A quasi-variational inequality problem arising in the modeling of growing sandpiles

DOI:10.1051/m2an/2012062 www.esaim-m2an.org

A QUASI-VARIATIONAL INEQUALITY PROBLEM ARISING IN THE MODELING OF GROWING SANDPILES

John W. Barrett

and Leonid Prigozhin

Abstract. Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized mixed formulation involving both the primal (sand surface) and dual (sand flux) variables.

We derive, analyse and compare two methods for the approximation, and numerical solution, of this mixed problem. We prove subsequence convergence of both approximations, as the mesh discretization parameters tend to zero; and hence prove existence of a solution to this mixed model and the associated regularized quasi-variational inequality problem. One of these numerical approximations, in which the flux is approximated by the divergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithm to compute not only the evolving pile surface, but also the flux of pouring sand.

Results of our numerical experiments confirm the validity of the regularization employed.

Mathematics Subject Classification. 35D30, 35K86, 35R37, 49J40, 49M29, 65M12, 65M60, 82C27.

Received March 6, 2012. Revised October 5, 2012.

Published online June 17, 2013.

1. Introduction

Let a cohesionless granular material (sand), characterized by its angle of reposeα, be poured out onto a rigid surfacey = w₀(x), where y is vertical, x∈Ω ⊂R^d,d = 1 or 2, andΩ is a domain with boundary ∂Ω. The support surface w₀ ∈W₀^1,∞(Ω) and the nonnegative density of the distributed sourcef ∈L²(0, T;L²(Ω)) are given. We consider the growing sandpile y=w(x, t) and set an open boundary conditionw|∂Ω = 0. Denoting byq(x, t) the horizontal projection of the flux of material pouring down the evolving pile surface, we can write the mass balance equation

∂w

∂t +∇. q=f. (1.1)

Keywords and phrases. Quasi-variational inequalities, critical-state problems, primal and mixed formulations, finite elements, existence, convergence analysis.

1 Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.j.barrett@imperial.ac.uk

2 Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 84990, Israel.

Article published by EDP Sciences c EDP Sciences, SMAI 2013

J.W. BARRETT AND L. PRIGOZHIN

The quasi-stationary model of sand surface evolution, see Prigozhin [15,17,18], assumes the flow of sand is confined to a thin surface layer and directed towards the steepest descent of the pile surface. Wherever the support surface is covered by sand, the pile slope should not exceed the critical value; that is, w > w₀ ⇒

|∇w| ≤k₀, wherek₀ = tanα is the internal friction coefficient. Of course, the uncovered parts of the support can be steeper. This model does not allow for any flow on the subcritical parts of the pile surface; that is,

|∇w|< k₀ ⇒ q= 0. These constitutive relations can be conveniently reformulated fora.e.(x, t)∈Ω×(0, T) as

|∇w| ≤M(w) and M(w)|q|+∇w . q= 0, (1.2) where for anyx∈Ω

M(w)(x) :=

k₀ w(x)> w₀(x),

max(k₀,|∇w₀(x)|) w(x)≤w₀(x). (1.3) Let us define, for anyη∈C(Ω), the closed convex non-empty set

K(η) :=

ϕ∈W₀^1,∞(Ω) : |∇ϕ| ≤M(η) a.e.inΩ

. (1.4)

SinceM(w)|q|+∇ϕ . q≥0 for anyϕ∈K(w), we have, on noting (1.2), thatw∈K(w) and∇(ϕ−w). q≥0.

A weak form of the latter inequality is: for a.a.t∈(0, T)

Ω

∇. q(w−ϕ) dx≥0 ∀ ϕ∈K(w). (1.5)

Combining (1.5) and (1.1) yields an evolutionary quasi-variational inequality for the evolving pile surface: find w∈K(w) such that fora.a. t∈(0, T)

Ω

∂w

∂t −f

(ϕ−w) dx≥0 ∀ ϕ∈K(w). (1.6)

Assuming there is no sand on the support initially, we set

w(·,0) =w₀(·). (1.7)

A solution w to this quasi-variational inequality problem, (1.6) and (1.7), if it exists, should be a monotoni- cally non-decreasing function in time for any f ≥0, see Section 3 in Prigozhin [18]. However, existence and uniqueness of a solution has only been proved for support surfaces with no steep slopes; that is,|∇w₀| ≤k₀, see Prigozhin [15,18]. In this caseK(w)≡K:=

ϕ∈W₀^1,∞(Ω) : |∇ϕ| ≤k₀ a.e.inΩ

and the quasi-variational inequality becomes simply a variational inequality. Independently, the variational inequality for supports without steep slopes has been derived and studied in Aronson, Evans and Wu [2] as thep→ ∞limit of the evolutionary p-Laplacian equation.

The quasi-variational inequality (1.6) can, of course, be considered not only with the initial condition (1.7).

However, ifw(·,0) =w₀(·)≥w₀(·) andw₀does not belong to the admissible setK(w₀), an instantaneous solu- tion reconstruction takes place. Such discontinuous solutions, interpreted as simplified descriptions of collapsing piles with overcritical slopes, were studied in the variational inequality case in Evans, Feldman and Gariepy [10], and Dumont and Igbida [8]. Since we assumed the initial condition (1.7) and, obviously,w₀∈K(w₀), one could expect a solution continuously evolving in time. However, for the quasi-variational inequality with the open boundary condition w|∂Ω = 0, an uncontrollable influx of material from outside can occur through the parts of the boundary where ∇w₀. ν ≥ k₀, where ν is the outward unit normal to ∂Ω. This makes the solution non-unique and, possibly, discontinuous. Such an influx is prevented in our model by assuming that

∇w₀. ν < k₀ on ∂Ω, (1.8)

which implies that∇w . ν < k₀on∂Ω fort >0.

A QUASI-VARIATIONAL INEQUALITY PROBLEM ARISING IN THE MODELING OF GROWING SANDPILES

For the variational inequality version of the sand model, equivalent dual and mixed variational formulations have recently been proposed; see,e.g., Barrett and Prigozhin [4] and Dumont and Igbida [7]. Such formulations are often advantageous, because they allow one to determine not only the evolving sand surface w but also the surface flux q, which is of interest too in various applications; see Prigozhin [16,17], and Barrett and Prigozhin [6]. In such formulations, and this is their additional advantage, the difficult to deal with gradient constraint is replaced by a simpler, although non-smooth, nonlinearity.

Here we will also use a mixed variational formulation of a regularized version of the growing sandpile model involving both variables. Instead of excluding the surface fluxqfrom the model formulation, as in the transition to (1.6) above, we now note that the first condition in (1.2) holds if fora.e.(x, t)∈Ω×(0, T)

M(w)|v|+∇w . v≥0 (1.9)

for any test fluxv. Hence we can reformulate the conditions (1.2) for a.a.t∈(0, T) as

Ω

M(w) (|v| − |q|)−w∇.(v−q)

dx≥0 (1.10)

for any test fluxv, and consider a mixed formulation of the sand model as (1.1) and (1.10).

The quasi-variational inequality (1.6) is a difficult problem; in particular, due to the discontinuity of the nonlinear operator M, which determines the gradient constraint in (1.4). Furthermore, the natural function space for the flux q is the space of vector-valued bounded Radon measures havingL² divergence. If q is such a measure, the discontinuity ofM(w) also makes it difficult to give a sense to the term

ΩM(w)|q|dxin the inequality (1.10) of the mixed formulation.

In this work we consider a regularized version of the growing sandpile model with a continuous operatorM_ε: C(Ω)→C(Ω), determined as follows. For a fixed smallε >0, we approximate the initial dataw₀∈W₀^1,∞(Ω) byw^ε₀∈W₀^1,∞(Ω)∩C¹(Ω), andM(·) by the continuous functionM_ε(·) such that for anyx∈Ω

M_ε(η)(x) :=

⎧⎪

⎪⎨

⎪⎪

⎩

k₀ η(x)≥w₀^ε(x) +ε,

k₁^ε(x) + (k₀−k^ε₁(x))

η(x)−w^ε₀(x) ε

η(x)∈[w^ε₀(x), w₀^ε(x) +ε], k₁^ε(x) := max(k₀,|∇w₀^ε(x)|) η(x)≤w₀^ε(x).

(1.11)

We note thatM_εis such that for allη₁, η₂∈C(Ω)

|Mε(η₁)−M_ε(η₂)|0,∞,Ω≤k_1,∞^ε −k₀

ε |η1−η₂|0,∞,Ω, (1.12)

where

k^ε_1,∞:= max

x∈Ωk^ε₁(x). (1.13)

In addition, it follows for anyx∈Ωthat

η₁(x)≥η₂(x) ⇒ 0< k₀≤M_ε(η₁(x))≤M_ε(η₂(x))≤k₁^ε(x). (1.14) We note that the analysis of the sand quasi-variational inequality problem studied in this paper is far more involved than that of the superconductivity quasi-variational inequality problem studied by the present authors in [6]. In the superconductivity context,M :R→[M₀, M₁]⊂R withM₀>0. In [6], we exploit the fact that

|∇w(x)| ≤M(w(x)) can be rewritten as|∇[F(w(x))]| ≤1 for allx∈Ω, where F(·) = [M(·)]⁻¹andF(0) = 0.

Clearly, such a reformulation is not applicable toM(·), (1.3), orM_ε(·) (1.11).

In addition, we note that in the very recent paper by Rodrigues and Santos [19] an existence result can be deduced for the primal quasi-variational inequality problem (1.6) for a continuous and positiveM(·), such as

J.W. BARRETT AND L. PRIGOZHIN

M_ε(·), andf ∈W^1,∞(Ω×(0, T)). Assumingw⁰ ∈K(w⁰)∩C₀(Ω), they show thatw∈L^∞(0, T;W₀^1,∞(Ω))∩ W^1,∞(0, T; [C₀(Ω)]). Their proof is based on the method of vanishing viscosity and constraint penalization.

The outline of this paper is as follows. In the next section we introduce two fully practical finite element approximations, (Q^h,τ_A ) and (Q^h,τ_B,r), to the regularized mixed formulation (1.1) and (1.10), whereM(·) is replaced byM_ε(·), and prove well-posedness and stability bounds. Herehandτare the spatial and temporal discretization parameters, respectively. In addition, r > 1 is a regularization parameter in replacing the non-differentiable nonlinearity | · | by the strictly convex function ¹_r| · |^r. The approximation (Q^h,τ_A ) is based on a continuous piecewise linear approximation forwand a piecewise constant approximation forq, whereas (Q^h,τ_B,r) is based on a piecewise constant approximation forwand the lowest order Raviart–Thomas element forq. In Section3we prove subsequence convergence of both approximations to a solution of a weak formulation of the regularized mixed problem. This is achieved by passing to the limit h → 0 first, then r → 1 in the case of (Q^h,τ_B,r), and finallyτ →0. In Section 4, we introduce iterative algorithms for solving the resulting nonlinear algebraic equations arising from both approximations at each time level. Finally, in Section5we present various numerical experiments. Even though the approximation (Q^h,τ_A ) is simpler and may seem more natural than (Q^h,τ_B,r), and its convergence proof is certainly more straightforward; these experiments show that only the approximation (Q^h,τ_B,r) leads to an efficient algorithm to approximate both the surfacewand the fluxq.

We end this section with a few remarks about the notation employed in this paper. Above and throughout we adopt the standard notation for Sobolev spaces on a bounded domainD with a Lipschitz boundary, denoting the norm of W^,s(D) ( ∈ N, s ∈ [1,∞]) by.,s,D and the semi-norm by | · |,s,D. Of course, we have that

| · |_0,s,D≡ · _0,s,D. We extend these norms and semi-norms in the natural way to the corresponding spaces of vector functions. Fors= 2,W^,2(D) will be denoted byH(D) with the associated norm and semi-norm written as, respectively, · ,D and| · |,D. We set W₀^1,s(D) :={η ∈W^1,s(D) :η= 0 on ∂D}, andH₀¹(D)≡W₀^1,2(D).

We recall the Poincar´e inequality for anys∈[1,∞]

|η|0,s,D≤C(D)|∇η|0,s,D ∀η∈W₀^1,s(D), (1.15) where the constantC(D) depends onD, but is independent ofs; seee.g.p. 164 in Gilbarg and Trudinger [13].

In addition,|D|will denote the measure ofD and (·,·)D the standard inner product onL²(D). WhenD ≡Ω, for ease of notation we write (·,·) for (·,·)Ω.

Form∈N, let (i)C^m(D) denote the Banach space of continuous functions with all derivatives up to order m continuous on D, (ii) C₀^m(D) denote the space of continuous functions with compact support in D with all derivatives up to order m continuous on D and (iii) C₀^m(D) denote the Banach space {η ∈ C^m(D) : η = 0 on∂D}. In the casem= 0, we drop the superscript 0 for all three spaces.

As one can identifyL¹(D) as a closed subspace of the Banach space of bounded Radon measures, M(D)≡ [C(D)],i.e.the dual ofC(D); it is convenient to adopt the notation

D

|μ| ≡ μ_M(D):= sup

η∈C(D)

|η|0,∞,D≤1

μ, η_C(D)<∞, (1.16)

where·,·_C(D)denotes the duality pairing on [C(D)]×C(D).

We introduce also the Banach spaces for a givens∈[1,∞]

V^s(D) :={v∈[L^s(D)]^d:∇. v∈L²(D)} and V^M(D) :={v∈[M(D)]^d:∇. v∈L²(D)}. (1.17) The condition∇. v∈L²(D) in (1.17) means that there existsu∈L²(D) such thatv,∇φ_C(D)=−(u, φ)D for anyφ∈C₀¹(D).

We note that if {μn}_n≥0 is a bounded sequence inM(D), then there exist a subsequence{μnj}nj≥0 and a μ∈ M(D) such that asn_j → ∞

μ_nj →μ weakly inM(D); i.e. μnj −μ, η_C(D)→0 ∀ η∈C(D). (1.18)

A QUASI-VARIATIONAL INEQUALITY PROBLEM ARISING IN THE MODELING OF GROWING SANDPILES

In addition, we have that

lim inf

nj→∞

D

|μ_nj| ≥

D

|μ|; (1.19)

seee.g.p. 223 in Folland [12].

We recall the following Sobolev interpolation theorem, see Theorem 5.8 in Adam and Fournier [1]. If η ∈ W^1,s(D), withs > d, thenη∈C(Ω) with the embedding being compact; and moreover,

|η|_0,∞,D≤C(s, D)η^α_1,s,D|η|^1−α_0,D with α= ds

ds+ 2(s−d) ∈(0,1). (1.20) We recall also the Aubin–Lions–Simon compactness theorem, see Corollary 4 in Simon [20]. LetB0,BandB1be Banach spaces,Bi,i= 0,1, reflexive, with a compact embeddingB0→ Band a continuous embeddingB→ B1. Then, forα >1, the embedding

{η∈L^∞(0, T;B₀) : ∂η

∂t ∈L^α(0, T;B₁)}→C([0, T];B) (1.21) is compact.

Finally, throughoutC denotes a generic positive constant independent of the regularization parameter,r∈ (1,∞), the mesh parameter h and the time step parameter τ. Whereas, C(s) denotes a positive constant dependent on the parameters.

2. Finite element approximation

We make the following assumptions on the data.

(A1) Ω⊂R^d,d= 1 or 2, has a Lipschitz boundary ∂Ω with outward unit normalν.f ∈L²(0, T;L²(Ω)) is a nonnegative source, andM_ε(·) is given by (1.11). In addition, the initial dataw^ε₀∈C₀¹(Ω) is such that

∇w^ε₀. ν < k₀ on∂Ω.

For ease of exposition, we shall assume thatΩis a polygonal domain to avoid perturbation of domain errors in the finite element approximation. We make the following standard assumption on the partitioning.

(A2) Ωis polygonal. Let{T^h}_h>0be a regular family of partitionings ofΩinto disjoint open simplicesσwith h_σ := diam(σ) andh:= max_σ∈Thh_σ, so thatΩ=∪_σ∈T^hσ.

Letν_∂σ be the outward unit normal to∂σ, the boundary ofσ. We then introduce the following finite element spaces

S^h:={η^h∈L^∞(Ω) :η^h|σ=a_σ∈R ∀σ∈ T^h}, (2.1a) S^h_≥0:={η^h∈L^∞(Ω) :η^h|σ=a_σ∈R_≥0 ∀σ∈ T^h}, (2.1b) S^h:={η^h∈[L^∞(Ω)]^d:η^h|σ=a_σ∈R^d ∀ σ∈ T^h}, (2.1c) U^h:={η^h∈C(Ω) :η^h|σ=a_σ. x+b_σ, a_σ∈R^d, b_σ∈R ∀σ∈ T^h}, (2.1d)

U₀^h:=U^h∩W₀^1,∞(Ω), (2.1e)

V^h:={v^h∈[L^∞(Ω)]^d:v^h|σ=a_σ+b_σx, a_σ∈R^d, b_σ ∈R ∀ σ∈ T^h

and (v^h|σ−v^h|σ). ν_∂σ= 0 on∂σ∩∂σ ∀σ, σ∈ T^h.} (2.1f) Here V^his the lowest order Raviart–Thomas finite element space.

J.W. BARRETT AND L. PRIGOZHIN

Let π^h : C(Ω) → U^h denote the interpolation operator such that π^hη(x_j) = η(x_j), j = 1, . . . , J, where {x_j}^J_j=1 are the vertices of the partitioningT^h. We note form= 0 and 1 that

|(I−π^h)η|m,s,σ≤C h^2−m|η|_2,s,σ ∀σ∈ T^h, for anys∈[1,∞], (2.2a)

h→0lim(I−π^h)ηm,∞,Ω = 0 ∀η∈C^m(Ω); (2.2b)

whereI is the identity operator. LetP^h: [L¹(Ω)]^d→S^h be such that P^hv|σ= 1

|σ|

σ

vdx ∀σ∈ T^h. (2.3)

We note that

|P^hv|0,s,σ≤ |v|0,s,σ ∀v∈[L^s(σ)]^d, s∈[1,∞], ∀σ∈ T^h, (2.4a)

h→0lim| |v| − |P^hv| |_0,∞,Ω≤ lim

h→0|v−P^hv|_0,∞,Ω= 0 ∀ v∈[C(Ω)]^d. (2.4b) Similarly, we defineP^h:L¹(Ω)→S^hwith the equivalent to (2.4a,b) holding.

In addition, we introduce the generalised interpolation operatorI^h: [W^1,s(Ω)]^d→V^h, wheres >1, satisfying

∂iσ

(v−I^hv). ν_∂

iσds= 0 i= 1,2,3, ∀σ∈ T^h; (2.5)

where∂σ≡ ∪³_i=1∂_iσandν_∂

iσ are the corresponding outward unit normals on∂_iσ. It follows that

(∇.(v−I^hv), η^h) = 0 ∀η^h∈S^h. (2.6) Moreover, we have for allσ∈ T^h and anys∈(1,∞] that

||v| − |I^hv||0,s,σ≤ |v−I^hv|0,s,σ ≤C h_σ|v|1,s,σ and |I^hv|1,s,σ≤C|v|1,s,σ, (2.7) e.g.see Lemma 3.1 in Farhloul [11] and the proof given there fors≥2 is also valid for anys∈(1,∞].

We introduce (η, χ)^h:=

σ∈T^h(η, χ)^h_σ, and (η, χ)^h_σ :=_d+1¹ |σ| ^d+1

j=1

η(x^σ_j)χ(x^σ_j) =

σ

π^h[η χ] dx ∀η, χ∈C(σ), ∀σ∈ T^h; (2.8)

where{x^σ_j}^d+1_j=1are the vertices ofσ. Hence (η, χ)^haverages the integrandη χover each simplexσat its vertices, and is exact ifη χis piecewise linear over the partitioningT^h. We recall the well-known results that

|η^h|²_0,Ω≤ |η^h|²_h:= (η^h, η^h)^h≤(d+ 2)|η^h|²_0,Ω ∀η^h∈U^h, (2.9a) (η^h, χ^h)−(η^h, χ^h)^h=((I−π^h)(η^hχ^h),1)≤ |(I−π^h)(η^hχ^h)|_0,1,Ω≤C h|η^h|_0,Ω|χ^h|_1,Ω ∀η^h, χ^h∈U^h, (2.9b) where we have noted (2.2a).

In order to prove existence of solutions to approximations of (1.10), we regularise the non-differentiable nonlinearity| · |by the strictly convex function ¹_r| · |^r forr >1. We note for alla, b∈R^d that

1 r

∂|a|^r

∂a_i =|a|^r−2a_i ⇒ |a|^r−2a .(a−b)≥¹_r [|a|^r− |b|^r]. (2.10)

A QUASI-VARIATIONAL INEQUALITY PROBLEM ARISING IN THE MODELING OF GROWING SANDPILES

Similarly to (2.9a), we have from the equivalence of norms and the convexity of| · |^r for anyr >1 and for any v^h∈V^h that

C(|v^h|^r,1)^h_σ≤

σ

|v^h|^rdx≤(|v^h|^r,1)^h_σ ∀ σ∈ T^h. (2.11) Furthermore, it follows from (2.7) and (2.8) for anyr >1 and anyσ∈ T^h that

|

σ

|I^hv|^rdx−(|I^hv|^r,1)^h_σ| ≤C r|σ| |I^hv|^r−1_0,∞,σ max

x, y∈σ|(I^hv)(x)−(I^hv)(y)|

≤C r h_σ|σ| v^r_1,∞,σ ∀v∈[W^1,∞(σ)]^d. (2.12) In addition, let 0 = t₀ < t₁ < . . . < t_N−1 < t_N =T be a partitioning of [0, T] into possibly variable time stepsτ_n:=t_n−t_n−1,n= 1, . . . , N. We set τ:= max_n=1,...,Nτ_n and introduce

fⁿ(·) := 1 τ_n

_tn

tn−1

f(·, t) dt∈L²(Ω) n= 1, . . . , N. (2.13) We note that

N n=1

τ_n|fⁿ|^s_0,s,Ω≤ _T

0 |f|^s_0,s,Ωdt for anys∈[1,2]. (2.14) Finally, on setting

w^ε,h₀ =P^h[π^hw₀^ε], (2.15)

we introduceM_ε^h:S^h→S^h approximatingM_ε:C(Ω)→C(Ω), defined by (1.11), for anyσ∈ T^h as

M_ε^h(η^h) :=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

k₀ η^h≥w₀^ε,h+ε,

k_1,σ^ε,h+ (k₀−k_1,σ^ε,h)

η^h−w₀^ε,h ε

η^h∈[w^ε,h₀ , w₀^ε,h+ε], k_1,σ^ε,h:= max(k₀,|∇π^hw^ε₀|σ|) η^h≤w₀^ε,h.

(2.16)

We note thatM_εis also well-defined onS^h withM_ε:S^h→L^∞(Ω), and we have the following result.

Lemma 2.1. For any η^h∈S^h, we have that

|M_ε(η^h)−M_ε^h(η^h)|0,∞,Ω ≤C(ε⁻¹) |(I−P^h)w^ε₀|0,∞,Ω+(I−π^h)w^ε₀1,∞,Ω

. (2.17)

Proof. It is convenient to rewrite (1.11) and (2.16) for anyη^h∈S^hand fora.e.x∈Ωas M_ε(η^h)(x) =k₀+

k₁^ε(x)−k₀ ε

min( max(w^ε₀(x) +ε−η^h(x),0), ε), (2.18a) M_ε^h(η^h)(x) =k₀+

k^ε,h₁ (x)−k₀ ε

min( max(w^ε,h₀ (x) +ε−η^h(x),0), ε); (2.18b) where

k₀≤M_ε^h(η^h)(x)≤k^ε,h₁ (x) := max(k₀,|∇π^hw^ε₀(x)|) fora.e.x∈Ω. (2.19)

J.W. BARRETT AND L. PRIGOZHIN

Since

|min(max(a,0), ε)−min(max(b,0), ε)| ≤ |a−b| and |min(max(a,0), ε)| ≤ε ∀a, b∈R, (2.20) it follows from (2.18a,b), (2.19), (2.15), (2.4a) and Assumption (A1) that

|Mε(η^h)−M_ε^h(η^h)|_0,∞,Ω≤ k_1,∞^ε −k₀

ε |w₀^ε−w^ε,h₀ |_0,∞,Ω+| |∇w₀^ε| − |∇π^hw^ε₀| |_0,∞,Ω

≤C(ε⁻¹) |(I−P^h)w₀^ε|0,∞,Ω+(I−π^h)w₀^ε1,∞,Ω

; (2.21)

and hence the desired result (2.17).

2.1. Approximation (Q^h,τ_A )

Our first fully practical finite element approximation is:

(Q^h,τ_A )Forn= 1, . . . , N, findW_Aⁿ∈U₀^h andQⁿ

A∈S^hsuch that W_Aⁿ−W_Aⁿ⁻¹

τ_n , η^h h

−(Qⁿ_A,∇η^h) = (fⁿ, η^h) ∀η^h∈U₀^h, (2.22a) (M_ε^h(P^hW_Aⁿ),|v^h| − |Qⁿ_A|) + (∇W_Aⁿ, v^h−Qⁿ_A)≥0 ∀v^h∈S^h; (2.22b) whereW_A⁰=π^hw^ε₀.

For anyχ^h∈U₀^h, we introduce the closed convex non-empty set

K^h(χ^h) :={η^h∈U₀^h:|∇η^h| ≤M_ε^h(P^hχ^h) a.e. inΩ}. (2.23) In Theorem2.3below, we will show that (Q^h,τ_A ), (2.22a,b), is equivalent to (P^h,τ_A ) and (M^h,τ_A ). The former is the approximation of the primal quasi-variational inequality:

(P^h,τ_A )Forn= 1, . . . , N, findW_Aⁿ∈K^h(W_Aⁿ) such that W_Aⁿ−W_Aⁿ⁻¹

τ_n , η^h−W_Aⁿ h

≥(fⁿ, η^h−W_Aⁿ) ∀ η^h ∈K^h(W_Aⁿ), (2.24) whereW_A⁰=π^hw^ε₀.

The latter, having obtained{W_Aⁿ}^N_n=1 from (P^h,τ_A ), is the minimization problem:

(M^h,τ_A )Forn= 1, . . . , N, findQⁿ

A∈Z^h,n such that

(M_ε^h(P^hW_Aⁿ),|Qⁿ_A|)≤(M_ε^h(P^hW_Aⁿ),|v^h|) ∀ v^h∈Z^h,n, (2.25) where

Z^h,n:=

v^h∈S^h: (v^h,∇η^h) =

W_Aⁿ−W_Aⁿ⁻¹ τ_n , η^h

h

−(fⁿ, η^h) ∀η^h∈U₀^h

. (2.26)

As∇U₀^h is a strict subset of S^h, it follows that the affine manifoldZ^h,n,n= 1, . . . , N, is non-empty.

We consider the following regularization of (Q^h,τ_A ) for a givenr >1:

(Q^h,τ_A,r)Forn= 1, . . . , N, findW_A,rⁿ ∈U₀^h andQⁿ

A,r ∈S^h such that W_A,rⁿ −W_A,rⁿ⁻¹

τ_n , η^h _h

−(Qⁿ_A,r,∇η^h) = (fⁿ, η^h) ∀η^h∈U₀^h, (2.27a)

A QUASI-VARIATIONAL INEQUALITY PROBLEM ARISING IN THE MODELING OF GROWING SANDPILES

(M_ε^h(P^hW_A,rⁿ )|Qⁿ_A,r|^r−2Qⁿ

A,r, v^h) + (∇W_A,rⁿ , v^h) = 0 ∀v^h∈S^h; (2.27b) whereW_A,r⁰ =π^hw^ε₀.

Associated with (Q^h,τ_A,r) is the corresponding approximation of a generalised p-Laplacian problem for p >1, where, here and throughout the paper, ¹_r+¹_p = 1:

(P^h,τ_A,p)Forn= 1, . . . , N, findW_A,rⁿ ∈U₀^h such that W_A,rⁿ −W_A,rⁿ⁻¹

τ_n , η^h _h

+

[M_ε^h(P^hW_A,rⁿ )]^−(p−1)|∇W_A,rⁿ |^p−2∇W_A,rⁿ ,∇η^h

= (fⁿ, η^h) ∀η^h∈U₀^h, (2.28) whereW_A,r⁰ =π^hw^ε₀.

Theorem 2.2. Let the Assumptions (A1) and (A2) hold. Then for all r ∈(1,2), for all regular partitionings T^h of Ω, and for all τ_n >0, there exists a solution, W_A,rⁿ ∈U₀^h and Qⁿ

A,r ∈S^h to the nth step of (Q^h,τ_A,r). In addition, we have that

n=0,...,Nmax |W_A,rⁿ |0,Ω+ N n=1

|W_A,rⁿ −W_A,rⁿ⁻¹|²_0,Ω+ N n=1

τ_n|Qⁿ_A,r|^r_0,r,Ω+ _N

n=1

τ_n|∇W_A,rⁿ |^p_0,p,Ω ¹_p

≤C (2.29)

where ¹_r+¹_p = 1. Moreover, (Q^h,τ_A,r),(2.27a,b), is equivalent to (P^h,τ_A,p),(2.28).

Proof. It follows immediately from (2.27b) that

∇W_A,rⁿ =−M_ε^h(P^hW_A,rⁿ )|Qⁿ

A,r|^r−2Qⁿ

A,r

⇔ Qⁿ

A,r=−[M_ε^h(P^hW_A,rⁿ )]^−(p−1)|∇W_A,rⁿ |^p−2∇W_A,rⁿ onσ, ∀σ∈ T^h. (2.30) Substituting this expression forQⁿ_A,rinto (2.27a) yields (2.28). Hence (P^h,τ_A,p), with (2.30), is equivalent to (Q^h,τ_A,r).

We now apply the Brouwer fixed point theorem to prove existence of a solution to (P^h,τ_A,p), and therefore to (Q^h,τ_A,r). LetF^h:U₀^h→U₀^h be such that for anyϕ^h∈U₀^h,F^hϕ^h∈U₀^hsolves

F^hϕ^h−W_A,rⁿ⁻¹ τ_n , η^h

_h +

[M_ε^h(P^hϕ^h)]^−(p−1)|∇F^hϕ^h|^p−2∇F^hϕ^h,∇η^h

= (fⁿ, η^h) ∀η^h∈U₀^h. (2.31) The well-posedness of the mappingF^hfollows from noting that (2.31) is the Euler–Lagrange system associated with the strictly convex minimization problem:

min

η^h∈U₀^hE_p^h,n(η^h), (2.32a)

whereE_p^h,n:U₀^h→Ris defined by E_p^h,n(η^h) := 1

2τ_n|η^h−W_A,rⁿ⁻¹|²_h+1 p

Ω

[M_ε^h(P^hϕ^h)]^−(p−1)|∇η^h|^pdx−(fⁿ, η^h); (2.32b) that is, there exists a unique element (F^hϕ^h)∈U₀^h solving (2.31). It follows immediately from (2.32a,b) that

1

2τ_n |F^hϕ^h−W_A,rⁿ⁻¹|²_h−(fⁿ, F^hϕ^h)≤E_p^h,n(F^hϕ^h)≤E_p^h,n(0) = 1

2τ_n |W_A,rⁿ⁻¹|²_h. (2.33)

J.W. BARRETT AND L. PRIGOZHIN

It is easily deduced from (2.33) and (2.9a) that

F^hϕ^h∈B_γ :={η^h∈U₀^h:|η^h|_0,Ω≤γ}, (2.34) where γ ∈R>0 depends on |W_A,rⁿ⁻¹|0,Ω, |fⁿ|0,Ω andτ_n. Hence F^h : B_γ →B_γ. In addition, it is easily verified that the mappingF^his continuous, asM_ε^h:S^h→S^his continuous. Therefore, the Brouwer fixed point theorem yields that the mappingF^hhas at least one fixed point in B_γ. Hence, there exists a solution to (P^h,τ_A,p), (2.28), and therefore to (Q^h,τ_A,r), (2.27a,b).

It follows from (2.30) and (2.19) that forn= 1, . . . , N

|∇W_A,rⁿ |^p_0,p,Ω=|[M_ε^h(P^hW_A,rⁿ )]^p−1Qⁿ

A,r|^r_0,r,Ω≤(k^ε,h_1,∞)^p−1(M_ε^h(P^hW_A,rⁿ ),|Qⁿ_A,r|^r); (2.35) where, on noting (2.19), (2.2b) and Assumption (A1),

k_1,∞^ε,h := max

x∈Ωk₁^ε,h(x)≤C. (2.36)

Choosingη^h=W_A,rⁿ ,v^h=Qⁿ

A,r in (2.27a,b), combining and noting the simple identity (a−b)a= 1

2 a²+ (a−b)²−b²

∀a, b∈R, (2.37)

we obtain forn= 1, . . . , N, on applying a Young’s inequality and (1.15), that for allδ >0

|W_A,rⁿ |²_h+|W_A,rⁿ −W_A,rⁿ⁻¹|²_h+ 2τ_n(M_ε^h(P^hW_A,rⁿ ),|Qⁿ_A,r|^r)

=|W_A,rⁿ⁻¹|²_h+ 2τ_n(fⁿ, W_A,rⁿ )

≤ |W_A,rⁿ⁻¹|²_h+ 2τ_n 1

rδ^−r|fⁿ|^r_0,r,Ω+1

pδ^p|W_A,rⁿ |^p_0,p,Ω

≤ |W_A,rⁿ⁻¹|²_h+ 2τ_n 1

rδ^−r|fⁿ|^r_0,r,Ω+1

p[δ C(Ω)]^p|∇W_A,rⁿ |^p_0,p,Ω

. (2.38) It follows on summing (2.38) fromn= 1 tom, with δ= 1/(C(Ω) [k_1,∞^ε,h ]^1r), and noting (2.35) and (2.36) that form= 1, . . . , N

|W_A,r^m |²_h+ m n=1

|W_A,rⁿ −W_A,rⁿ⁻¹|²_h+ m n=1

τ_n(M_ε^h(P^hW_A,rⁿ ),|Qⁿ_A,r|^r)≤ |W_A,r⁰ |²_h+ 2 [C(Ω)]^rk_1,∞^ε,h m n=1

τ_n|fⁿ|^r_0,r,Ω. (2.39) The desired result (2.29) follows immediately from (2.39), (2.9a), (2.14), (2.19), (2.36) and (2.35).

Theorem 2.3. Let the Assumptions (A1) and (A2) hold. Then for all regular partitionings T^h of Ω, and for allτ_n>0, there exists a solution,W_Aⁿ∈U₀^h andQⁿ

A∈S^h to the nthstep of (Q^h,τ_A ). In addition, we have that

n=0,...,Nmax |W_Aⁿ|_0,Ω+ N n=1

|W_Aⁿ−W_Aⁿ⁻¹|²_0,Ω+ N n=1

τ_n|Qⁿ_A|_0,1,Ω+ max

n=0,...,NW_Aⁿ_1,∞,Ω≤C. (2.40) Moreover, (Q^h,τ_A ),(2.22a,b), is equivalent to (P^h,τ_A ),(2.24), and (M^h,τ_A ),(2.25). Furthermore, forn= 1, . . . , N, having obtained W_Aⁿ, then Qⁿ

A = −λⁿ_A∇W_Aⁿ, where λⁿ_A ∈ S_≥0^h is the Lagrange multiplier associated with the gradient inequality constraint in (P^h,τ_A ).