝐾 will of course depend on the considered reconstruction formula, however to simplify notations we consider that the value 𝑏𝐾 at 𝑥𝐾 always belongs to the set of local values. Those local values represent all the knowledge we have in practice of the field 𝑏, and once again its Laplacian cannot be expected to be computable. For typical applications such as stratigraphic modelling it consists in cell values complemented by vertex or face values, thus conditioning the schemes we can effectively use. We assume that we are given a gradient reconstruction operator ∇𝐾 : 𝑋𝐾 ÞÝÑ R2, and we define a reconstruction formula in each cell through the operator Π𝐾 : 𝑋𝐾ÞÝÑ P1p𝐾q, where P1p𝐾q is the set of first order polynomials on 𝐾 and:
Π𝐾p𝐵𝐾q p𝑥q “ 𝑏𝐾` ∇𝐾p𝐵𝐾q ¨ p𝑥 ´ 𝑥𝐾q
and thus ∇Π𝐾p𝐵𝐾q “ ∇𝐾p𝐵𝐾q. Such gradient reconstructions are the usual basis of modern finite volume methods (see [4,5,12]), and many formulae can be found in the literature. Using those reconstructions, it is tempting to define the flux operator 𝐹𝐾,𝜎p𝐵𝐾q by setting:
𝐹𝐾,𝜎p𝐵𝐾q “ ´|𝜎|𝜆𝐾∇𝐾p𝐵𝐾q ¨ 𝑛𝐾,𝜎.
As consistency of the flux will of course rely on the consistency of the gradient reconstruction operators, the above definition will indeed lead to consistent fluxes, however to construct our generalized MFD algorithms conservativity will play a crucial role. This means that we will require the conservativity of the flux:
ÿ
𝐾P𝒯𝜎
𝐹𝐾,𝜎p𝐵𝐾q “ 0
which cannot be satisfied in general (except by the two-point fluxes) with such a simple formula. This is the reason why, following the ideas of [4], we introduce a stabilized gradient operator ∇𝐾,𝜎 : 𝑋𝐾 ˆ R ÞÝÑ R by setting:
∇𝐾,𝜎p𝐵𝐾, 𝜌𝜎q “ ∇𝐾p𝐵𝐾q ` 𝜂 𝑑𝐾,𝜎
pp𝜌𝜎´ 𝑏𝐾q ´ ∇𝐾p𝐵𝐾q ¨ p𝑥𝜎´ 𝑥𝐾qq 𝑛𝐾,𝜎
where 𝜂 ą 0 is a stabilization parameter. Then, for any 𝐾 P 𝒯 and any 𝜎 P ℱ𝐾, we define an intermediate flux operator ˆ𝐹𝐾,𝜎: 𝑋𝐾ˆ R ÞÝÑ R by setting:
ˆ
𝐹𝐾,𝜎p𝐵𝐾, 𝜌𝜎q “ ´|𝜎|𝜆𝐾∇𝐾,𝜎p𝐵𝐾, 𝜌𝜎q ¨ 𝑛𝐾,𝜎. Obviously, for any 𝜎 P ℱint, we would like to define ˆ𝑏𝜎 as the solution of
ÿ 𝐾P𝒯𝜎 ˆ 𝐹𝐾,𝜎 ´ 𝐵𝐾, ˆ𝑏𝜎 ¯ “ 0. (4.1)
Fortunately, as 𝜆 is positive almost everywhere we immediately deduce that such a ˆ𝑏𝜎 is always defined and is given by: ˆ 𝑏𝜎“ 1 ˜ ÿ 𝑀 P𝒯𝜎 𝜂𝜆𝑀 𝑑𝑀,𝜎 ¸ ˜ ÿ 𝐾P𝒯𝜎 𝜂𝜆𝐾 𝑑𝐾,𝜎 𝑏𝐾` ÿ 𝐾P𝒯𝜎 ∇𝐾p𝐵𝐾q ¨ ˆ 𝜂𝜆𝐾 𝑑𝐾,𝜎 p𝑥𝜎´ 𝑥𝐾q ´ 𝜆𝐾𝑛𝐾,𝜎 ˙¸ . (4.2)
where ˆ𝑏𝜎 is the unique solution of (4.1), while for any 𝜎 P ℱext, we simply set:
𝐹𝐾,𝜎p𝐵𝜎q “ ´|𝜎|𝜆𝐾∇𝐾p𝐵𝐾q ¨ 𝑛𝐾,𝜎. (4.4)
Remark 4.1. It is tempting to recover conservativity using the following simpler formula for ˆ𝐹𝐾,𝜎p𝐵𝜎q: ˆ
𝐹𝐾,𝜎p𝐵𝜎q “ 1
2p𝐹𝐾,𝜎p𝐵𝐾q ´ 𝐹𝐿,𝜎p𝐵𝐿qq .
The main drawback of this alternative formula can be understood in cases where 𝜆 is discontinuous, where this would obviously lead to a very coarse approximation of the flux along the lines of discontinuity of 𝜆. Provided those discontinuities are resolved by the mesh, our formula for ˆ𝐹𝐾,𝜎p𝐵𝜎q will instead behave as the usual harmonic mean. It is thus a more robust and versatile choice.
4.2. Consistency properties of the flux
Denoting 𝐵p𝑥, 𝜉q the ball centered at 𝑥 of radius 𝜉 and 𝐵Ωp𝑥, 𝜉q “ 𝐵p𝑥, 𝜉q X Ω, we recall the usual definition of strong consistency for gradient reconstruction operators:
Definition 4.2 (Strong consistency). The gradient reconstruction operator ∇𝐾 is strongly consistent if and only if there exists 𝐶 ą 0 independent on ℎ such that for any 𝜙 P 𝐶2´𝐵
Ωp𝑥𝐾, 𝜉q ¯ , every 0 ă 𝜉 ď 2ℎ𝐾 and every 𝑥 P 𝐵Ωp𝑥𝐾, 𝜉q: |∇𝜙p𝑥q ´ ∇𝐾p𝒟𝐾p𝜙qq| ď 𝐶𝜉||𝜙||𝑊2,8p𝐵 Ωp𝑥𝐾,𝜉qXΩq.
As an immediate consequence of the above definition, we have, using Taylor’s expansion and the density of 𝐶8´𝐵
Ωp𝑥𝐾, 𝜉q ¯
in 𝑊2,8p𝐵Ωp𝑥, 𝜉qq that for any 𝜙 P 𝑊2,8pΩq there exists 𝐶 ą 0 depending on 𝜙 but not on ℎ such that for every ℎ𝐾ď 𝜉 ď 2ℎ𝐾, any 𝐾 P 𝒯 and almost every 𝑥 P 𝐵Ωp𝑥𝐾, 𝜉q:
|∇𝜙p𝑥q ´ ∇𝐾p𝒟𝐾p𝜙qq| ď 𝐶𝜉||𝜙||𝑊2,8p𝐵 Ωp𝑥𝐾,𝜉qq
and
|𝜙p𝑥q ´ Π𝐾p𝒟𝐾p𝜙qq p𝑥q| ď 𝐶𝜉2||𝜙||𝑊2,8p𝐵Ωp𝑥𝐾,𝜉qq.
Another immediate consequence is the consistency of our discrete fluxes:
Proposition 4.3. Assume that assumption (A1) is satisfied and that the gradient reconstruction operator is strongly consistent. Then, there exists 𝐶 ą 0 independent on ℎ such that for any 𝐾 P 𝒯 , any 𝜎 P ℱ𝐾, any 𝑏 P 𝑊2,8pΩq and any 𝜆 P 𝑊1,8pΩq, we have:
ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 𝐿8p𝐵Ωp𝑥𝐾,2ℎ𝐾qq ď 𝐶ℎ𝐾||𝜆||𝑊1,8pΩq||𝑏||𝑊2,8pΩq. (4.5)
Proof. By density of 𝐶8`Ω˘ in 𝑊1,8pΩq and 𝑊2,8pΩq, it is clear that it suffices to establish the result for 𝜆 P 𝐶8`Ω˘ and 𝑏 P 𝐶8`Ω˘. Assuming such regularity, we start by establishing that there exists 𝐶 ą 0 independent on ℎ such that for any 𝜎 P ℱint:
|ˆ𝑏𝜎´ 𝑏 p𝑥𝜎q | ď 𝐶ℎ2𝐾||𝑏||𝑊2,8pΩq and |ˆ𝑏𝜎´ Π𝐾p𝐵𝐾q p𝑥𝜎q | ď 𝐶ℎ2𝐾||𝑏||𝑊2,8pΩq @𝐾 P 𝒯𝜎. (4.6)
Immediately, using the strong consistency of the gradient operator we get that for all 𝐾 P 𝒯𝜎: |Π𝐾p𝐵𝐾q p𝑥𝜎q ´ 𝑏 p𝑥𝜎q | ď 𝐶ℎ2𝐾||𝑏||𝑊2,8p𝐵
Ωp𝑥𝐾,ℎ𝐾qq
and thus for all 𝐾 P 𝒯𝜎:
1 ˜ ÿ 𝑀 P𝒯𝜎 𝜂𝜆𝑀 𝑑𝑀,𝜎 ¸ ÿ 𝐿P𝒯𝜎 𝜂𝜆𝐿 𝑑𝐿,𝜎 |Π𝐿p𝐵𝐿q p𝑥𝜎q ´ 𝑏 p𝑥𝜎q | ď 𝐶 ÿ 𝐿P𝒯𝜎 𝜂𝜆𝐿 𝑑𝐿,𝜎 ℎ2𝐿 ˜ ÿ 𝑀 P𝒯𝜎 𝜂𝜆𝑀 𝑑𝑀,𝜎 ¸ ||𝑏||𝑊2,8p𝐵 Ωp𝑥𝐾,ℎ𝐾qqď 𝐶ℎ 2 𝐾||𝑏||𝑊2,8pΩq
as under assumption (A1), we know (see [3,7]) that there exists 𝐶 ą 0 such that for any p𝐾, 𝑀 q P 𝒯2such that ℱ𝐾X ℱ𝑀 ‰ H, we have ℎ𝑀{ℎ𝐾 ď 𝐶. Moreover, as the gradient operator is strongly consistent, we know that:
|𝜆𝐾∇𝐾p𝐵𝐾q ´ 𝜆 p𝑥𝜎q ∇𝑏 p𝑥𝜎q | ď 𝐶ℎ ` ||𝜆||𝑊1,8pΩq||𝑏||𝑊1,8pΩq` ||𝜆||𝐿8pΩq||𝑏||𝑊2,8pΩq˘ . Using: ÿ 𝐾P𝒯𝜎 𝜆 p𝑥𝜎q ∇𝑏 p𝑥𝜎q ¨ 𝑛𝐾,𝜎“ 0 we get: ˇ ˇ ˇ ˇ ˇ ÿ 𝐾P𝒯𝜎 𝜆𝐾∇𝐾p𝐵𝐾q ¨ 𝑛𝐾,𝜎 ˇ ˇ ˇ ˇ ˇ ď 𝐶ℎ`||𝜆||𝑊1,8pΩq||𝑏||𝑊1,8pΩq` ||𝜆||𝐿8pΩq||𝑏||𝑊2,8pΩq˘ .
Meanwhile, we clearly get using again assumption (A1): 1 ˜ ÿ 𝑀 P𝒯𝜎 𝜂𝜆𝑀 𝑑𝑀,𝜎 ¸ ď 1 𝜂𝜆´ 1 ˜ ÿ 𝑀 P𝒯𝜎 1 𝑑𝑀,𝜎 ¸ ď 1 𝜂𝜆´card p𝒯𝜎q max 𝑀 P𝒯𝜎 𝑑𝑀,𝜎ď 1 𝜂𝜆´ max 𝑀 P𝒯𝜎 ℎ𝑀 ď 𝐶ℎ𝐾 𝜂𝜆´
which finally establishes that:
|ˆ𝑏𝜎´ 𝑏 p𝑥𝜎q | ď 𝐶ℎ2𝐾||𝑏||𝑊2,8pΩq
and (4.6) immediately follows using the triangular inequality and the strong consistency of the gradient operator. Then, for any 𝑥 P 𝐵Ωp𝑥𝐾, 2ℎ𝐾q and any 𝜎 P ℱint:
1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆p𝑥q∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎“ 𝜆𝐾p∇𝑏p𝑥q ´ ∇𝐾p𝐵𝐾qq ¨ 𝑛𝐾,𝜎` p𝜆p𝑥q ´ 𝜆𝐾q ∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎 ´𝜂𝜆𝐾 𝑑𝐾,𝜎 ´ ˆ 𝑏𝜎´ Π𝐾p𝐵𝐾q p𝑥𝜎q ¯ . Immediately, as the gradient reconstruction operator is strongly consistent we get:
|𝜆𝐾p∇𝑏p𝑥q ´ ∇𝐾p𝐵𝐾qq ¨ 𝑛𝐾,𝜎| ď 𝐶𝜆`ℎ𝐾||𝑏||𝑊2,8pΩq
while Taylor’s expansion immediately gives:
|p𝜆p𝑥q ´ 𝜆𝐾q ∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎| ď 𝐶ℎ𝐾||𝜆||𝑊1,8pΩq||𝑏||𝑊1,8.
Under assumption (A1), we have ℎ𝐾{𝑑𝐾,𝜎ď ℎ𝐾{𝜌𝐾 ď 1{𝜌 which combined with (4.6) gives the desired result for 𝜎 P ℱint. For 𝜎 P ℱext, we directly get:
1
5. A multiple flow direction algorithm on general meshes
5.1. The generalized MFD algorithm
Given an approximation p𝐵𝐾q𝐾P𝒯 of the topography and p𝑓𝐾q𝐾P𝒯 of the source term as data, and using both upwinding for the water height and our discrete fluxes, the most natural discrete formulation of the Manning–Strickler model consists in finding
´ p𝑢𝐾q𝐾P𝒯 , p𝑢𝜎q𝜎Pℱ𝒟 ext,in ¯ such that: ˇ ˇ ˇ ˇ ˇ ˇ ˇ ÿ 𝜎Pℱ𝐾 𝑢up𝜎 𝐹𝐾,𝜎p𝐵𝜎q “ |𝐾|𝑓𝐾 @𝐾 P 𝒯 𝑢𝜎“ 0 @𝜎 P ℱext,in𝒟 (5.1)
where the upwinding formula 𝑣up
𝜎 for any set of cell values p𝑣𝐾q𝐾P𝒯 is defined by:
𝑣𝜎up“ ˇ ˇ ˇ ˇ ˇ 𝑣𝐾 if 𝐹𝐾,𝜎p𝐵𝜎q ě 0 𝑣𝐿 if 𝐹𝐾,𝜎p𝐵𝜎q ă 0 @𝜎 P ℱint, 𝒯𝜎“ t𝐾, 𝐿u and 𝑣𝜎up“ ˇ ˇ ˇ ˇ ˇ 𝑣𝐾 if 𝐹𝐾,𝜎p𝐵𝜎q ě 0 0 if 𝐹𝐾,𝜎p𝐵𝜎q ă 0 @𝜎 P ℱext, 𝒯𝜎“ t𝐾u
and where we have defined the influx boundary associated with the discrete fluxes by setting ℱext,in𝒟 “ t𝜎 P ℱext| 𝐹𝐾,𝜎p𝐵𝜎q ă 0, 𝒯𝜎 “ t𝐾u u .
By construction of the fluxes, for any 𝜎 P ℱint we have that 𝐹𝐿,𝜎p𝐵𝜎q “ ´𝐹𝐾,𝜎p𝐵𝜎q. Mimicking what we have done for the TPFA scheme, we gather faces by upwinding kind and use the fact that 𝑢up
𝜎 “ 0 for all 𝜎 P ℱext,in𝒟 , leading to: ¨ ˝ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qě0 𝐹𝐾,𝜎p𝐵𝜎q ˛ ‚𝑢𝐾´ ÿ 𝜎Pℱ𝐾Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0 𝑢𝐿𝐹𝐿,𝜎p𝐵𝜎q “ |𝐾|𝑓𝐾. Defining: 𝑠𝐾“ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qě0 𝐹𝐾,𝜎p𝐵𝜎q and r𝑞𝐾 “ 𝑠𝐾𝑢𝐾
and noticing that 𝑠𝐿ą 0 as soon as there exists 𝜎 P ℱ𝐿such that 𝐹𝐿,𝜎p𝐵𝜎q ą 0, we see that the above equation can be rewritten: r 𝑞𝐾´ ÿ 𝜎Pℱ𝐾Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0 r 𝑞𝐿 𝑠𝐿 𝐹𝐿,𝜎p𝐵𝜎q “ |𝐾|𝑓𝐾.
For any 𝜎 P ℱext,in𝒟 , we set:
𝑠𝜎“ ´
ÿ
𝐾P𝒯𝜎,𝐹𝐾,𝜎p𝐵𝜎qă0
𝐹𝐾,𝜎p𝐵𝜎q and 𝑞r𝜎“ 𝑠𝜎𝑢𝜎.
Notice that the above system is set for the unknowns ´
p𝑞r𝐾q𝐾P𝒯 , p𝑞r𝜎q𝜎Pℱ𝒟 ext,in
¯
, thus it has the same number of unknowns than the original system (5.1). We still reconstruct a water flux vector cellwise by setting:
𝑄𝐾 “ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qą0 r 𝑞𝐾 |𝐾|𝑠𝐾 𝐹𝐾,𝜎p𝐵𝜎q p𝑥𝜎´ 𝑥𝐾q ` ÿ 𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0 r 𝑞𝐿 |𝐾|𝑠𝐿 𝐹𝐾,𝜎p𝐵𝜎q p𝑥𝜎´ 𝑥𝐾q (5.3)
the consistent water discharge being given by 𝑞𝐾 “ ||𝑄𝐾||. Finally, for any cell 𝐾 P 𝒯 , we set 𝑢𝐾 as we have done for the TPFA case:
𝑢𝐾 “ ˇ ˇ ˇ ˇ ˇ ˇ r 𝑞𝐾 𝑠𝐾 if 𝑠𝐾 ‰ 0 0 otherwise (5.4)
while for any 𝜎 P ℱext,in𝒟 , 𝑢𝜎“ 0. Notice that this new system allows to correctly define the numerical method, while the system (5.1) does not uniquely define the water height on cells where 𝑠𝐾 “ 0. Contrary to system (3.4), system (5.2) admits no obvious renumbering of mesh elements that makes the system triangular. However, as it is anyway necessary to enable parallelism, we say that the generalized MFD algorithm will consists in solving the linear system (5.2) using any linear solver of the literature. Notice that as we are using the unknownr𝑞 rather than 𝑢, the discrete system (5.2) takes the form of a perturbation of the identity matrix. Thus, we expect its condition number to be relatively good (depending of course of the coefficient 𝜆) and in any case much better than if had tried to solve (5.2) directly.
5.2. On existence and uniqueness for the generalized MFD algorithm
Existence and uniqueness of a solution to the above system are much less obvious than in the case of the TPFA scheme. To explain this fact, let us denote
𝒯˚
“ t𝐾 P 𝒯 | 𝑠𝐾 ą 0u .
Using the definition of 𝒯˚, summing (5.2) over 𝐾 P 𝒯 , after a straightforward manipulation of the last sum we get: ÿ 𝐾P𝒯 z𝒯˚ r 𝑞𝐾` ÿ 𝐾P𝒯˚ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qą0 r 𝑞𝐾 𝑠𝐾 𝐹𝐾,𝜎p𝐵𝜎q ´ ÿ 𝐿P𝒯˚ ÿ 𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0 r 𝑞𝐿 𝑠𝐿 𝐹𝐿,𝜎p𝐵𝜎q “ ÿ 𝐾P𝒯 |𝐾|𝑓𝐾 and consequently: ÿ 𝐾P𝒯 z𝒯˚ r 𝑞𝐾` ÿ 𝐾P𝒯˚ ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 r 𝑞𝐾 𝑠𝐾 𝐹𝐾,𝜎p𝐵𝜎q “ ÿ 𝐾P𝒯 |𝐾|𝑓𝐾. (5.5)
Thus, the existence of a solution for any second member p𝑓𝐾q𝐾P𝒯 requires that: 𝒜 “ p𝒯 z𝒯˚
q Y 𝒯˚˚‰ H where 𝒯˚˚
“ t𝐾 P 𝒯˚| t𝜎 P ℱ𝐾X ℱext| 𝐹𝐾,𝜎p𝐵𝜎q ą 0u ‰ Hu . (5.6) In the case of the TPFA scheme, condition (5.6) is necessarily satisfied. If condition (5.6) is not satisfied, then (5.2) can have a solution only if
ÿ
𝐾P𝒯
|𝐾|𝑓𝐾 “ 0.
discrete level, that is to say the flux family p𝐹𝐾,𝜎p𝐵𝜎qq𝐾P𝒯 ,𝜎Pℱ𝐾. Without any further assumption on the mesh, far from the asymptotic regime consistency alone cannot be expected to ensure well-posedness for coarse meshes. This means that well-posedness will in general be controlled by the structure of the discrete flux family rather than by the properties of 𝑏. As in practice the Laplacian of 𝑏 cannot be computed in general, it is anyway better to have an existence result that uses only computable quantities. In fact, not only the necessary condition (5.6) must be satisfied, but it also requires that there does not exist what we will call a flux cycle. For any subset 𝒞 of 𝒯 , we denote:
ℱint𝒞 “ t𝜎 P ℱint| 𝒯𝜎 Ă 𝒞u .
We say that a set of cells 𝒞 Ă 𝒯 form a flux cycle if and only if for any 𝐾 P 𝒞 𝐹𝐾,𝜎p𝐵𝜎q ď 0 @𝜎 P ℱ𝐾z`ℱ𝐾X ℱint𝒞 ˘ and ÿ 𝜎Pℱ𝐾Xℱint𝒞 ,𝐹𝐾,𝜎p𝐵𝜎qą0 𝐹𝐾,𝜎p𝐵𝜎q ą 0 and ÿ 𝜎Pℱ𝐾Xℱint𝒞 ,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q ă 0.
The first condition implies that water can enter the cycle but not leave it, while the second condition implies that water will run through the cycle without stopping anywhere. Whether such configurations can exist on coarse meshes for consistent flux families is a difficult question. However, from the physical point of view flux cycles are completely unrealistic as they represent regions where water will at some point go from a topographically low region to a topographically higher one. Moreover, under the positivity hypothesis on the Laplacian of the topography, we know that for the exact fluxes we have:
ÿ
𝜎Pℱ𝐾
ż
𝜎
´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎ą 0
and thus no flux cycle can exist. Thus, the presence of flux cycles should be considered as anomalous and a sign that a too coarse mesh has been used. A natural sufficient condition of existence and uniqueness for (5.2) should then be one that is satisfied by the continuous fluxes and immediately implies that no flux cycle exist. Proposition 5.1. We say that there exists a flowing path from 𝐾 P 𝒯 z𝒜 to 𝒜 if there exists r𝐾 P 𝒜 such that
D 𝑛𝐾 P N, 𝑛𝐾ą 0 and p𝜎𝑖q0ď𝑖ď𝑛𝐾´1 such that 𝜎𝑖P ℱint @0 ď 𝑖 ď 𝑛𝐾´ 1
and 𝒯𝜎𝑖 “ t𝐾𝑖, 𝐾𝑖`1u @0 ď 𝑖 ď 𝑛𝐾´ 1 where 𝐾 “ 𝐾0 𝐾 “ 𝐾r 𝑛𝐾 and 𝐹𝐾𝑖,𝜎𝑖p𝐵𝜎𝑖q ą 0.
If 𝒜 ‰ H and for any cell 𝐾 P 𝒯 z𝒜 there exists a flowing path to 𝒜, then (5.2) is well-posed.
Proof. As system (5.2) is linear, it suffices to establish uniqueness to ensure the existence of solutions. Thus, assume that 𝑞 “r ´p𝑞r𝐾q𝐾P𝒯 , pr𝑞𝜎q𝜎Pℱ𝒟
ext,in
¯
is solution of (5.2) with zero right-hand side. Then, taking the modulus of (5.2) and summing over 𝐾 P 𝒯 we get:
Injecting this into (5.7), we obtain: ÿ 𝐾P𝒯 z𝒯˚ |𝑞r𝐾| ` ÿ 𝐾P𝒯˚ ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 |𝑞r𝐾| 𝑠𝐾 𝐹𝐾,𝜎p𝐵𝜎q ď 0
which immediately implies that r𝑞𝐾 “ 0 for any 𝐾 P 𝒜. Next, let us denote 𝒯0“ 𝒯 and 𝒜0“ 𝒜, and define 𝒯𝑖 for any 𝑖 ě 1 by setting:
𝒯𝑖“ 𝒯 z ď 0ď𝑘ď𝑖´1 𝒜𝑘 where 𝒜𝑖“ p𝒯𝑖z𝒯𝑖˚q Y 𝒯𝑖˚˚ with 𝒯˚ 𝑖 “ t𝐾 P 𝒯𝑖 | 𝑠𝐾 ą 0u and 𝒯𝑖˚˚“ 𝐾 P 𝒯𝑖˚| 𝜎 P ℱ𝐾X ℱext𝑖 | 𝐹𝐾,𝜎p𝐵𝜎q ą 0 ( ‰ H( and for any 𝑖 ě 1:
ℱ𝑖“ t𝜎 P ℱ | 𝒯𝜎X 𝒯𝑖‰ Hu and ℱint𝑖 “ t𝜎 P ℱ𝑖X ℱint | 𝒯𝜎Ă 𝒯𝑖u and ℱext𝑖 “ ℱ𝑖zℱint𝑖 .
Clearly, as 𝒜 ‰ H we have 𝒯1Ĺ 𝒯0or 𝒯1“ H. We now proceed by induction. Assume that for 𝑛 ě 1, we have established that r 𝑞𝐾 “ 0 for all 𝐾 P ď 0ď𝑘ď𝑛´1 𝒜𝑘 and 𝒯𝑖Ĺ 𝒯𝑖´1 or 𝒯𝑖 “ H @1 ď 𝑖 ď 𝑛 ´ 1.
If 𝒯𝑛 “ H, then there is nothing to prove. Otherwise, as for any 𝐾 P 𝒯𝑛, there exists a flow path to 𝒜, then 𝒯˚˚
𝑛 ‰ H. Thus, summing over 𝒯𝑛 and proceeding as above we obtain that: ÿ 𝐾P𝒯𝑛z𝒯𝑛˚ |r𝑞𝐾| ` ÿ 𝐾P𝒯𝑛˚ ÿ 𝜎Pℱ𝐾Xℱext𝑛 ,𝐹𝐾,𝜎p𝐵𝜎qą0 |𝑞r𝐾| 𝑠𝐾 𝐹𝐾,𝜎p𝐵𝜎q ď 0.
Thus, we obtain that𝑞r𝐾 “ 0 for any 𝐾 P 𝒜𝑛 and 𝒜𝑛‰ H which implies 𝒯𝑛`1Ĺ 𝒯𝑛. Then, it is clear that the sequence p𝒯𝑖q𝑖ě0 is strictly decreasing in the sense that it satisfies 𝒯𝑖Ĺ 𝒯𝑖´1 or 𝒯𝑖“ H because of the flow path assumption. Thus, there exists 𝑛𝒯 ą 0 such that 𝒯𝑛𝒯 “ H and thusr𝑞 “ 0, which concludes the proof.
6. Convergence analysis
The main idea underlying our convergence analysis is that for all consistent fluxes, if the topography 𝑏 is regular enough numerical fluxes will converge to the continuous ones. The major originality of the present analysis regarding for instance finite volumes or discontinuous Galerkin methods for steady-advection diffusion problems of the form (3.2) is that here the coefficients 𝛽 and 𝜇 are approximated through discrete differential operators applied to the topography 𝑏. Thus, this additional approximation has to be handled carefully to recover the correct convergence estimates.
To establish precise error estimates we first need to choose a discrete norm. To this end, we define 𝜔𝐾,𝜎“ 1 2 ` |𝐹𝐾,𝜎p𝐵𝜎q | ` |𝜎| ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝐵p𝑥𝐾,2ℎ𝐾qq ˘
and set for 𝑣 “ p𝑣𝐾q𝐾P𝒯: ||𝑣||2ℎ“ 𝛼𝜆´ 2 ÿ 𝐾P𝒯 |𝐾|𝑣2𝐾` 1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝜔𝐾,𝜎|𝑣𝐾|2` 1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝜔𝐾,𝜎p𝑣𝐾´ 𝑣up𝜎 q 2.
when the continuous fluxesş𝜎𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 cancel creating technical difficulties, which 𝜔𝐾,𝜎avoids. Immediately, we get that: |𝜔𝐾,𝜎´ |𝐹𝐾,𝜎p𝐵𝜎q | | “ 1 2 ˇ ˇ|𝐹𝐾,𝜎p𝐵𝜎q ´ |𝜎| ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝐵p𝑥𝐾,2ℎ𝐾qq ˇ ˇ and thus: |𝜔𝐾,𝜎´ |𝐹𝐾,𝜎p𝐵𝜎q | | ď 1 2𝑅𝐾,𝜎 (6.1)
where for any 𝐾 P 𝒯 and any 𝜎 P ℱ𝐾 we have denoted: 𝑅𝐾,𝜎“ max ˆˇ ˇ ˇ ˇ 𝐹𝐾,𝜎p𝐵𝜎q ` ż 𝜎 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˇ ˇ ˇ ˇ ,ˇˇ|𝐹𝐾,𝜎p𝐵𝜎q | ´ |𝜎| ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝐵p𝑥 𝐾,2ℎ𝐾qq ˇ ˇ ˙ . (6.2)
Notice that from (4.5), we get that there exists 𝐶 ą 0 such that for any 𝐾 P 𝒯 and any 𝜎 P ℱ𝐾, we have: 𝑅𝐾,𝜎ď 𝐶|𝜎|ℎ𝐾||𝜆||𝑊1,8pΩq||𝑏||𝑊2,8pΩq. (6.3)
The main objective of this section is to establish the following explicit 𝐿2 error estimate:
Theorem 6.1. Assume that the gradient operator underlying the fluxes is consistent. Further assume that the mesh satisfies assumption (A1), that 𝜆 P 𝑊1,8pΩq and that the solution 𝑢 of (3.1) belongs to 𝐻1
pΩq. Let 𝛾pℎq “ max ˜ sup 𝐾P𝒯 ,𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝑅𝐾,𝜎 𝜔𝐾,𝜎 , sup 𝐾P𝒯 ,𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎q‰0 𝑅𝐾,𝜎 𝜔𝐾,𝜎 ¸
with 𝑅𝐾,𝜎 defined by (6.2). Then, there exists ℐℎp𝑢q P 𝐿2pΩq such that for any 𝐾 P 𝒯 , ℐℎp𝑢q|𝐾 “ ℐ𝐾p𝑢q is constant on 𝐾 and:
|𝑢 ´ ℐ𝐾p𝑢q|2𝐿2p𝐾q` ℎ
1{2
𝐾 |𝑢 ´ ℐ𝐾p𝑢q|2𝐿2pB𝐾qď 𝐶polyℎ𝐾|𝑢|𝐻1p𝐾q.
Moreover if 𝛾pℎq Ñ 0 when ℎ Ñ 0, then there exists ℎ0 ą 0 and 𝐶 ą 0 depending on 𝑢, 𝜆, 𝑏 and the mesh parameters such that for any ℎ ď ℎ0:
||𝑢 ´ ℐℎp𝑢q||ℎď 𝐶 maxpℎ, 𝛾pℎqq1{2||𝑢||𝐻1pΩq (6.4)
where 𝑢 is reconstructed by (5.4) from the solution of (5.2).
Notice that the existence of ℐℎp𝑢q in the above result is an immediate consequence of assumption (A1) and (2.3). As a corollary, we will then be able to establish an explicit error estimate for the water discharge: Corollary 6.2. Under the assumptions of Proposition 6.1, there exists ℎ0 ą 0 and 𝐶 ą 0 depending on 𝑢, 𝜆, 𝑏 and the mesh parameters such that for any ℎ ď ℎ0:
˜ ÿ 𝐾P𝒯 || ´ 𝜆𝑢∇𝑏 ´ 𝑄𝐾||2𝐿2p𝐾q2 ¸1{2 ď 𝐶 maxpℎ, 𝛾pℎqq1{2p1 ` maxpℎ, 𝛾pℎqq1{2q||𝑢||𝐻1pΩq. (6.5)
The error estimates use the surprising factor 𝛾pℎq instead of ℎ as one could legitimately expect. This comes from the fact that we use approximate fluxes, which is equivalent in some sense to using a quadrature formula for the coefficients of (3.2). As the natural norm for the discrete problem uses those approximate fluxes, it seems unfortunately unavoidable that this impacts the error estimate. In the case where for any 𝐾 P 𝒯 and any 𝜎 P ℱ𝐾 that is indeed present in the norm || ¨ ||ℎ, the 𝜔𝐾,𝜎’s are bounded by below by a constant independent on ℎ, which will happen most of the time in practice, then 𝛾pℎq will behave as ℎ and we retrieve a convergence at rate ℎ1{2. Controlling the lower bound for 𝜔
more than 𝐾 in the definition of 𝜔𝐾,𝜎. If this supremum is zero, most gradient reconstruction operators and in particular those described in our numerical experiments will provide a discrete gradient that aligns with the continuous one and the discrete flux will be zero too, avoiding most of the problematic cases for the asymptotic behavior of 𝜔𝐾,𝜎. In the case of exact fluxes it is known that for the water height 𝑢 the optimal convergence rate to 𝑢 is ℎ1{2, even if superconvergence at rate ℎ is very often observed in practice, as revealed by [6,19] (see also [7]). Thus, as additional flux consistency terms could only decrease the convergence rate, it seems clear that our estimate for the water height cannot be expected to be improved in terms of order of convergence.
The proof of Theorem6.1being rather lengthy, we will try to decompose it as much as possible. Let us notice that in the special case where ∇𝑏 is constant, the discrete flux are exact. Then our problem corresponds to a piecewise constant version of the discontinuous Galerkin method described in [20], and we can in principle follow the steps of their proof. Denoting ||𝑣||2
ℎ “ ř
𝐾P𝒯 ||𝑣|| 2
𝐾,ℎ with obvious notations, the first step of their proof is to establish a local error identity for ||𝑢𝐾´ 𝑣𝐾||𝐾,ℎ for any 𝑣𝐾 in R, using the exactness of the flux. Then, taking 𝑣𝐾 “ ℐ𝐾p𝑢q and summing over 𝐾 P 𝒯 , they obtain a global error estimate where the residual terms only involve 𝑢 ´ ℐp𝑢q, which can be straightforwardly estimated using polynomial approximation results. The general guideline of our proof is the same, however we have to endure some additional technicalities due to the non exactness of the flux. We first start by establishing the equivalent of the local error identity of [20], but keeping our approximate flux in the result. Thus, an additional residual term standing for the consistency error of the flux will appear on the right hand side of our identity. Then, to match the definition of the || ¨ ||ℎ norm, each time our estimates will involve the sum of the flux over the full set ℱ𝐾, we will replace it by the Laplacian of 𝑏, and put the difference in a residual term. For an isolated flux term, we will do the same using this time either 𝜔𝐾,𝜎if the term is involved in the || ¨ ||ℎnorm or by the continuous flux if it is directly a residual term. Thus, when summing the local error identities over 𝐾 P 𝒯 to obtain the global error estimate, residual terms involving the discrete counterpart of the Laplacian will be handled through a technical result described in the following subsection, while other residual terms will be estimated directly using the strong consistency of the flux. The reason for doing so is to obtain an error estimate involving only the || ¨ ||ℎ norm on the left-hand side and residual terms involving either 𝑢 ´ ℐp𝑢q or flux consistency error terms on the right-hand side. Finally, each residual term will be estimated using polynomial approximation results and the consistency of the flux.
6.1. Local error identity and approximation results for the Laplacian
Let us now establish an abstract local error identity, following the lines of [20].
Lemma 6.3. For any 𝐾 P 𝒯 , any 𝑣𝐾 P R and any 𝜂 “ p𝜂𝜎q𝜎Pℱ𝐾 P 𝐿2pB𝐾q constant on each 𝜎 P ℱ𝐾, we have: 1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qą0 𝐹𝐾,𝜎p𝐵𝜎q p𝑢𝐾´ 𝑣𝐾q 2 `1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑢up𝜎 ´ 𝜂𝜎q 2 ´1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q pp𝑢𝐾´ 𝑣𝐾q ´ p𝑢up𝜎 ´ 𝜂𝜎qq 2 `1 2 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q p𝑢𝐾´ 𝑣𝐾q 2 “ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qą0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ 𝑣𝐾q p𝑢𝐾´ 𝑣𝐾q ` ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ 𝜂𝜎q p𝑢𝐾´ 𝑣𝐾q ´ ÿ 𝜎Pℱ𝐾 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢 p𝑢𝐾´ 𝑣𝐾q (6.6)
Proof. This proof is a direct adaptation of its counterpart in [20]. For any 𝑤𝐾 P R and any 𝜉 “ p𝜉𝜎q𝜎Pℱ𝐾 P 𝐿2pB𝐾q constant on each 𝜎 P ℱ𝐾, consider:
ℐ𝐾“ ´ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑤𝐾´ 𝜉𝜎q 𝑤𝐾` ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑤2𝐾.
Using the identity 𝑎p𝑎 ´ 𝑏q “ 1 2`𝑎 2 ´ 𝑏2` p𝑎 ´ 𝑏q2˘, we obtain: ℐ𝐾 “ ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑤𝐾2 ´ 1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q 𝑤2𝐾` 1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q 𝜉𝜎2 ´1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑤𝐾´ 𝜉𝜎q 2 and thus: ℐ𝐾 “ 1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qą0 𝐹𝐾,𝜎p𝐵𝜎q 𝑤2𝐾` 1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q 𝜉𝜎2 ´1 2 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑤𝐾´ 𝜉𝜎q 2 `1 2 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑤𝐾2.
On the other hand, from (5.1) multiplying by 𝑤𝐾 and using the definition of 𝑢up𝜎 we have by construction:
´ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑢𝐾´ 𝑢up𝜎 q 𝑤𝐾` ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑢𝐾𝑤𝐾“ ż 𝐾 𝑓 𝑤𝐾. Let us now set 𝑤𝐾 “ 𝑢𝐾´ 𝑣𝐾 and 𝜉𝜎“ 𝑢up𝜎 ´ 𝜂𝜎 for all 𝜎 P ℱ𝐾. We obtain:
ℐ𝐾 “ ´ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝜂𝜎´ 𝑣𝐾q 𝑤𝐾` ż 𝐾 𝑓 𝑤𝐾´ ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑣𝐾𝑤𝐾.
Using ´div p𝑢𝜆∇𝑏q “ 𝑓 , this leads to: ℐ𝐾 “ ´ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝜂𝜎´ 𝑣𝐾q 𝑤𝐾` ż 𝐾 ´div p𝑢𝜆∇𝑏q 𝑤𝐾´ ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑣𝐾𝑤𝐾. Integrating by parts, we get:
ż 𝐾 ´div p𝑢𝜆∇𝑏q 𝑤𝐾“ ´ ÿ 𝜎Pℱ𝐾 ż 𝜎 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎𝑢𝑤𝐾 “ ÿ 𝜎Pℱ𝐾 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q 𝑢𝑤𝐾´ ÿ 𝜎Pℱ𝐾 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢𝑤𝐾. Then: ℐ𝐾“ ´ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝜂𝜎´ 𝑣𝐾q 𝑤𝐾` ÿ 𝜎Pℱ𝐾 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ 𝑣𝐾q 𝑤𝐾 ´ ÿ 𝜎Pℱ𝐾 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢𝑤𝐾 “ ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ 𝜂𝜎q 𝑤𝐾` ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qą0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ 𝑣𝐾q 𝑤𝐾 ´ ÿ 𝜎Pℱ𝐾 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢𝑤𝐾
In the above local error identity, one can see that the left hand side strongly looks like the 𝐾-term of the || ¨ ||ℎ norm. However, to match the || ¨ ||ℎ norm, we need to replace the sum of the fluxes over ℱ𝐾 by the Laplacian of 𝑏, as well as isolated flux by 𝜔𝐾,𝜎. If this last operation straightforwardly leads to a residual term because of (6.1), however the following lemma precise what kind of link we can expect between the discrete Laplacian and the true Laplacian:
Lemma 6.4. For any p𝑤𝐾q𝐾P𝒯, one has: ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑤𝐾“ ´ ÿ 𝐾P𝒯 ż 𝐾 𝜆∆𝑏𝑤𝐾` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ ˆ p𝑤𝐾´ 𝑤𝜎upq ` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑤𝐾.
Proof. Let us first notice that: ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑤up𝜎 “ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝐹𝐾,𝜎p𝐵𝜎q 𝑤up𝜎 .
Indeed, using 𝐹𝐾,𝜎p𝐵𝜎q ` 𝐹𝐿,𝜎p𝐵𝜎q “ 0 for any 𝜎 P ℱint, 𝒯𝜎 “ t𝐾, 𝐿u we have ÿ
𝐾P𝒯 ÿ
𝜎Pℱ𝐾Xℱint
𝐹𝐾,𝜎p𝐵𝜎q 𝑤up𝜎 “ 0
while for 𝜎 P ℱext such that 𝐹𝐾,𝜎p𝐵𝜎q ą 0, 𝒯𝜎“ t𝐾u, 𝑤up𝜎 “ 0 immediately implies that: ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q 𝑤up𝜎 “ 0. Thus, we have: ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑤𝐾 “ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑤𝐾´ 𝑤𝜎upq ` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝐹𝐾,𝜎p𝐵𝜎q 𝑤𝐾
using the definition of 𝑤up
𝜎 . In the same way: ÿ 𝐾P𝒯 ż 𝐾 ´𝜆∆𝑏𝑤𝐾“ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 ż 𝜎 ´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎𝑤𝐾 “ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 ż 𝜎 ´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎p𝑤𝐾´ 𝑤𝜎upq ` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 ż 𝜎 ´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎𝑤𝐾.
Combining those expressions gives the expected result.
6.2. Global error estimate
Lemma 6.5. Using the hypothesis and notations of Proposition6.1, we have: ||𝑢 ´ ℐℎp𝑢q||2ℎď ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ ℐ𝜎upp𝑢qq pp𝑢 up 𝜎 ´ ℐ up 𝜎 p𝑢qq ´ p𝑢𝐾´ ℐ𝐾p𝑢qqq ` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ ℐ𝐾p𝑢qq p𝑢𝐾´ ℐ𝐾p𝑢qq ` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ ℐ𝜎upp𝑢qq pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq ´ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢 p𝑢𝐾´ ℐ𝐾p𝑢qq ´1 2 ˜ ÿ 𝐾P𝒯 ż 𝐾 𝜆∆𝑏|𝑢𝐾´ ℐ𝐾p𝑢q|2` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q |𝑢𝐾´ ℐ𝐾p𝑢q|2 ¸ ´1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 p|𝐹𝐾,𝜎p𝐵𝜎q | ´ 𝜔𝐾,𝜎q pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq 2 ´1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 p|𝐹𝐾,𝜎p𝐵𝜎q | ´ 𝜔𝐾,𝜎q |𝑢𝐾´ ℐ𝐾p𝑢q|2.
Proof. We start from (6.6) with 𝑣𝐾 “ ℐ𝐾p𝑢q and 𝜂𝜎 “ ℐ𝜎upp𝑢q. Summing over 𝐾 P 𝒯 the second term of the left hand side of (6.6) leads to:
1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qq 2 “ ´1 2 ÿ 𝐿P𝒯 ÿ 𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0 𝐹𝐿,𝜎p𝐵𝜎q p𝑢𝐿´ ℐ𝐿p𝑢qq 2 `1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q p𝑢𝜎´ ℐ𝜎p𝑢qq 2 “ ´1 2 ÿ 𝐿P𝒯 ÿ 𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0 𝐹𝐿,𝜎p𝐵𝜎q p𝑢𝐿´ ℐ𝐿p𝑢qq 2
Consequently, summing over 𝐾 P 𝒯 the entire left hand side of (6.6) leads to: ÿ 𝐾P𝒯 ℐ𝐾 “ 1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q |𝑢𝐾´ ℐ𝐾p𝑢q|2 ´1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝐹𝐾,𝜎p𝐵𝜎q pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq 2 `1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝐹𝐾,𝜎p𝐵𝜎q |𝑢𝐾´ ℐ𝐾p𝑢q|2“ 𝐹1` 𝐹2` 𝐹3.
The first term of the above expression rewrites: 𝐹1“ 1 2 ÿ 𝐾P𝒯 ż 𝐾 ´𝜆∆𝑏|𝑢𝐾´ ℐ𝐾p𝑢q|2 `1 2 ˜ ÿ 𝐾P𝒯 ż 𝐾 𝜆∆𝑏|𝑢𝐾´ ℐ𝐾p𝑢q|2` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q |𝑢𝐾´ ℐ𝐾p𝑢q|2 ¸ “ 𝐿0` 𝐿1
while the second term and third term give: 𝐹2` 𝐹3“ 1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝜔𝐾,𝜎pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq 2 `1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝜔𝐾,𝜎|𝑢𝐾´ ℐ𝐾p𝑢q|2 `1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 p|𝐹𝐾,𝜎p𝐵𝜎q | ´ 𝜔𝐾,𝜎q pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ𝜎upp𝑢qqq 2 `1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 p|𝐹𝐾,𝜎p𝐵𝜎q | ´ 𝜔𝐾,𝜎q |𝑢𝐾´ ℐ𝐾p𝑢q|2“ 𝐿2` 𝐿3` 𝐿4` 𝐿5
with obvious notations. Immediately, using the hypothesis on 𝜆 and 𝑏 we have: 𝐿0ě 𝛼𝜆´ 2 ÿ 𝐾P𝒯 |𝐾| p𝑢𝐾´ ℐ𝐾p𝑢qq 2 .
Then, noticing that the second member of the sum over 𝐾 P 𝒯 of (6.6) can be decomposed into three terms 𝑇1` 𝑇2` 𝑇3 with obvious notations and combining the above results, we have that
ř
𝐾P𝒯 ℐ𝐾 “ 𝑇1` 𝑇2` 𝑇3 is equivalent to ||𝑢 ´ ℐℎp𝑢q||2ℎ ď 𝑇1` 𝑇2` 𝑇3´ 𝐿1´ 𝐿4´ 𝐿5 as 𝐿0` 𝐿2` 𝐿3 exactly gives the square of the || ¨ ||ℎ norm. Using the fact that both 𝑢𝜎 and ℐ𝜎p𝑢q cancels for 𝜎 P ℱext,in𝒟 , we obtain:
𝑇1` 𝑇2“ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ ℐ𝜎upp𝑢qq pp𝑢 up 𝜎 ´ ℐ up 𝜎 p𝑢qq ´ p𝑢𝐾´ ℐ𝐾p𝑢qqq ` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ ℐ𝐾p𝑢qq p𝑢𝐾´ ℐ𝐾p𝑢qq ` ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎| ż 𝜎 𝐹𝐾,𝜎p𝐵𝜎q p𝑢 ´ ℐ𝜎upp𝑢qq pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq
6.3. Estimation of the residual terms
Establishing Proposition 6.1 now just consists in estimating each term appearing in the second member of the above global estimate. From Lemma6.5, we see that ||𝑢 ´ ℐℎp𝑢q||2ℎď
ř7
𝑖“1𝑇𝑖 with obvious notations. The first three terms account for the polynomial approximation error, while the four other terms account for the consistency error of the flux. We now estimate those two families of residual terms separately.
Proof of Proposition6.1: terms accounting for polynomial approximation error. Using Cauchy–Schwarz inequality and the fact that p𝑢up
𝜎 ´ ℐ𝜎upp𝑢qq ´ p𝑢𝐾´ ℐ𝐾p𝑢qq is constant on 𝜎 gives: |𝑇1|2ď ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0 ˆż 𝜎 1 |𝜎||𝐹𝐾,𝜎p𝐵𝜎q | ||𝑢 ´ ℐ up 𝜎 p𝑢q| 2 ˙ ||𝑢 ´ ℐℎp𝑢q||2ℎ.
As the gradient operator underlying the fluxes is consistent, we know from Proposition 4.5 that there exists 𝐶 ą 0 such that: ˇ ˇ ˇ ˇ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ˇ ˇ ˇ ˇ ď || 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ´ 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝜎q` ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝜎q ď 𝐶ℎ||𝜆||𝑊1,8pΩq||𝑏||𝑊2,8pΩq` ||𝜆||𝐿8pΩq||𝑏||𝑊2,8pΩq.
As soon as ℎ ď ℎ0 for some fixed ℎ0 this leads to the existence of some 𝐶 ą 0 independent on ℎ such that |𝑇1|2ď 𝐶||𝜆||𝑊1,8pΩq||𝑏||𝑊2,8 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0 ˆż 𝜎 |𝑢 ´ ℐ𝜎upp𝑢q| 2 ˙ ||𝑢 ´ ℐℎp𝑢q||2ℎ.
Then, using the definition of ℐℎp𝑢q, we get ż 𝜎 |𝑢 ´ ℐ𝜎upp𝑢q| 2 ď 𝐶ℎ𝐾𝜎up|𝑢| 2 𝐻1p𝐾up 𝜎 q
where, if 𝜎 P ℱint, 𝐾𝜎up “ 𝐾 if 𝐹𝐾,𝜎p𝐵𝜎q ě 0 and 𝐾𝜎up “ 𝐿 if 𝐹𝐾,𝜎p𝐵𝜎q ă 0, and 𝐾𝜎up “ 𝐾 if 𝜎 P ℱext. From [7], we know that assumption (A1) implies that card pℱ𝐾q is bounded by a constant 𝜌ℱ depending on the mesh parameters but independent on ℎ, and thus there exists 𝐶 ą 0 such that |𝑇1|2ď 2ℎ𝐶||𝜆||𝑊1,8pΩq||𝑏||𝑊2,8|𝑢|2𝐻1pΩq||𝑢 ´ ℐℎp𝑢q||2ℎ. For 𝑇2, obviously we have using Cauchy–Schwarz inequal-ity: |𝑇2|2ď ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 ˆż 𝜎 1 |𝜎||𝐹𝐾,𝜎p𝐵𝜎q | ||𝑢 ´ ℐ𝐾p𝑢q| 2 ˙ ||𝑢 ´ ℐℎp𝑢q||2ℎ
and thus, proceeding the same way than for 𝑇1, we obtain |𝑇2|2ď 2ℎ𝐶||𝜆||𝑊1,8pΩq||𝑏||𝑊2,8|𝑢|2𝐻1pΩq||𝑢 ´ ℐℎp𝑢q||2ℎ. The situation is more delicate for 𝑇3. Indeed, ℐ𝜎upp𝑢q “ 0 for 𝜎 P ℱext,in𝒟 , while 𝑢 “ 0 on 𝜎 P ℱext,in, thus ℐ𝜎upp𝑢q is not directly a good approximation of 𝑢. For any 𝜎 P ℱext,in𝒟 X ℱext,in, ℐ𝜎upp𝑢q “ 𝑢 “ 0 and thus the contribution to 𝑇3 is zero. Then, it just remains to consider the case where 𝜎 P ℱext,in𝒟 and 𝜎 R ℱext,in. In this case we have ´ş𝜎𝜆∇𝑏 ¨ 𝑛𝐾,𝜎ě 0 while 𝐹𝐾,𝜎p𝐵𝜎q ă 0. However, by definition of the residual
ˇ ˇ𝐹𝐾,𝜎p𝐵𝜎q ` ş 𝜎𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˇ ˇď 𝑅𝐾,𝜎 which immediately implies that
ˇ ˇ ş
𝜎𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˇ
ˇ ď 𝑅𝐾,𝜎 and |𝐹𝐾,𝜎p𝐵𝜎q| ď 𝑅𝐾,𝜎. Consequently, we get using Cauchy–Schwarz inequality:
Using the definition of 𝛾pℎq and the trace inequality on Ω, this leads to: 𝑇3ď 𝛾pℎq1{2 ¨ ˝ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎|𝑅𝐾,𝜎 ż 𝜎 𝑢2 ˛ ‚ 1{2 ||𝑢 ´ ℐℎp𝑢q||ℎď 𝐶ℎ1{2𝛾pℎq1{2||𝑢||𝐻1pΩq||𝑢 ´ ℐℎp𝑢q||ℎ. It finally remains to estimate the terms accounting for the consistency error on the fluxes.
Proof of Proposition6.1: terms accounting for flux consistency error. For 𝑇4, let us first remark that: 𝑇4“ ´ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱint ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢 pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq ´ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢 p𝑢𝐾´ ℐ𝐾p𝑢qq .
Using the fact that both 𝑢𝜎 and ℐ𝜎p𝑢q cancel for 𝜎 P ℱ𝐾X ℱext and 𝐹𝐾,𝜎p𝐵𝜎q ă 0, this rewrites: 𝑇4“ ´ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢 pp𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq ´ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 ż 𝜎 ˆ 1 |𝜎|𝐹𝐾,𝜎p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ˙ 𝑢 p𝑢𝐾´ ℐ𝐾p𝑢qq “ 𝑇4,1` 𝑇4,2. Cauchy–Schwarz inequality then leads to:
|𝑇4,1| ď ¨ ˝ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 1 |𝜎|𝑅𝐾,𝜎 ż 𝜎 𝑢2 ˛ ‚ 1{2 ˆ ¨ ˝ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝑅𝐾,𝜎 ´ p𝑢𝐾´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qq 2¯ ˛ ‚ 1{2 .
Using (6.3), (2.2) and the definition of 𝛾pℎq, we easily get:
|𝑇4,1| ď 𝐶𝛾pℎq1{2 ¨ ˝ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 ℎ𝐾 ż 𝜎 𝑢2 ˛ ‚ 1{2 ||𝑢 ´ ℐℎp𝑢q||ℎď 𝐶𝛾pℎq1{2||𝑢||𝐻1pΩq||𝑢 ´ ℐℎp𝑢q||ℎ.
In the same way, Cauchy–Schwarz inequality and this time the trace inequality on Ω directly gives:
|𝑇4,2| ď 𝐶𝛾pℎq1{2 ¨ ˝ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 ℎ𝐾 ż 𝜎 𝑢2 ˛ ‚ 1{2 ||𝑢 ´ ℐℎp𝑢q||ℎď 𝐶𝛾pℎq1{2ℎ1{2||𝑢||𝐻1pΩq||𝑢 ´ ℐℎp𝑢q||ℎ.
Using p𝑎2
´ 𝑏2q “ p𝑎 ´ 𝑏qp𝑎 ` 𝑏q and Cauchy–Schwarz inequality, we obtain:
|𝑇5| ď 𝛾pℎq1{2||𝑢 ´ ℐℎp𝑢q||ℎ ¨ ˝ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝑅𝐾,𝜎pp𝑢𝐾´ ℐ𝐾p𝑢qq ` p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qqq 2 ˛ ‚ 1{2 ` 𝛾pℎq1{2||𝑢 ´ ℐℎp𝑢q||ℎ ¨ ˝ ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝑅𝐾,𝜎p𝑢𝐾´ ℐ𝐾p𝑢qq 2 ˛ ‚ 1{2 .
Using (6.3) and assumption (A1), we immediately obtain that: ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝑅𝐾,𝜎p𝑢𝐾´ ℐ𝐾p𝑢qq 2 ď 𝐶𝜌ℱ ÿ 𝐾P𝒯 |𝐾| p𝑢𝐾´ ℐ𝐾p𝑢qq 2
and in the same way, using the fact that 𝑢up
𝜎 “ and ℐ𝜎upp𝑢q “ 0 on any 𝜎 P ℱext such that 𝐹𝐾,𝜎p𝐵𝜎q ă 0 and the commensurability of ℎ𝐾 and ℎ𝐿 if ℱ𝐾X ℱ𝐾 ‰ H under assumption (A1):
ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 𝑅𝐾,𝜎p𝑢up𝜎 ´ ℐ up 𝜎 p𝑢qq 2 ď 𝐶 ÿ 𝐿P𝒯 ÿ 𝜎Pℱ𝐿,𝐹𝐿,𝜎p𝐵𝜎qą0 |𝜎|ℎ𝐿pℎ𝐿´ ℐ up 𝐿 p𝑢qq 2 ď 𝐶𝜌ℱ ÿ 𝐾P𝒯 |𝐾| p𝑢𝐾´ ℐ𝐾p𝑢qq 2 and also: ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝑅𝐾,𝜎p𝑢𝐾´ ℐ𝐾p𝑢qq 2 ď 𝐶 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 |𝜎|ℎ𝐾p𝑢𝐾´ ℐ𝐾p𝑢qq 2 ď 𝐶𝜌ℱ ÿ 𝐾P𝒯 |𝐾| p𝑢𝐾´ ℐ𝐾p𝑢qq2.
Gathering those intermediate results, we get that |𝑇5| ď 𝐶𝛾pℎq1{2||𝑢 ´ ℐℎp𝑢q||2ℎ. Finally, we have by definition of 𝛾pℎq and || ¨ ||ℎ that |𝑇6` 𝑇7| ď 𝐶𝛾pℎq1{2||𝑢 ´ ℐℎp𝑢q||2ℎ. Then, using the hypothesis that 𝛾pℎq goes to zero when ℎ goes to zero, for any 𝐶 ą 0 independent on ℎ there exists ℎ0ą 0 such that 1 ´ 𝐶 max pℎ, 𝛾pℎqq1{2ą 1{2,
and the result then immediately follows.
6.4. Error estimate for the water discharge
To establish Corollary6.2, we will need an auxiliary discrete stability result: Lemma 6.6. The following estimates holds:
1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾 𝐹𝐾,𝜎p𝐵𝜎q 𝑢2𝐾` 1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0 𝐹𝐾,𝜎p𝐵𝜎q 𝑢2𝐾 ´1 2 ÿ 𝐾P𝒯 ÿ 𝜎Pℱ𝐾,𝐹𝐾,𝜎p𝐵𝜎qă0 p𝑢𝐾´ 𝑢up𝜎 q 2 𝐹𝐾,𝜎p𝐵𝜎q “ ÿ 𝐾P𝒯 |𝐾|𝑓𝐾𝑢𝐾. (6.7)
Proof. To establish (6.7), using the definition of 𝑢up
