THE POISSON PROBLEM IN A THREE-DIMENSIONAL PLATE

Alexandre L. Madureira

Laboratório Nacional de Computação Científica (LNCC)

Abstract. This note was the basis of a short course presented on June 23, 2000, at University of Pavia, Italy. Here we define and analyse dimension reduction models for the Poisson problem in a thin, three-dimensional plate. After introducing the Poisson problem, and a variational approach for dimension reduction, we present the asymptotic expansion for the exact and model solutions. Then we estimate the modeling error. This work is based on the thesis [3], see the web site http://www.math.psu.edu/dna/education.html#students

1 – Introduction

For a positive number ε < 1, define the three-dimensional plate P^{ε} = Ω×(−ε, ε) and its boundaries ∂P_{L}^{ε} = ∂Ω × (−ε, ε) and ∂P_{±}^{ε} = Ω × {−ε, ε}.

Ω  P^{ε} ·  x^{ε} = (x_{∼}, x_{3}^{ε}) 2ε

Assume that u^{ε} ∈ H^{1}(P^{ε}) satisfies (in the weak sense)

(1)  \begin{matrix}Δ u^{ε} = −f^{ε}  in P^{ε}, \\ \frac{∂u^{ε}}{∂n} = 0  on ∂P_{±}^{ε}, \\ u^{ε} = 0  on ∂P_{L}^{ε}, \end{matrix}

where f^{ε} : P^{ε} → ℝ. In general, the solution of (1) will depend on ε in a nontrivial way. In fact the above problem is a singularly perturbed one, and as ε goes to zero it “loses” ellipticity. This causes the onset of boundary layers, as we make clear below.

It is possible to characterize the solution of (1) in an alternative way, as the minimizer of the associate energy functional, i.e.,

u^{ε} = arg min_{v∈V (P^{ε})} J (v), where J (v) = \frac{1}{2} ∫_{P^{ε}} | ∇ v|^{2} dx − ∫_{P^{ε}} f^{ε}v dx,

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This work has been partially supported by IAN-CNR, Italy

and V (P^{ε}) = {v ∈ H^{1}(P^{ε}) : v = 0 on ∂P_{L}^{ε}}.

Aiming to find a “good” approximation for u^{ε}, we search for

(2)  u^{ε}(p) = arg min_{v∈H̊^{1}(Ω;ℙ_{p}(−ε,ε))} J (v),

where the notation is as follows. For an integer p and a positive real number a, we define ℙ_{p}(−a, a) as the space of polynomials of degree p in (−a, a). So H̊^{1}(Ω; ℙ_{p}(−a, a)) denotes the space of polynomials of degree p with coefficients in H̊^{1}(Ω). The space H̊^{1}(Ω) is the set of functions in the usual Sobolev space H^{1}(Ω) with zero trace on ∂Ω. It follows from its definition that u^{ε}(p) is the Ritz projection of u^{ε} into H̊^{1}(Ω; ℙ_{p}(−ε, ε)) and such model is a minimum energy one. Observe that the use of higher polynomial degrees yield higher order models, actually leading to a hierarchy of models that furnish increasingly better solutions.

Rewriting (2) in weak form, it is not hard to check that if u^{ε}(1)(x) = ω_{0}(x_{∼}) +

ω_{1}(x_{∼})x_{3}^{ε}, then

(3) Δ_{2D} ω_{0} = −\frac{1}{2}f^{0}, \frac{2ε^{2}}{3} Δ_{2D} ω_{1} − 2ω_{1} = −f^{1} in Ω,

ω_{0} = ω_{1} = 0 on ∂Ω,

where Δ_{2D} = ∂_{11} + ∂_{22}, and

(4)  f^{0}(x_{∼}^{ε}) = \frac{1}{ε} ∫_{−ε}^{ε} f^{ε}(x_{∼}^{ε}, x_{3}^{ε}) dx_{3}^{ε}, f^{1}(x_{∼}^{ε}) = \frac{1}{ε} ∫_{−ε}^{ε} f^{ε}(x_{∼}^{ε}, x_{3}^{ε})x_{3}^{ε} dx_{3}^{ε}.

Note that the equations (3) are independent of each other. We can express in a unique way any function defined on P^{ε} as a sum of its even and odd parts with respect to x_{3}^{ε}. The even part of f^{ε} appears only in the equation for ω_{0}, and the odd part of f^{ε} shows up in the equation for ω_{1}. Also, the equation determining ω_{1} is singularly perturbed, but this is not the case for the equation determining ω_{0}. If higher order methods were used, we would have two singularly perturbed independent systems of equations, corresponding to the even and odd parts of u^{ε}(p). Similar splitting also occurs in plate models for linearized elasticity, where, for an isotropic plate, the equations decouple into two independent problems corresponding to bending and stretching of a plate.

The natural question of how close u^{ε}(p) is to u^{ε} is not easy to answer due to the complex influence of ε in both the original and model solutions. We resolve this, not by comparing the exact and model solutions directly, but rather by first looking at the difference between the solutions and their truncated asymptotic expansions, and then comparing both asymptotic expansions. This is possible because the same projection used to define each model can be used to find the first terms of the asymptotic expansion of the model. This allows us to compare corresponding terms of the expansions. Schematically, this is how it works:

u^{ε}  Asymptotic Expansion of u^{ε} Asymptotic Expansion of u^{ε}(p) u^{ε}(p)

Next we develop then the asymptotic expansion for the exact solution, and then do the same for the model solution. Finally we compare them both to find upper bounds for the modeling error.

2 – The asymptotic expansion for the original 3D solution

Our first step to show the influence of ε explicitly is to rewrite (1) in the scaled domain P = Ω × (−1, 1). Let ∂P_{L} = ∂Ω × (−1, 1) and ∂P_{±} = Ω × {−1, 1}. Also, x = (x_{∼}, x_{3}) is a typical point of P, with x_{∼} = x_{∼}^{ε} and x_{3} = ε^{−1}x_{3}^{ε}.

P^{ε}  x_{3} = ε^{−1}x_{3}^{ε} ·  x^{ε} = (x_{∼}, x_{3}^{ε})  P  1 −1 · x = (x_{∼}, x_{3})

In this new domain we define u(ε)(x) = u^{ε}(x^{ε}), and f(x) = f^{ε}(x^{ε}). We conclude from (1) that

(5)  \begin{matrix}(Δ_{2D} +ε^{−2}∂_{33})u(ε) = −f  in P, \\ \frac{∂u(ε)}{∂n} = 0  on ∂P_{±}, \\ u(ε) = 0  on ∂P_{L}. \end{matrix}

We assume that f is independent of ε.

Consider the asymptotic expansion

(6)  u(ε) ∼ u^{0} + ε^{2}u^{2} + ε^{4}u^{4} + · · · .

Formally substituting (6) in (5) and grouping together terms with same power in ε we have

ε^{−2}∂_{33}u^{0} + [Δ_{2D} u^{0} + ∂_{33}u^{2}] + ε^{2}[Δ_{2D} u^{2} + ∂_{33}u^{4}] + ⋯ = −f,

\frac{∂u^{0}}{∂n} + ε^{2} \frac{∂u^{2}}{∂n} + ε^{4} \frac{∂u^{4}}{∂n} + ⋯ = 0 on ∂P_{±}.

It is then natural to require that

∂_{33}u^{0} = 0, (7)

∂_{33}u^{2} = −f − Δ_{2D} u^{0}, (8)

∂_{33}u^{2k} = − Δ_{2D} u^{2k−2}, for all k > 1, (9)

along with the boundary conditions

(10)  \frac{∂u^{2k}}{∂n} = δ_{k1}g on ∂P_{±},  for all k ∈ ℕ.

Equations (7)–(10) define a sequence of Neumann problems in the interval x_{3} ∈ (−1, 1) parametrized by x_{∼} ∈ Ω. If the data for these problems is compatible then the solution can be written as

(11)  u^{2k}(x) = u^{∘}^{2k}(x) + ζ^{2k}(x_{∼}),  for all k ∈ ℕ,

where

(12)  ∫_{−1}^{1} u^{∘}^{2k}(x_{∼}, x_{3}) dx_{3} = 0,

with u^{∘}^{2k} uniquely determined, but ζ^{2k} an arbitrary function of x_{∼} only. From the Dirichlet boundary condition in (5), it would be natural to require that u^{2k} = 0 on ∂P_{L}. This is equivalent to imposing

ζ^{2k} = 0 on ∂Ω, (13)

u^{∘}^{2k} = 0 on ∂P_{L}. (14)

However, in general, only (13) can be imposed and (14) will not hold. We shall correct this discrepancy latter. Now we show that the functions ζ^{2k}, u^{∘}^{2k} (and so u^{2k}) are uniquely determined from (7)–(13). In fact, (7) and (10) yields u^{∘}^{0} = 0. From the compatibility of (8) and (10) we see that

(15)  Δ_{2D} ζ^{0}(x_{∼}) = −\frac{1}{2} ∫_{−1}^{1} f(x_{∼}, x_{3}) dx_{3},

which together with (13), determines ζ^{0} and then, from (11), u^{0}. In view of the compatibility condition (15), u^{∘}^{2} is fully determined by (8) and (10). Next, the Neumann problem (9), (10) admits a solution for k > 1 if and only if ∂_{11}ζ^{2k−2} = 0. But in view of (13), this means ζ^{2k−2} = 0, for k > 1, and then u^{∘}^{2k} is uniquely determined from (9), (10). Note that u^{0} = ζ^{0} and u^{2k} = u^{∘}^{2k} for k ≥ 1.

Observe that u^{0} satisfies all the boundary conditions imposed since u^{∘}^{0} = 0 and so (14) holds for k = 0. In general this is not the case for u^{2}, u^{4}, etc, as they do not vanish on the lateral boundary of the domain (although their vertical integrals do). We introduce then, formally, the boundary corrector

(16)  U ∼ ε^{2}U^{2} + ε^{3}U^{3} + ε^{4}U^{4} + ⋯,

to correct the values of u^{2}, u^{4}, etc. on ∂P_{L}. We expect also that

(17)  \begin{matrix}(Δ_{2D} +ε^{−2}∂_{33})U = 0  in P, \\ \frac{∂U}{∂n} = 0  on ∂P_{±}. \end{matrix}

We hope to pose a boundary corrector problem that is independent of ε. In a two-dimensional beam, it is enough to define a stretched coordinate in the horizontal direction and pose the corrector problem in the semi-infinite strip Σ = ℝ^{+}×(−1, 1). We proceed here analogously, using a new system of (boundary-fitted) horizontal coordinates. In this new system, a point close to the boundary ∂Ω has as coordinates its distance to ∂Ω and the arclength along the boundary. We are able then to define a horizontal stretched coordinate, in the normal direction, and pose, after some work, a sequence of problems in, once more, Σ. The geometry of Ω plays an important role, as we see below.

Following the notation of Chen [2], suppose that ∂Ω is arclength parametrized by z_{∼}(θ) = (X(θ), Y (θ)). Let s_{∼} = (X′, Y′), n_{∼} = (Y′, −X′) denote the tangent and the outward normal of ∂Ω, and define

Ω_{b} = {z_{∼} − ρn_{∼} : z_{∼} ∈ ∂Ω, 0 < ρ < ρ_{0}},

where ρ_{0} is a positive number smaller than the minimum radius of curvature of ∂Ω. With L denoting the arclength of ∂Ω, then

x_{∼}(ρ, θ) = z_{∼}(θ) − ρn_{∼}(θ).

is a diffeomorphism between (0, ρ_{0}) × ℝ/L and Ω_{b}. Extending n_{∼} and s_{∼} to Ω_{b} by

(18)  n_{∼}(ρ, θ) = n_{∼}(θ),  s_{∼}(ρ, θ) = s_{∼}(θ),

then, for α = 1, 2:

∂_{α}θ = \frac{s_{α}}{J(θ)}, ∂_{α}ρ = −n_{α},

where J(ρ, θ) = 1 − ρκ(θ), and κ is the curvature of ∂Ω. Finally, the change of coordinates yields

∂_{α}f = ∂_{θ}f∂_{α}θ + ∂_{ρ}f∂_{α}ρ, for α = 1, 2.

The expression for the Laplacian in these new coordinates follows:

(19)  \begin{matrix}Δ_{2D} U = ∂_{ρρ}U − \frac{κ}{J} ∂_{ρ}U + \frac{1}{J^{2}} ∂_{θθ}U + \frac{ρκ′}{J^{3}} ∂_{θ}U \\ = ∂_{ρρ}U + ∑_{j=0}^{∞} ρ^{j} (a_{1}^{j}∂_{ρ}U + a_{2}^{j}∂_{θθ}U + a_{3}^{j}∂_{θ}U) , \end{matrix}

where we formally replace each coefficient with its respective Taylor expansion, see [1], and

a_{1}^{j} = −[κ(θ)]^{j+1}, a_{2}^{j} = (j + 1)[κ(θ)]^{j}, a_{3}^{j} = \frac{j(j + 1)}{2}[κ(θ)]^{j−1}κ′(θ).

Defining the new variable ρ̂ = ε^{−1}ρ and using the same name for functions different only up to this change of coordinates, we have from (19) that

(20)  Δ_{2D} U = ε^{−2}∂_{ρ̂ρ̂}U + ∑_{j=0}^{∞}(ερ̂)^{j} (a_{1}^{j}ε^{−1}∂ρ̂U + a_{2}^{j}∂_{θθ}U + a_{3}^{j}∂_{θ}U) ,

Aiming to solve (17), we formally use (16) and (20), collect together terms with same order of ε and for k ≥ 2, pose the following sequence of problems parametrized by θ:

(21)

(∂_{ρ̂ρ̂} + ∂_{33})U^{k} = F_{k} in Σ,

\frac{∂U^{k}}{∂n} = 0 on ∂Σ_{±},

U^{k}(0, θ, x_{3}) = u^{k}(0, θ, x_{3}) for x_{3} ∈ (−1, 1),

where

F_{k} = ∑_{j=0}^{k−2} ρ̂^{j} (a_{1}^{j}∂ρ̂U^{k−j−1} + a_{2}^{j}∂_{θθ}U^{k−j−2} + a_{3}^{j}∂_{θ}U^{k−j−2}) ,

The next lemma shows that U^{k} decays to zero exponentially. See [3] for a proof.

Lemma 1. Let U^{k} be defined by (21) for any positive positive k. Then

∫_{t}^{∞} ∫_{−1}^{1}(U^{k})^{2} + (∂ρ̂U^{k})^{2} + (∂_{θ}U^{k})^{2} dx_{3} dρ̂ ≤ c_{k}(f)e^{−α_{k}t}.

Combining (6) and the boundary layer expansion we write

(22) u^{ε}(x^{ε}) ∼ ζ^{0}(x_{∼}^{ε}) + ∑_{k=1}^{∞} ε^{2k}u^{2k}(x_{∼}^{ε}, ε^{−1}x_{3}^{ε}) − χ(ρ) ∑_{k=2}^{∞} ε^{k}U^{k}(ε^{−1}ρ, θ, ε^{−1}x_{3}^{ε}),

where χ(ρ) is a smooth cutoff function identically one if 0 ≤ ρ ≤ ρ_{0}/3 and identically zero if ρ ≥ 2ρ_{0}/3.

We omit many of the details in the results below. Using the convenient hypothesis that f is “smooth”, we will bound their (arbitrarily high) Sobolev norms by a general constant C(f). This allows a simplification of the estimates and no major loss of information occurs. We present the convergence estimates of the truncated asymptotic expansion in the H^{1}(P^{ε}) norm without a proof. Let

e_{N} = u^{ε} − ∑_{k=0}^{N} ε^{2k}u^{2k}(x_{∼}^{ε}, ε^{−1}x_{3}^{ε}) + χ(ρ) ∑_{k=2}^{2N} ε^{k}U^{k}(ε^{−1}ρ, θ, ε^{−1}x_{3}^{ε})

Then we have the result below.

Theorem 2. For any nonnegative integer N, there exists a constant C(f) such that the difference between the truncated asymptotic expansion and the original solution measured in the original domain is bounded as follows:

∥e_{0}∥_{H^{1}(P^{ε})} ≤ C(f)ε^{3/2}, ∥e_{N}∥_{H^{1}(P^{ε})} ≤ C(f)ε^{2N+1}.

Remark. If ∫_{−1}^{1} f(x_{∼}, x_{3}) dx_{3} ≢ 0, then

∥u^{ε}∥_{H^{1}(P^{ε})} ≥ C(f)ε^{1/2},

and then

\frac{∥e_{0}∥_{H^{1}(P^{ε})}}{∥u^{ε}∥_{H^{1}(P^{ε})}} = O(ε), \frac{∥e_{N}∥_{H^{1}(P^{ε})}}{∥u^{ε}∥_{H^{1}(P^{ε})}} = O(ε^{2N+1/2}).

We compile in the table below the estimates for the error between the original solution and the truncated asymptotic expansion, for N ≥ 1.

Table 1. Convergence rates of the truncated asymptotic expansion

u^{ε}  BL  e_{N}(N ≥ 1)  Relative Error
∥ · ∥_{L^{2}(P^{ε})}  ε^{1/2}  ε^{3} ε^{2N+2}(ε^{2N+5/2}) ε^{2N+3/2}(ε^{2N+2})
∥∂_{ρ} · ∥_{L^{2}(P^{ε})} ε^{1/2}  ε^{2} ε^{2N+1}(ε^{2N+5/2}) ε^{2N+1/2}(ε^{2N+2})
∥∂_{θ} · ∥_{L^{2}(P^{ε})} ε^{1/2}  ε^{3} ε^{2N+2}(ε^{2N+5/2}) ε^{2N+3/2}(ε^{2N+2})
∥∂_{x_{3}^{ε}} · ∥_{L^{2}(P^{ε})} ε^{3/2}  ε^{2} ε^{2N+1}(ε^{2N+3/2}) ε^{2N−1/2}(ε^{2N})
∥ · ∥_{H^{1}(P^{ε})}  ε^{1/2}  ε^{2} ε^{2N+1}(ε^{2N+3/2}) ε^{2N+1/2}(ε^{2N+1})

3 – Asymptotic expansion for u^{ε}(p)

To develop an asymptotic expansion for the solution of the hierarchical models, we reason as before, but use weak equations instead of their strong form. We start by posing a problem for the solution of the minimum energy model in the scaled domain P. If we define u(p)(x) = u^{ε}(p)(x^{ε}), then (23)

∫_{P} ∇_{∼} u(p) ∇_{∼} v + ε^{−2}∂_{3}u(p)∂_{3}v dx = ∫_{P} fv dx for all v ∈ H̊^{1}(Ω; ℙ_{p}(−1, 1)).

Considering the asymptotic expansion

(24)  u^{0}(p) + ε^{2}u^{2}(p) + ε^{4}u^{4}(p) + ⋯,

and formally substituting it for u(p) in (23), we can easily conclude that for all v ∈ H̊^{1}(Ω; ℙ_{p}(−1, 1)),

(25)  \begin{matrix}∫_{P} ∂_{3}u^{0}(p)∂_{3}v dx_{∼} = 0, \\ ∫_{P} ∂_{3}u^{2}(p)∂_{3}v dx_{∼} = ∫_{P} (f + Δ_{2D} u^{0}(p))v dx_{∼}, \\ ∫_{P} ∂_{3}u^{2k}(p)∂_{3}v dx_{∼} = ∫_{P} Δ_{2D} u^{2k−2}(p)v dx_{∼},  for k > 1. \end{matrix}

Let ℙ̂_{p}(−1, 1) be the space of polynomials of degree p in (−1, 1) with zero average. Repeating the arguments of the expansion for the exact solution, we set u^{0}(p)(x) = ζ^{0}(x_{∼}) and u^{2}(p)(x_{∼}, ·) as the Galerkin projection of u^{2}(x_{∼}, ·) into ℙ̂_{p}(−1, 1) for almost every x_{∼} ∈ Ω, i.e., (26)

\begin{aligned}∫_{−1}^{1} ∂_{3}u^{2}(p)(x_{∼}, x_{3})∂_{3}v(x_{3}) dx_{3} = ∫_{−1}^{1}[f(x_{∼}, x_{3}) + Δ_{2D} & ζ^{0}(x_{∼})]v(x_{3}) dx_{3}, \\ & for all v ∈ ℙ̂_{p}(−1, 1). \end{aligned}

For any integer k ≥ 2, we define u^{2k}(p)(x_{∼}, ·) ∈ ℙ̂_{p}(−1, 1) by

(27)  ∫_{−1}^{1} ∂_{3}u^{2k}(p)(x_{∼}, x_{3})∂_{3}v(x_{3}) dx_{3} = ∫_{−1}^{1} Δ_{2D} u^{2k−2}(p)(x_{∼}, x_{3})v(x_{3}) dx_{3},

for all v ∈ ℙ̂_{p}(−1, 1), and for almost every x_{∼} ∈ Ω.

The ansatz (24) does not satisfy the Dirichlet boundary conditions at ∂P_{L} and we use then boundary correctors U^{k}(p). These functions are polynomial in the transverse direction, and are defined in the semi-infinite strip Σ. We need to define the spaces

V (Σ, p) = {v ∈ D′(ℝ^{+}; ℙ_{p}(−1, 1)) : ∥ ∇_{∼} v∥_{L^{2}(Σ)} + ∥v(0, ·)∥_{L^{2}(−1,1)} < ∞},

V_{0}(Σ, p) = {v ∈ V (Σ, p) : v(0, ·) = 0}.

For any positive integer k, let U^{k}(p) ∈ V (Σ, p) be the solutions of

(28)

∫_{Σ} ∇_{∼} U^{k}(p) · ∇_{∼} v dρ̂dx_{3} = ∫_{Σ} F_{k}(p)v dρ̂dx_{3} for all v ∈ V_{0}(Σ, p),

U^{k}(p)(0, θ, x_{3}) = u^{k}(p)(0, θ, x_{3}) for all x_{3} ∈ (−1, 1),

F_{k}(p) = ∑_{j=0}^{k−1} ρ̂^{j} (a_{1}^{j}∂ρ̂U^{k−j−1}(p) + a_{2}^{j}∂_{θθ}U^{k−j−2}(p) + a_{3}^{j}∂_{θ}U^{k−j−2}(p)) ,

where u^{k} = 0 for k odd and U^{0}(p) = U^{1}(p) = 0.

A result guaranteeing existence, uniqueness and exponential decay holds for U^{k}(p).

We have finally that

u^{ε}(p)(x^{ε}) ∼ ζ^{0}(x_{∼}^{ε}) + ∑_{k=1}^{∞} ε^{2k}u^{2k}(p)(x_{∼}^{ε}, ε^{−1}x_{3}^{ε}) − χ(ρ) ∑_{k=2}^{∞} ε^{k}U^{k}(p)(ε^{−1}ρ, θ, ε^{−1}x_{3}^{ε}),

where ζ^{0} solves (15).

We present next an estimate, in the H^{1}(P^{ε}) norm, of u^{ε}(p) minus its truncated asymptotic expansion. We would like to remark that this result gives a bound that is uniform in p.

Theorem 3. For any positive integer N, let

e_{2N}(p) = u^{ε}(p) − ∑_{k=0}^{N} ε^{2k}u^{2k}(p) + χ(ρ) ∑_{k=2}^{2N} ε^{k}U^{k}(p).

Then there exists a constant C(f) such that ∥e_{2N}(p)∥_{H^{1}(P^{ε})} ≤ C(f)ε^{2N+1}, for all p ∈ ℕ.

4 – Estimating the error

We start by comparing some terms of the asymptotic expansion of both u^{ε} and u^{ε}(p).

The first three estimates in the lemma below hold since u^{2}(p)(x_{∼}, ·) is the Galerkin projection of u^{2}(x_{∼}, ·) into ℙ̂_{p}(−1, 1) for x_{∼} ∈ Ω. The convergence of the boundary correctors is more involved. It follows from the definition of U^{2} and U^{2}(p). The slow rate of convergence follows from the the singular behaviour of U^{2} close to the corners of the plate (it has a singularity of the type r^{γ} log r, for a certain power γ and where r indicates the distance to the corners).

Lemma 4. For any nonnegative real number s, and for any arbitrarily small δ > 0, there exists a constant C such that

∥u^{2} − u^{2}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{1/2}p^{−2−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))},

∥ ∇_{∼} u^{2} − ∇_{∼} u^{2}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{1/2}p^{−2−s}∥f∥_{H^{1}(Ω;H^{s}(−1,1))},

∥∂_{x_{3}^{ε}}u^{2} − ∂_{x_{3}^{ε}}u^{2}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{−1/2}p^{−1−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))}.

Also, for the first boundary corrector,

∥∂_{x_{3}^{ε}}{χ[U^{2} − U^{2}(p)]}∥_{L^{2}(P^{ε})} + ∥∂_{ρ}{χ[U^{2} − U^{2}(p)]}∥_{L^{2}(P^{ε})}

≤ C_{δ}p^{−6+δ}∥f∥_{L^{2}(∂Ω;H^{5}(−1,1))}.

Finally, we present the convergence results for the model defined by (2). Let P_{0}^{ε} = Ω_{0} × (−ε, ε), where Ω_{0} is an open domain such that Ω̄_{0} ⊂ Ω.

Theorem 5. For any nonnegative real number s, and for any arbitrarily small δ > 0, there exist constants C and C(f) such that the error between u^{ε} and the approximation u^{ε}(p) given by the SP(p) model is bounded as

∥u^{ε} − u^{ε}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{5/2}p^{−2−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))} + C(f)ε^{3},

∥∂_{ρ}[u^{ε} − u^{ε}(p)]∥_{L^{2}(P^{ε})} ≤ C_{δ}ε^{2}p^{−6+δ}∥f∥_{L^{2}(∂Ω;H^{5}(−1,1))} + C(f)ε^{5/2},

∥∂_{θ}[u^{ε} − u^{ε}(p)]∥_{L^{2}(P^{ε})} ≤ Cε^{5/2}p^{−2−s}∥f∥_{H^{1}(Ω;H^{s}(−1,1))} + C(f)ε^{3},

∥ ∇_{∼} u^{ε} − ∇_{∼} u^{ε}(p)∥_{L^{2}(P_{0}^{ε})} ≤ Cε^{5/2}p^{−2−s}∥f∥_{H^{1}(Ω;H^{s}(−1,1))} + C(f)ε^{9/2},

∥∂_{x_{3}^{ε}}u^{ε} − ∂_{x_{3}^{ε}}u^{ε}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{3/2}p^{−1−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))} + C(f)ε^{2},

∥u^{ε} − u^{ε}(p)∥_{H^{1}(P^{ε})} ≤ Cε^{3/2}p^{−1−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))} + C(f)ε^{2}.

Proof. We prove the fifth estimate. Using the triangle inequality, the following holds:

(29)  \begin{matrix}∥ & ∂_{x_{3}^{ε}}u^{ε} − ∂_{x_{3}^{ε}}u^{ε}(p)∥_{L^{2}(P^{ε})} ≤ ∥u^{ε} − ζ^{0} − ε^{2}u^{2} + ε^{2}χU^{2}∥_{H^{1}(P^{ε})} \\ + ε^{2}(∥∂_{x_{3}^{ε}}u^{2} − ∂_{x_{3}^{ε}}u^{2}(p)∥_{L^{2}(P^{ε})} + |χ[U^{2} − U^{2}(p)]|_{H^{1}(P^{ε})}) \\ + ∥u^{ε}(p) − ζ^{0} − ε^{2}u^{2}(p) + ε^{2}χU^{2}(p)∥_{H^{1}(P^{ε})}. \end{matrix}

From Theorems 2 and 3, we have that

∥u^{ε} − ζ^{0} − ε^{2}u^{2} + ε^{2}χU^{2}∥_{H^{1}(P^{ε})}

+ ∥u^{ε}(p) − ζ^{0} − ε^{2}u^{2}(p) + ε^{2}χU^{2}(p)∥_{H^{1}(P^{ε})} ≤ C(f)ε^{3}, and from the exponential decay of the boudary correctors,

|χ[U^{2} − U^{2}(p)]|_{H^{1}(P^{ε})} ≤ C(f).

Finally we apply Lemma 4 to bound ∥∂_{x_{3}^{ε}}u^{2} − ∂_{x_{3}^{ε}}u^{2}(p)∥_{L^{2}(P^{ε})}, and substituting in (29) we have the result. The other estimates follow from similar arguments. □

We summarize the convergence results in the table below. We present only the leading terms of the errors and in parenthesis we show interior estimates if those are better than the global ones.

We need the following notation. For a nonnegative real number s, let

a_{s} = ∥f∥_{L^{2}(Ω;H^{s}(−1,1))}, a_{s}^{1} = ∥f∥_{H^{1}(Ω;H^{s}(−1,1))}, a^{b} = ∥f∥_{L^{2}(∂Ω;H^{5}(−1,1))},

Table 2. Convergence estimates for the SP(p) models

u^{ε} − u^{ε}(p)  Relative Error
∥ · ∥_{L^{2}(P^{ε})} ε^{5/2}p^{−2−s}a_{s} ε^{2}p^{−2−s}a_{s}
∥∂_{ρ} · ∥_{L^{2}(P^{ε})} ε^{2}p^{−6+δ}a^{b} (ε^{5/2}p^{−2−s}a_{s}^{1}) ε^{3/2}p^{−6+δ}a^{b} (ε^{2}p^{−2−s}a_{s}^{1})
∥∂_{θ} · ∥_{L^{2}(P^{ε})}  ε^{5/2}p^{−2−s}a_{s}^{1}  ε^{2}p^{−2−s}a_{s}^{1}
∥∂_{x_{3}^{ε}} · ∥_{L^{2}(P^{ε})}  ε^{3/2}p^{−1−s}a_{s}  p^{−1−s}a_{s}
∥ · ∥_{H^{1}(P^{ε})}  ε^{3/2}p^{−1−s}a_{s}  εp^{−1−s}a_{s}

Section 5 – An alternative variational approach

We now present the SP′(p) models for (1) and the results related to it. Let V′(P^{ε}) = L^{2}(P^{ε}) and S′_{0}(P^{ε}) = {σ ∈ H(div, P^{ε}) : σ · n = 0 on ∂P_{±}^{ε}}. Then we have the following principle.

SP′: (u^{ε}, σ^{ε}) is the unique critical point of

L′(v, τ) = \frac{1}{2} ∫_{P^{ε}} |τ|^{2} dx^{ε} + ∫_{P^{ε}} f^{ε}v dx^{ε} + ∫_{P^{ε}} div τv dx^{ε}

in V′(P^{ε}) × S′_{0}(P^{ε}). Looking for critical points in the spaces V′(P^{ε}, p) = {v ∈

V′(P^{ε}) : deg_{3} v ≤ p} and S′_{0}(P^{ε}, p) = {τ ∈ S′_{0}(P^{ε}) : deg_{3} τ_{∼} ≤ p, deg_{3} τ_{3} ≤

p − 1} we derive the SP′_{1}(p) models. Another option is to choose S′_{0}(P^{ε}, p) = {τ ∈ S′_{0}(P^{ε}) : deg_{3} τ_{∼} ≤ p, deg_{3} τ_{3} ≤ p + 1} instead, yielding the SP′_{2}(p) models. For both SP′_{1}(p) and SP′_{2}(p) models, div S′_{0}(P^{ε}, p) = V (P^{ε}, p) and σ^{ε}(p) is the minimizer of the complementary energy

J_{c}(τ) = \frac{1}{2} ∫_{P^{ε}} |τ|^{2} dx_{∼}

over all τ ∈ S′_{0}(P^{ε}, p) such that div τ = −π_{V′}f^{ε}, where π_{V′}f^{ε} is the orthogonal L^{2} projection on f^{ε} into V′(P^{ε}, p).

We present next some results regarding the SP′_{2}(p) models, omiting the motivations and the proofs.

For the SP′_{2}(p) methods, the asymptotic expansions for u^{ε}(p) and σ_{∼}^{ε}(p) are

u^{ε}(p)(x^{ε}) ∼ ζ^{0}(x_{∼}^{ε}) + ∑_{k=1}^{∞} ε^{2k}u^{2k}(p)(x_{∼}^{ε}, ε^{−1}x_{3}^{ε}) + boundary correctors,

σ(p)(x^{ε}) ∼ ( \begin{matrix}∇_{∼} ζ^{0} \\ 0 \end{matrix} ) (x_{∼}^{ε}) + ∑_{k=1}^{∞} ε^{2k} ( \begin{matrix}σ_{∼}^{2k}(p) \\ ε^{−1}σ_{3}^{2k}(p) \end{matrix} ) (x_{∼}^{ε}, ε^{−1}x_{3}^{ε})

+ boundary correctors.

Equations (13), (15) define ζ^{0}. The other terms are determined as below. With x_{∼} ∈ Ω as a parameter, u^{2}(p)(x) ∈ ℙ̂_{p}(−1, 1) and σ_{3}^{2}(p)(x_{∼}, ·) ∈ ℙ̊_{p+1}(−1, 1), where ℙ̊_{p}(−1, 1) are the polynomials of degree p that vanish on {−1, 1}. Then

\begin{aligned}∫_{−1}^{1} σ_{3}^{2}(p)(x_{∼}, x_{3})τ_{3}(x_{3}) dx_{3} + ∫_{−1}^{1} u^{2}(p)(x_{∼}, x_{3})∂_{3} & τ_{3}(x_{3}) dx_{3} = 0 \\ & for all τ_{3} ∈ ℙ̊_{p+1}(−1, 1), \end{aligned}

\begin{aligned}∫_{−1}^{1} ∂_{3}σ_{3}^{2}(p)(x_{∼}, x_{3})v(x_{3}) dx_{3} = − ∫_{−1}^{1}[f(x_{∼}, x_{3}) + Δ_{2D} & ζ^{0}(x_{∼})]v(x_{3}) dx_{3} \\ & for all v ∈ ℙ̂_{p}(−1, 1). \end{aligned}

Note that u^{2}(p), σ_{3}^{2}(p) are mixed method approximations of u^{2}, ∂_{3}u^{2} (with x_{∼} ∈ Ω as a parameter).

For all integers k ≥ 2, let σ_{3}^{2k}(p)(x_{∼}, ·) ∈ ℙ̊_{p+1}(−1, 1) and u^{2k}(p)(x_{∼}, ·) ∈ ℙ̂_{p}(−1, 1) be such that

\begin{matrix}∫_{−1}^{1} σ_{3}^{2k}(p)(x_{∼}, x_{3})τ_{3}(x_{3}) dx_{3} + ∫_{−1}^{1} u^{2k}(p) & (x_{∼}, x_{3})∂_{3}τ_{3}(x_{3}) dx_{3} = 0 \\ for all τ_{3} ∈ ℙ̊_{p+1}(−1, 1), \end{matrix}

\begin{matrix}∫_{−1}^{1} ∂_{3}σ_{3}^{2k}(p)(x_{∼}, x_{3})v(x_{3}) dx_{3} = − ∫_{−1}^{1} Δ_{2D} u^{2k−2}(p) & (x_{∼}, x_{3})v(x_{3}) dx_{3} \\ for all v ∈ ℙ̂_{p+1}(−1, 1). \end{matrix}

Also, σ_{∼}^{2k} = ∇_{∼} u^{2k}(p). We present some details regarding the boundary corrector problem. We expect a pair of correctors U(p), Ξ(p) with trace U_{0}(p) on ∂P_{L}^{ε} to satisfy (30)

∫_{P^{ε}} Ξ(p) · τ + U(p) div τ dx = ∫_{∂P_{L}^{ε}} U_{0}(p)τ · n dx_{∼} dx_{3} for all τ ∈ S′_{0}(P^{ε}, p),

∫_{P^{ε}} div Ξ(p)v dx = 0 for all v ∈ V′(P^{ε}, p).

We use (18) to define

Ξ_{n}(p)(x^{ε}) = Ξ_{∼}(p)(x^{ε}) · n_{∼}(x_{∼}^{ε}), Ξ_{s}(p)(x^{ε}) = Ξ_{∼}(p)(x^{ε}) · s_{∼}(x_{∼}^{ε}),

τ_{n}(p)(x^{ε}) = τ_{∼}(p)(x^{ε}) · n_{∼}(x_{∼}^{ε}), τ_{s}(p)(x^{ε}) = τ_{∼}(p)(x^{ε}) · s_{∼}(x_{∼}^{ε}),

in Ω_{b}. Then, a long but straightforward computation shows that

div Ξ_{∼}(p) = ∂_{ρ}Ξ_{n}(p) + \frac{1}{J} ∂_{θ}Ξ_{s}(p) − \frac{κ}{J} Ξ_{n}(p).

Hoping that the correctors will decay very quickly, we, in a first step, pose (30) in Ω_{b} using the boundary fitted coordinates (ρ, θ, x_{3}^{ε}). Next, we use the “stretched” (in the normal and vertical directions) variables (ρ̂, θ, x_{3}) in order to pose a ε-independent sequence of corrector problems, and define

Ξ̂_{n}(p)(ρ̂, θ, x_{3}) = εΞ_{n}(p)(ρ, θ, x_{3}^{ε}),  Ξ̂_{s}(p)(ρ̂, θ, x_{3}) = Ξ_{s}(p)(ρ, θ, x_{3}^{ε}),
Ξ̂_{3}(p)(ρ̂, θ, x_{3}) = εΞ_{3}(p)(ρ, θ, x_{3}^{ε}).

Similar definitions hold for τ̂_{n}(p), τ̂_{s}(p) and τ̂_{3}(p). The motivation for multiplying Ξ_{n}(p) and Ξ_{3}(p) by ε is that we expect them to “behave” as ε^{−1}, after all they approximate ∂_{ρ}U and ∂_{3}U in P^{ε}. All the above described transformations lead to

\begin{aligned}_{∫Q̂} & [ε^{−2}Ξ̂_{n}(p)τ̂_{n} + Ξ̂_{s}(p)τ̂_{s} + ε^{−2}Ξ̂_{3}(p)τ̂_{3} + U(p)(ε^{−2}∂ρ̂τ̂_{n} + \frac{1}{J} ∂_{θ}τ̂_{s} + ε^{−2}∂_{3}τ̂_{3})]J \\ & − ε^{−1}κU(p)τ̂_{n} dQ̂ = ∫_{0}^{2π} ∫_{−1}^{1} U_{0}(p)(0, θ, x_{3})τ̂_{n}(0, θ, x_{3}) dx_{3} dθ, \end{aligned} ; ∫Q̂[ε^{−2}∂ρ̂Ξ̂_{n}(p) + \frac{1}{J} ∂_{θ}Ξ̂_{s} + ε^{−2}∂_{3}Ξ̂_{3} − ε^{−1} \frac{κ}{J} Ξ̂_{n}(p)]vJ dQ̂ = 0,

where Q̂ = ℝ^{+} × (0, 2π) × (−1, 1) is a semi-infinite quadrilateral domain with the union of its top and bottom boundaries given by ∂Q̂_{±} = ℝ^{+} × (0, 2π) × {−1, 1}, and

τ ∈ {τ ∈ H(div, Q̂) : τ_{3} = 0 on ∂Q̂_{±}, deg_{3} τ_{∼} ≤ p, deg_{3} τ_{3} ≤ p + 1},

v ∈ {v ∈ L^{2}(Q̂) : deg_{3} v ≤ p}.

Replacing τ̂_{n} by τ̂_{n}/J, τ̂_{3} by τ̂_{3}/J, and v by v/J, formally substituting the Taylor series of the coefficients and

U(p)(x) ∼ ε^{2}U^{2}(p)(x) + ε^{3}U^{3}(p)(x) + ε^{4}U^{4}(p)(x) + …,
Ξ̂(p)(x) ∼ ε^{2}Ξ̂^{2}(p)(x) + ε^{3}Ξ̂^{3}(p)(x) + ε^{4}Ξ̂^{4}(p)(x) + …
U_{0}(p) ∼ ε^{2}u^{2}(p) + ε^{3}u^{3}(p) + ε^{4}u^{4}(p) + …,

where u^{k}(p) = 0 for k odd, we arrive at the following sequence of problems, parametrized by θ ∈ ℝ/L and defined in the semi-infinite strip Σ:

∫_{Σ} Ξ̂_{n}^{k}(p)τ̂_{n} + Ξ̂_{3}^{k}(p)τ̂_{3} + U^{k}(p) (∂ρ̂τ̂_{n} + ∂_{3}τ̂_{3}) dρ̂dx_{3}

= − ∫_{−1}^{1} u^{k}(p)(0, θ, x_{3})τ̂_{n}(0, x_{3}) dx_{3} for all τ_{∼} ∈ S_{∼}′_{0}(Σ, p),

∫_{Σ}[∂ρ̂Ξ̂_{n}^{k}(p) + ∂_{3}Ξ̂_{3}^{k}(p)]v dρ̂dx_{3} = ∫_{Σ} G_{k}(p)v dρ̂dx_{3}  for all v ∈ V′(Σ, p),

Ξ̂_{s}^{k}(p) = ρ̂κ(θ)Ξ̂_{s}^{k−1}(p) + ∂_{θ}U^{k}(p),

G_{k}(p) = ∑_{j=0}^{k−2} ρ̂^{j} (a_{1}^{j}Ξ̂_{n}^{k−j−1}(p) + a_{2}^{j}∂_{θθ}U^{k−j−2}(p) + a_{3}^{j}∂_{θ}U^{k−j−2}(p)) .

THE POISSON PROBLEM IN A THREE-DIMENSIONAL PLATE 13

Finally,

u^{ε}(p)(x^{ε}) ∼ ζ^{0}(x_{∼}^{ε}) + ∑_{k≥1} ε^{2k}u^{2k}(p)(x_{∼}^{ε}, ε^{−1}x_{3}^{ε}) − χ(ρ) ∑_{k≥2} ε^{k}U^{k}(p)(ε^{−1}ρ, θ, ε^{−1}x_{3}^{ε}),

\begin{aligned}σ^{ε}(p)(x^{ε}) ∼ ( \begin{matrix}∇_{∼} ζ^{0} \\ 0 \end{matrix} ) (x_{∼}^{ε}) & + ∑_{k≥1} ε^{2k} ( \begin{matrix}σ_{∼}^{2k}(p) \\ ε^{−1}σ_{3}^{2k}(p) \end{matrix} ) (x_{∼}^{ε}, ε^{−1}x_{3}^{ε}) \\ & − χ(ρ) ∑_{k≥2} ε^{k} ( \begin{matrix}ε^{−1}Ξ̂_{n}^{k}(p)n_{∼} + Ξ̂_{s}^{k}(p)s_{∼} \\ ε^{−1}Ξ̂_{3}^{k}(p) \end{matrix} ) (ε^{−1}ρ, θ, ε^{−1}x_{3}^{ε}). \end{aligned}

We present next the various error estimates. Theorem 6. For any nonnegative integer N, there exists a constant C(f) such that

∥u^{ε}(p) − ζ^{0} − ∑_{k=1}^{N} ε^{2k}u^{2k}(p) + χ(ρ) ∑_{k=2}^{2N} ε^{k}U^{k}(p)∥_{L^{2}(P^{ε})}

+ ∥σ^{ε}(p) − ( \begin{matrix}∇_{∼} ζ^{0} \\ 0 \end{matrix} ) − ∑_{k=1}^{N} ε^{2k} ( \begin{matrix}σ_{∼}^{2k}(p) \\ ε^{−1}σ_{3}^{2k}(p) \end{matrix} )

+ χ(ρ) ∑_{k=2}^{2N} ε^{k} ( \begin{matrix}ε^{−1}Ξ̂_{n}^{k}(p)n_{∼} + Ξ̂_{s}^{k}(p)s_{∼} \\ ε^{−1}Ξ̂_{3}^{k}(p) \end{matrix} )∥_{L^{2}(P^{ε})} ≤ C(f)ε^{2N+1}

The next two lemmas estimate the difference between the first terms of the asymptotic expansion of u^{ε} and u^{ε}(p). Lemma 7. For any nonnegative real number s, and for any arbitrarily small δ > 0, there exists a constant C such that

∥u^{2} − u^{2}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{1/2}p^{−2−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))},

∥ ∇_{∼} u^{2} − ∇_{∼} u^{2}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{1/2}p^{−2−s}∥f∥_{H^{1}(Ω;H^{s}(−1,1))},

∥σ_{3}^{2} − σ_{3}^{2}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{1/2}p^{−1−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))}, ∥∂_{x_{3}^{ε}}σ_{3}^{2} − ∂_{x_{3}^{ε}}σ_{3}^{2}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{−1/2}p^{−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))},

|χ[Ξ̂_{n}^{2} − Ξ̂_{n}^{2}(p)]|_{L^{2}(P^{ε})} + |χ[Ξ̂_{3}^{2} − Ξ̂_{3}^{2}(p)]|_{L^{2}(P^{ε})} ≤ C_{δ}εp^{−5+δ}∥f∥_{L^{2}(∂Ω;H^{4}(−1,1))}, where σ_{3}^{2}(x) = ∂_{x_{3}}u^{2}(x), Ξ̂_{3}^{2}(x) = ∂_{x_{3}}U^{2}(x).

We end this note by presenting the convergence results for the SP′(p) model. Theorem 8. For any nonnegative real number s, and for any arbitrarily small δ > 0, there exist constants C and C(f) such that the following bounds hold:

∥u^{ε} − u^{ε}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{5/2}p^{−2−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))} + C(f)ε^{3},

∥σ_{∼}^{ε} · n_{∼} − σ_{∼}^{ε}(p) · n_{∼}∥_{L^{2}(P^{ε})} ≤ C_{δ}ε^{2}p^{−5+δ})∥f∥_{L^{2}(∂Ω;H^{4}(−1,1))} + C(f)ε^{5/2},

∥σ_{∼}^{ε} · s_{∼} − σ_{∼}^{ε}(p) · s_{∼}∥_{L^{2}(P^{ε})} ≤ Cε^{5/2}p^{−2−s}∥f∥_{H^{1}(Ω;H^{s}(−1,1))} + C(f)ε^{3},

∥σ_{∼}^{ε} − σ_{∼}^{ε}(p)∥_{L^{2}(P_{0}^{ε})} ≤ Cε^{5/2}p^{−2−s}∥f∥_{H^{1}(Ω;H^{s}(−1,1))} + C(f)ε^{9/2},

∥σ_{3}^{ε} − σ_{3}^{ε}(p)∥_{L^{2}(P^{ε})} ≤ Cε^{3/2}p^{−1−s}∥f∥_{L^{2}(Ω;H^{s}(−1,1))} + C(f)ε^{2},

where σ^{ε}(x^{ε}) = ∇ u^{ε}(x^{ε}).

References

[1] D. N. Arnold and R. S. Falk, Asymptotic analysis of the boundary layer for the Reissner– Mindlin plate model, SIAM J. Math. Anal. 27 (1996), 468–514.

[2] C. Chen, Asymptotic Convergence Rates for the Kirchhoff Plate Model, Ph.D. Dissertation, The Pennsylvania State University, University Park, Pa., 1995.

[3] A. L. Madureira, Asymptotics and Hierarchical Modeling of Thin Plates, Ph.D. Dissertation, The Pennsylvania State University, University Park, Pa., 1999.

Laboratório Nacional de Computação Científica, Petrópolis, Rio de Janeiro, Brazil. CEP: 25651-070, Caixa Postal: 95113. http://www.lncc.br/∼alm

E-mail address: alm@lncc.br