The mathematical modeling of the dynamics of granular materials is a fascinating subject as evidenced by the numerous studies which treat this subject in the literature. In this work, we are interested in the formation of a pile of sand under the effect of a vertical source. Regarding this phenomenon, there are mainly two models based on partial differential equations. The first model is that of Hadeler and Kuttler (see    ) where the authors model the pile of sand using two layers: a fixed layer above which a mobile layer moves. The second approach to which this work relates is the variational model of Prigozhin (see   ) where the surface flow of sand is supposed to exist only at critical slope, that is the maximal admissible slope for any stationary configuration of sand (here for simplicity it is assumed ).
In the particular case of the Dirichlet boundary condition; for that model there are a sufficiently developed theory and efficient numerical schemes which use duality arguments (see    ). However, very few results exist in the literature on the model of Prighozin in the case where one considers other types of boundary conditions other than the Dirichlet boundary condition, since it raises many difficult questions: how to define the notion of the solution for the model and how to approach the problem numerically. We have recently partially provided solutions to these questions where we study the Prigozhin model with Neumann and Robin boundaries conditions (see  ). These types of conditions are interpreted by the presence of an obstacle for example a wall at the boundary and which prevents the sand escaping. In this work, we look at the model with a non-local boundary condition which generalizes the Neumann and Robin boundary condition:
with a bounded open domain with a boundary such that . The solution u is the height of the surface andf is the source, and a is a function defined on .
In this work, we consider a mixed boundary condition: Dirichlet, Neumann and a non-local boundaries condition:
· on , we apply homogeneous Dirichlet boundary condition. In other words, we assume that at this boundary, the sand falls down, that is
· on , there is a wall that prevents the sand escaping:
· we control the outgoing flow through the boundary , which results in the following non-local condition:
Beside the mathematical interest of non-local conditions, it seems that this type of boundary condition is also encountered in other physical applications. For example, in petroleum engineering model for well modelling in a 3D stratified petroleum reservoir with arbitrary geometry; this kind of boundary condition also arises in petroleum engineering, in the simulation of wells performance, since a nonlinear relation exists between the performance pressure tangential gradient and the fluid velocity along the well (see   for details). Another application of this type of the boundary condition is in the study of the heat conduction within linear thermoelasticity (see    ), and for the reaction-diffusion equation (see   ). One could find other applications in other fields of physics in the papers   .
The present work is organized as follows. In the next section, we use the nonlinear semi-group theory (see  ) to get the existence and uniqueness of a variational solution of (1) and the convergence of the approximate Euler discretization in time solutions to problem (1). In Section 3, we show how to compute the solution of Euler implicit time discretization of (1) using duality argument and in the last section, some results of numerical results are given.
2. Global Existence
Given , we say that is an -discretization for problem (1) if with , , such that
Definition 1. For any , is an -approximate solution of (1), if there exists an -discretization of problem (1) such that
and solves the following Euler implicit time discretization of (1):
The generic problem associated with (3) is given by
with , .
For convenience, we note the scalar product by and the euclidean norm on by . We introduce the convex set :
We also introduce the function defined on
The sub-differential of is defined in by
Remark. In order to define our notions of variational solutions for the above problems, we make the following observations:
(1) Assuming that is a solution of (4) in the following sense: there exists a measurable function m satisfying the properties , a.e. in and
then, v is also a solution of the following optimization problem
Indeed, taking as test function in (5), we get
Thus, using the fact that a.e. , we get
and taking in (7), we obtain
So, adding (8) and (9) and using the fact that a.e. in , we obtain
(2) It is not clear that there exists such that (5) is fulfilled, however we can find , solution of the problem (6), which is a consequence of the following result.
Lemma 2. is a maximal monotone graph in .
Proof. The proof is the same as in our previous paper.
We are now in a position to define our notion of solution to problems (4) and (1).
Definition 3 (see  ). For given , , we say that v is a variational solution of (4) if and
Definition 4 (see  ). Given , and , a variational solution (resp. -approximate solution) of (1) is a function satisfying for any , and
for any (resp. given by (2) with a variational solution of (3)).
Since is a maximal monotone graph in , then, thanks to  for any there exists a unique solution v of
which is equivalent to saying that (4) admits a unique variational solution. Moreover, if is the solution corresponding to for , then
As in the works of Igbida and al, we get the following result
Theorem 5. Let , , . Then,
1. for any and any -discretization of (1), there exists a unique -approximate variational solution of (1).
2. There exists such that and as ,
3. The function u given by (2) is the unique variational solution of (1). Moreover, if for , is a solution corresponding to , then
In particular, if , then a.e. in .
3. Numerical Analysis
Following the ideas developed in our previous work we show in this section how to approximate the solution of problem (1). In our previous works, we are introduced a numerical method based on Fenchel duality theory (see  ) for numerical approximation of problem (4), in the special case where the homogeneous Dirichlet boundary condition is applied. In order to use the same approach, we introduce the following result.
Lemma 6. The problem (4) is equivalent to: find solution of the following of
Proof. Let be a solution of (4). Then, we have
We use the well know inequality
From (15), we deduce that
Which end the first part of the proof. Now assume v is the solution of the minimization problem (14). Let , since is convex, then for any , . Then, we have
Which implies that
Hence, by letting in (17), we obtain
for any , so v is a solution of (4).
Inspired by the works of Igbida and al, we consider as a dual problem associated with problem (14), the following optimization problem
In the following, we prove the connection between problems (18) and (14). We also present a method for numerical approximation of the solution of problem (4) by computing . We first show the following result.
Lemma 7. Let , , and , we have
Proof. Taking , and using the fact that , we get
Since , we have
As z belongs to , it follows that
The connection between the problem (14) and (18) is given by the following results.
Theorem 8. Let , and v the variational solution of (14). Then, there exists a sequence in , such that, as ,
Moreover, we have
Let recall that, denote the space of all bounded Radon measures in .
For the proof of this result, we analyze the following problem
where for any , is given by
and satisfies the following properties.
(i) for any , ;
(ii) there exists and such that for any and ;
(iii) for any and , .
Lemma 9. There exists a unique weak solution to problem (26) in the sense that , and for all ,
(1) is bounded in .
(2) is bounded in .
(3) For any Borel set ,
Proof. (1) We define the operator by
It’s not difficult to see that the operator is monotone, coercive, hemicontinous and bounded.
Defining defined by
then thanks to , there exists such that
Which end the proof of the existence. Now, if (26) admits two solutions u and v, then we subtract the two equations obtained by replacing respectively by u and v in (27) to get
We take as a test function in (29) to obtain
Thus, using the property (i) of , we deduce that . a.e.
(1) Taking as test function in (27), we have
Since is constant on , there exists such that
Using the following trace inequality (see  )
where is a positive constant. One obtains
On the other hand
Combining (32) and property (ii) of , for any , we get
Thus, adding (31) and (33), we obtain
Using the Young’s inequality, we get
Then is bounded in .
(2) Using (32) and the property (iii) of , we have
and since is bounded in , it follows that is bounded in .
(3) Now, let be a fixed Borel set. We have
Letting and using the fact that is bounded in , we obtain
Proof of the Theorem 8. Thanks to Lemma 9, there exists in , and a subsequence that we denote again by , such that
Hence, by problem (26), we deduce that
We define the set by , with . Using the fact that in -weak as , it follows that
Taking in Lemma 7, we obtain
which implies that . Since is arbitrary, then, it follows that a.e. in . Let us now show that is also a solution of (4). For all , we have
i.e. is also a solution of (4). Since (4) admits a unique solution then . Now, we must show that satisfies (22). By (iii), we have
Moreover taking v as test function in (27) and having in mind that , we get
Combining (39) and (40), we get
Thanks to the Lemma 7 we have , hence from (22) and (23), we deduce that, as , .
Remark. As consequences of Theorem 8, we have another characterization of the solution v of the problem (4):
Moreover, we have another characterization of which will be useful for the numerical part:
where L is the variational solution to the Laplace problem
Consequently, the dual problem (18) is equivalent to
For the numerical approximation of the problem (45), we use the finite element method, then we assume that:
· is an open and bounded subset of ;
· will be a regular partitioning of by n disjoint open simplex , with .
Let be a finite dimensional subspace of , with a finite dimension equal to , where is the space of lowest-order Raviart-Thomas finite elements (see  ) defined by
where represents the outward unit normal to , the boundary of .
By we denote the interpolation operator onto given in ( , Theorem 6.1). Then, thanks to , for all , we have
We have the following convergence properties as h tends to 0.
Theorem 10. Let , , v a solution to the minimization problem (14) and a solution of the optimization problem
Then, as ,
The proof of this result is similar to the proof of Theorem 3.8 in the paper of Igbida et al.; therefore, we omit it.
4. Numerical Simulations
We start by approximating the term on each element of the partitioning . We take
where represents the area of simplex and is one of the vertices of . Using this approximation, at each time , and the time step, the solution of (47) is a minimizer of the non-differentiable functional ,
The minimization of the functional is done according to the Gauss-Seidel type algorithm.
In all the tests the discretization in time and convergence criterion , . For the discretization of the problem, we use the Raviart-Thomas elements of the lowest order  on a regular square grid.
After having obtained the minimizer of the functional , the solution of the Euler implicit time discretization of (1) is computed by using extremality relation (43) in a weak sense with piecewise finite elements .
In the first test, the source is a constant equal to 1, for all time and for all . The Neumann boundary conditions are considered on the boundary which simulated the existence of a wall on the boundary. The outgoing flow across the border at any time t is is .
Figure 1 shows the configuration of the sandpile at time when the solution u becomes stationary. On sees also the effect of the non-local boundary condition of the configuration of the sandpile.
In the second test (see Figure 2), one analyzes the impact of the condition of Neumann on the configuration of the sandpile in the absence of the source f. We can also see that the flux is concentrated around . This example allows us to conclude that the function g behaves likes a source of sand.
Figure 1. Final configuration of the sandpile surface at for , and .
Figure 2. Sandpile surface and the flux at for , and .
Figure 3. Sandpile surface and the flux at for , and .
Figure 4. Sandpile surface and the flux at for , and .
of the sandpile and the behavior of the flow. We can see here that the final configuration of the sandpile is slightly different from that of the test 1 (see Figure 1). This is due to the sign of the total outflow through . We also remark that the flux has the same direction as the gradient of the surface and almost zero on the axis .
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