Journal of Interpolation and Approximation in Scientific Computing 2015 No.2 (2015) 128-136 Available online at www.ispacs.com/jiasc Volume 2015, Issue 2, Year 2015 Article ID jiasc-00084, 9 Pages doi:10.5899/2015/jiasc-00084 Research Article
The Effect of Numerical Integration on the Finite Element Approximation of a Second Order Elliptic Equation with Highly Oscillating Coefficients Bienvenu Ondami∗ Universite´ Marien NGOUABI, Faculte´ des Sciences et Techniques, BP. 69, Brazzaville, Congo. c Bienvenu Ondami. This is an open access article distributed under the Creative Commons Attribution License, which permits Copyright 2015 ⃝ unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract In this paper, we have studied the effect of numerical integration on the Finite Element Method (FEM) based on the usual Ritz approximation using continuous piecewise linear functions, in the context of a class of second order elliptic boundary value problems with highly periodically oscillating coefficients. An error estimate depending on ε the parameter involved in the periodic homogenization and h the mesh size is established. Numerical results for one dimensional problem are presented. It is shown that when εh is a positive integer then the method gives different results depending on the shape of the coefficients and the numerical integration. Specifically we obtain perfectly correct results in some cases and completely false in other cases. Keywords: Numerical Integration, Homogenization, Elliptic Equations, Finite Elements.
1
Introduction
There are many practical computational problems with highly oscillatory solutions e.g. computation of flow in heterogeneous porous media for petroleum and groundwater reservoir simulation (see, e.g.,[8] and the bibliographies therein). If a porous medium with a periodic structure is considered, with the size of the period is small enough compared to the size of the reservoir, and denoting their ratio by ε an asymptotic analysis, as ε −→ 0, is required. Using the homogenization tools (see, e.g.,[3], [4], [9], [12]) the original equation describing this problem can be replaced by an effective or homogenized equation modeling some average quantity without the oscillations. Whenever effective equations are applicable they are very useful for computational purposes. There are however many situations for which ε is not sufficiently small so that the effective equations are not practical. In this cases the original equation has to be approximated directly. When the approximation is done by a finite element method, numerical integration is almost unavoidable. In this paper we will study the effect of numerical integration when a finite element method is used to approximate the original equation. Specific problems considered here include a linear elliptic equation in divergence form with highly periodically oscillating coefficients. The purpose of this paper is to show that even when εh is an integer the numerical solution with numerical integration effects can be correct in some cases depending on the shape of coefficients. The numerical approximation partial differential equations with highly oscillating coefficients has been a problem ∗ Corresponding
author. Email address:
[email protected], Tel: +242 069242063
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Journal of Interpolation and Approximation in Scientific Computing 2015 No.2 (2015) 128-136 http://www.ispacs.com/journals/jiasc/2015/jiasc-00084/
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of interest for many years and many methods have been developed (see, e.g., [1], [6], [7], [10], [11], [13] and the bibliographies therein). The paper is organized as follows. In section 2 we have given a short description of the boundary value problem used in this study and the classical homogenization result related to this problem. In section 3, the conforming finite element method with numerical integration of the problem is presented as well as an error estimate. Numerical simulations for the one-dimensional problem comparing the approximation obtained by that method and the approximation of the homogenized problem are presented in section 4. Lastly, some concluding remarks are presented in section 5. 2 Preliminaries and notations Let Ω ⊂ Rn (n = 1, 2) be a bounded polygonal convex domain with a periodic structure and smooth boundary Γ. More precisely, we shall scale this periodic structure by a parameter ε which represents the ratio of the cell size to the n
size of the whole region Ω and we assume that 0 < ε 6 < B ξ , ξ >, ∀ ξ ∈ Rn , where < ., . > denotes the standard inner product on Rn . We define the set of bounded, measurable, uniformly positive definite tensor on Ω by M (α , β ; Ω) := {A ∈ L∞ (Ω; S) / α I ≤ A ≤ β I a.e. in Ω} , where α and β are positive constants such that α ≤ β . Throughout the paper, we use the Sobolev spaces { } H m (Ω) = v ∈ L2 (Ω) , ∂ α v ∈ L2 (Ω) , ∀ |α | ≤ m , n
with the multi-index (α1 , α2 , ..., αn ) , |α | := ∑ αi , and ∂ α := ∂ α1 ∂ α2 ...∂ αn . The H m −norm and semi-norm of any v ∈ H m (Ω) are respectively defined by
i=1
∥v∥2m,Ω := |v|2m,Ω := while the
L2 −
∑
∥∂ α v∥2L2 (Ω) ,
∑
∥∂ α v∥2L2 (Ω) ,
|α |≤m
|α |=m
norm is ∥v∥2L2 (Ω)
∫
=
Ω
|v|2 dx.
In addition, we denote by H01 (Ω) ⊂ H 1 (Ω) the subspace consisting of zero-trace functions. We also use the space W 1,∞ (Ω) equipped with the norm ∥.∥∞,Ω . We consider the following elliptic boundary value problem: { −div (K ε (x) ▽uε ) = f in Ω, (Pε ) uε = 0 on Γ, where f is a function in L2 (Ω). If K ∈ M (α , β ; Ω) then K ε ∈ M (α , β ; Ω). By homogenization theory (see, e.g.,[4],[9], [12] ), it follows that uε ⇀ u
in
H01 (Ω) weakly,
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Journal of Interpolation and Approximation in Scientific Computing 2015 No.2 (2015) 128-136 http://www.ispacs.com/journals/jiasc/2015/jiasc-00084/
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where u satisfies the following homogenized equation, { −div (K ∗ ▽u) = f (P0 ) u = 0
in Ω, on Γ,
and the entries of the matrix K ∗ are given by (K ∗ )i j =
1 |Y |
∫ Y
− − K(y) [∇wi + → ei ] . [∇w j + → ej ]
1 ≤ i, j ≤ n,
with w j , j = 1, . . . , n is the solution of the so-called local or cell problem defined by: { w j ∈ Hp1 (Y )/R, − −div [K (y) (∇w j + → e j )] = 0 in Y. − Here → e is the jth standard basis vector of Rn . We denote by C∞ (Y ) the space of infinitely differentiable functions in j
p
Rn that are periodic of period Y. Then H p1 (Y ) is the completion for the norm of H 1 (Y ) of C∞ p (Y ). The effective tensor K ∗ is still symmetric and positive definite but in general cases, even with the permeability at the microscopic scale in the porous medium being isotropic, we may have an effective tensor which is significantly anisotropic. In porous medium flow, the problem (Pε ) results from Darcy’s law and continuity for a single phase, incompressible flow through a horizontal heterogeneous porous medium with periodic structure. 3 The Finite Element Approximation with Numerical Integration The variational form of the problem (Pε ) is given as { uε ∈ H01 (Ω) , ∫ ∫ a(uε , v) = Ω K ε (x)uε vdx = Ω f vdx
(3.1)
∀v ∈ H01 (Ω) .
We will make the following assumptions: (A1) K ε ∈ M (α , β ; Ω) , (A2) K ε ∈ W 1,∞ (Ω) , (A3) f ∈ L2 (Ω) . It is well known that (3.1) has a unique solution uε ∈ H 2 (Ω) ∩ H01 (Ω) . The finite element method: Let (Th )h>0 be a regular triangulation of Ω where h is the mesh size, and let ( ) Vh = {vh ; vh ∈ C0 Ω , vh is affine on each T ∈ Th , vh = 0 on Γ} be the finite element subspace. The finite element approximation to the solution uε ∈ H01 (Ω) of (3.1) is given by the finite method (FEM), { uε h ∈ Vh , (3.2) a(uε h , vh ) = f (vh ) ∀vh ∈ Vh where f (vh ) =
∫
Ω
f vh dx. The following error estimate is well-known, see, e.g., [1], [10]: ( ) 1 |uε − uε h |1,Ω ≤ C h 1 + . ε
(3.3)
Note that a(uε h , vh ) and f (vh ) contain definite integrals that can be computed numerically. Consequently, the FEM with numerical integration is given by { ∗ uε h ∈ Vh , (3.4) a∗ (uε∗h , vh ) = f ∗ (vh ) ∀vh ∈ Vh
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Journal of Interpolation and Approximation in Scientific Computing 2015 No.2 (2015) 128-136 http://www.ispacs.com/journals/jiasc/2015/jiasc-00084/
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where a∗ (., .) and f ∗ (.) are the computation results by numerical integration of a(uε h , vh ) and f (vh ) respectively. In almost all finite element compilation, numerical integration is unavoidable. Consequently, uε∗h instead of uε h is available. Throughout the paper, we denote by C generic constants, even if they take different values at different places. We now state the main result of this section. Theorem 3.1. Assume that the quadrature rules for computing a∗ (., .) and f ∗ (.) are exact on all polynomials of degrees less than or equal to 1 and let f ∈ H 2 (Ω) . Then the following estimate is valid: ∥uε − uε∗h ∥1,Ω
( ) 1 2 ≤C h 1+ ε
(3.5)
where C is a constant independent of ε and h. Proof. We do the proof for n = 2 (two-dimensional). The proof of this result is obtained by using [2, Theorem 2.1] and [5, Theorem 4.1.6]. Indeed, these theorems imply ( ) 2
∗ ε
∥uε − u ∥ ≤ Ch ∑ Ki j ∥uε ∥ + ∥ f ∥ (3.6) ε h 1,Ω
i, j=1
∞,Ω
2,Ω
2,Ω
where C is a constant independent of ε and h. Now we are going to estimate the term 2
∑
i, j=1
Using [1, Theorm 2], we deduce that
ε
Ki j ∥uε ∥ . 2,Ω ∞,Ω
( ) 1 ∥uε ∥2,Ω ≤ C 1 + ε
where C is a constant independent of ε . By the definition of ∥.∥∞,Ω we have ( ) 2
∂ K x1 , x2
( )
ε 2
ij ε ε x1 x2 2
Ki j = Ki j ,
∞ +
∞,Ω ε ε ∂ x1 L (Ω)
L∞ (Ω)
( ) 2
∂ K x1 , x2
ij ε ε
+
∂ x2
∞
, ∀ i, j = 1, 2.
L (Ω)
Therefore we get
ε 2
ε 2
ε 2
ε 2
∂K ∂K 1
Ki j
Ki j ∞ + 1 i j
ij = + , ∀ i, j = 1, 2,
2 2 ∞,Ω L (Ω) ε ∂ x1 L∞ (Ω) ε ∂ x2 L∞ (Ω) ( ) ( )
ε 2 1 1 2
Ki j ≤C 1+ 2 ≤C 1+ . ∞,Ω ε ε Finally, we deduce that
( )
ε 1
Ki j ≤C 1+ . ∞,Ω ε
Thus, we obtain
( )
ε 1 2
∥u ∥ K ≤ C 1 + . ∑ i j ∞,Ω ε 2,Ω ε i, j=1 2
We can now deduce that ( )
ε 1 2
∥u ∥ ∥ ∥ + f ≤ C 1 + . K ∑ i j ∞,Ω ε 2,Ω 2,Ω ε i, j=1 2
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( ) 1 2 ∥uε − uε∗h ∥1,Ω ≤ Ch 1 + , ε
which is the desired result. 2 Remark 3.1. For the one-dimensional problem, the result of theorem 3.1 could be obtained by simple calculations. Furthermore, the error estimate for higher order approximation is easy to be derived like in [10]. 4 Numerical Results In this section we present numerical results for one-dimensional problem comparing the finite element method described in this paper and the example of exact solution and homogenized solution(HOMOG). In this case the homogenized coefficient is computed as the harmonic mean. In the first example, we compare the approximate solution to the exact solution and the homogenized solution.The results obtained are pefectly correct even when εh is a positive integer. In the second example, we compare the approximate solution to the homogenized solution. The results obtained show that the approximate solution is false when εh is a positive integer. The first test problem involved simulation with the coefficient: K (y) =
1 , a + b sin(2π y)
0 < b < a, y ∈ [0, 1],
and f = 1. In this case, the exact solution is: ( )] [ ) ( )] [ ( ε −ax2 bε 2π x bε 2π x 2π x cos x− C1ε cos + sin +C2ε , uε (x) = + aC1ε + 2 2π ε 2π ε 2π ε where C1ε
=
a 2
(4.7)
[ ( ) ( )] − 2bπε cos 2επ − 2επ sin 2επ [ ( )] , a + 2bπε 1 − cos 2επ
and C2ε =
bε ε C . 2π 1
The homogenized solution is: u(x) =
ax (1 − x) . 2
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Journal of Interpolation and Approximation in Scientific Computing 2015 No.2 (2015) 128-136 http://www.ispacs.com/journals/jiasc/2015/jiasc-00084/
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Figure 1: Test problem 1: a = 1, b = 1/2. Those results show the convergence of the numerical solution toward the exact solution when well known, see, e.g., [1], [10].
h ε
