Moving Mesh Method with Error-Estimator-Based Monitor and Its ...

Journal of Scientific Computing, Vol. 21, No. 1, August 2004 (© 2004)

Moving Mesh Method with Error-Estimator-Based Monitor and Its Applications to Static Obstacle Problem R. Li, 1 W. B. Liu, 2 and H. P. Ma 3 Received May 27, 2003; accepted (in revised form) July 22, 2003 The main objective of this work is to demonstrate that sharp a posteriori error estimators can be employed as appropriate monitor functions for moving mesh methods. We illustrate the main ideas by considering elliptic obstacle problems. Some important issues such as how to derive the sharp estimators and how to smooth the monitor functions are addressed. The numerical schemes are applied to a number of test problems in two dimensions. It is shown that the moving mesh methods with the proposed monitor functions can effectively capture the free boundaries of the elliptic obstacle problems and reduce the numerical errors arising from the free boundaries. KEY WORDS: Finite element method; moving mesh method; a posteriori error estimator; obstacle problem.

1. INTRODUCTION Moving mesh methods have important applications for a variety of physical and engineering problems that require extremely fine meshes in a small portion of physical domain. Several properties are responsible for the increasing popularity of the moving mesh methods. Successful implementation of the adaptive strategy can increase the accuracy of the numerical approximation and also decrease the computational cost. Moreover, 1

School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China. E-mail: [email protected] 2 Institute of Computational and Applied Mathematics, Xiangtan University, Hunan, People’s Republic of China. E-mail: [email protected] 3 Department of Mathematics, Shanghai University, Shanghai 200436, People’s Republic of China. E-mail: [email protected] 31 0885-7474/04/0800-0031/0 © 2004 Plenum Publishing Corporation

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developing robust moving mesh methods for multi-dimensional problems becomes necessary since much less storage is required. There are several important moving mesh strategies for solving partial differential equations, including the variational approach [32, 7], moving mesh finite element methods [27, 11, 5], moving mesh PDE method [28, 21, 6], and moving mesh methods based on harmonic mapping [12, 23]. So far most of moving mesh techniques are designed to handle problems which develop dynamically singular or nearly singular solutions in fairly localized regions. Therefore, a monitor function (or indicator) associated with the solution gradient, such as M=`1+a 1 |u| 2+a 2 |Nu| 2, has been used most frequently in the moving mesh computations. With proper choice of the constants a 1 and a 2 , the gradient-based monitor can handle a variety of physical problems. How should we choose a monitor function when the singular region is of different nature? In such a case, the gradient-based monitor may not work, as the solution region requiring adaptivity may not have large gradient. For example, numerical schemes may loose accuracy when approximating problems with free boundaries, and as a result it is desirable to move more points to these free boundaries in order to reduce the leadingorder errors arising from them. Since the solution gradients may not large near the free boundaries, the traditional gradient-based monitors sometimes will not work well. In recent years adaptive mesh refinement for finite element approximation have been extensively investigated, which ensure a higher density of nodes in certain area of the given domain where the solution is more difficult to approximate (such as large solution gradients, free/moving boundaries). At the heart of any adaptive finite element refinement schemes are some appropriate a posteriori error estimators. The decision of whether further refinement of meshes is necessary is based on the estimate of the discretization error. If further refinement is to be performed then the a posteriori error estimators are used as a guide as to show the refinement might be accomplished most efficiently. In these h-version adaptive finite element computations, some mathematically justified a posteriori error estimators play the role as the monitor (indicator); in particular, a posteriori error estimators of residual type are widely used in adaptive mesh refinement for various problems, see, e.g., [2] and [31]. Now the natural question is can we use them as monitor functions in the moving mesh methods? The main objective of this article is to address the above question. More precisely, we will consider the following: • (Q1): Can some error estimators of residual type be used as monitor function for moving mesh methods?

Moving Mesh Method and Applications to Obstacle Problem

33

• (Q2): If the answer to (Q1) is yes, then how do we efficiently implement such an error estimator as a useful monitor function? To better illustrate our approaches to the above questions, we consider the following elliptic obstacle problem:

˛

− div(A Nu) \ f, u \ f,

(1.1)

(−div(A Nu) − f)(u − f)=0, where A is a given matrix function, and f and f are given functions. The above problem is a classical one for which several a posteriori error estimators have been derived, see, e.g., [1, 3, 19, 9, 26]. These error estimators are reliable in the sense ||u − u h || [ Cg˜

(1.2)

where g˜ is the error estimator in terms of computational quantities related to the discrete solution u h . A sub-class of the error estimators are not only upper-bounds but also lower-bounds: cg+O(E) [ ||u − u h || [ Cg,

(1.3)

where O(E) ° g. We call g an equivalent error estimator. For the test problem (1.1), we try to address the questions (Q1) and (Q2) raised above. We will demonstrate that the moving mesh computations with upperbounds monitor may not efficiently move more elements to the free boundary regions of the obstacle problem (1.1). However, equivalent a posteriori error estimators can indeed play a role as appropriate monitor functions in the moving mesh methods. These will be illustrated by our numerical experiments. We will also show how to derive the equivalent error-estimates in this work. The details of how to implement the moving mesh methods with the error-estimator-type monitors will be also provided. It is pointed out that some similar ideas have been used by the authors to deal with distributed elliptic optimal control problems [22]. The plan of the paper is as follows: In Sec. 2, we formulate finite element approximation for the obstacle problems and briefly discuss the moving mesh methods to be used. In Sec. 3, an equivalent error estimator will be derived for the obstacle problem (1.1). In Sec. 4, some details of the computational implementations will be discussed. Several numerical examples are considered in Sec. 5.

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2. PRELIMINARIES Let W be a bounded open set in R 2 with a Lipschitz boundary “W. In this paper we adopt the standard notation W m, q(W) for Sobolev spaces q on W with norm || · ||m, q, W and semi-norm | · |m, q, W . We set W m, 0 (W) — {w ¥ m, q m, 2 m W (W) : w|“W =0} and denote W (W) by H (W) with norm || · ||m, W and semi-norm | · |m, W . In addition c or C denotes a general positive constant independent of the maximum diameter of the finite elements. We first review some known results for variational inequalities and in particular, the obstacle problem (1.1). Let f ¥ L 2(W), g ¥ H 1(W), and Vg ={v ¥ H 1(W) : v − g ¥ H 10 (W)}, a(u, v)=F (A Nu, Nv)R 2 ,

-u, v ¥ H 1(W),

W

(f, v)=F fv,

-v ¥ H 1(W),

W

where A( · )=(ai, j ( · )) is a 2 × 2 matrix function, with aij ¥ L .(W). Assume that there is a constant c > 0 such that X tAX \ c ||X|| 2R 2

-X ¥ R 2.

(2.1)

Let K be a closed convex set in Vg . Then the following inequality is commonly referred to as variational inequality: For given f, g, find u ¥ K such that a(u, v − u) \ (f, v − u),

-v ¥ K.

(2.2)

When the matrix A is symmetric, the above variational inequality problem is equivalent to the following minimization problem (see, e.g., [18])): min 12 a(u, u) − (f, u).

(2.3)

u¥K

It is well known that the variational inequality (2.2) has a unique solution in K. In particular, if f ¥ H 1(W) satisfying (g − f)|“W \ 0, then the following variational inequality is a weak formulation of the obstacle problem (1.1) with the boundary condition u=g on “W: For given f, f, g, find u ¥ Kg (f) such that a(u, v − u) \ (f, v − u),

-v ¥ Kg (f),

where Kg (f)={v ¥ Vg : v \ f}.

(2.4)


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It is well known that the variational inequality (2.4) has a unique solution u ¥ Kg (f), and furthermore that the solution is regular, say in H 2(W), if the data are so. If u ¥ H 2(W), then (2.4) is equivalent to (1.1) (see, e.g., [15]). In the rest of the paper we assume that the solution of (2.4) is continuous, which implies that the non coincidence set of the obstacle problem (1.1) W+={x ¥ W : u(x) > f(x)} is open, and the coincidence set W −={x ¥ W : u(x)=f(x)} is closed. The boundary of the coincidence set in W is referred to as free boundary. For ease of exposition we only consider a special obstacle problem where f=0: For given f, g, find u ¥ Kg (0) such that a(u, v − u) \ (f, v − u),

-v ¥ Kg (0).

(2.5)

Note that here we have g|“W \ 0. More general obstacle problems may be transferred into this form by using the transformation u g=u − f. Let W h be a polygonal approximation to W with boundary “W h. Let h ¯ h= T be a partitioning of W h into disjoint regular triangular y, so that W h 1 y ¥ T h ¯y . Each element has at most one face on “W , and ¯y and ¯y Œ have either only one common vertex or a whole face if y and yŒ ¥ T h. We further require that Pi ¥ “W h S Pi ¥ “W where {Pi } (i=1 · · · J) is the vertex set associated with the triangulation T h. Let h y denote the maximum diameter of the element y in T h. We shall only discuss the piecewise linear element due to the limited higher order regularity for the solution of a variational inequality. Asso¯ h), such that q|y ciated with T h is a finite dimensional subspace V h of C 0(W h h are affine functions for all q ¥ V and y ¥ T . For ease of exposition we will assume that W h=W. Let V h0 ={q ¥ V h : q(Pi )=0, -Pi ¥ “W}, V hg ={q ¥ V h : q(Pi )=g(Pi ), -Pi ¥ “W}. We now consider the finite element approximation for (2.5). The finite element approximation of (2.5) reads: Find u h ¥ K hg (0) such that a(u h , vh − u h ) \ (f, vh − u h )

-vh ¥ K hg (0).

(2.6)

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Li, Liu, and Ma

It is known that there is a unique solution to this approximation scheme which converges to the solution of (2.5) as h Q 0, if K hg (0) converges to Kg (0) in the sense given in, say, [16]. In the following we will briefly review the moving mesh methods based on finite element approximations. The details of the methods can be found in Li et al. [23, 24], where the methods were proposed to solve timedependent PDEs. In this work, the moving mesh methods introduced in [23] are modified to solve the time-independent variational inequalities. The basic idea of the moving mesh method (also called r-method) is to equidistribute some functional (i.e., monitor) of the numerical solution on the mesh. Let (x, y) and (a, b) denote the physical and computational coordinates, respectively. A one-to-one coordinate transformation from the computational domain Wc to physical domain Wp is provided by the minimizer of a functional of the following form: E(a, b)=12 F

Wp

(Na TG 11 Na+Nb TG 2−1 Nb) dx dy

where N :=(“x , “y ) T. The monitor functions Gk are given symmetric positive definite matrices. In general, the monitor functions depend on the underlying solutions to be adapted. The simplest mesh map in two-dimensions is determined by the Euler–Lagrange equation of the above functional, with Gk =MI, k=1, 2, and I an identity matrix: (M −1a x )x +(M −1a y )y =0, (M −1bx )x +(M −1by )y =0,

(2.7)

where the scalar function M is the monitor function. In the first step we prepare the mesh on the logical domain Wc , which will be fixed throughout the computations. By solving above equations (or some variation of them), we will obtain a new mesh in the physical domain. Then we need to update the values of the approximate solutions on the new mesh by some interpolation procedure (a natural updating procedure different from the traditional interpolation was introduced in [23]). With these updated values, the monitor function M will be changed, and as a result we can solve (2.7) again to obtain new location of node points in Wp and then to update the numerical solutions. A few such iterations lead to a satisfactory mesh distribution in Wp . Then on this new mesh we solve the variational inequality problem (2.6) to obtain an approximate solution u h , which will be used to move the mesh based on the above iterative procedure. The mesh-moving is stopped if the difference between the solution for (2.7) and the initial mesh in Wc is sufficiently small. The detail implementation procedure can be found in [23, 24].


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3. CONSTRUCTION OF ERROR-ESTIMATOR-BASED MONITORS It is quite common to use some gradient-based functions in the moving mesh methods, see, e.g. [4, 8, 29]. However, our numerical experiments for variational inequality problem indicate that larger approximation errors occur around the free boundary, and the approximation errors of moving mesh methods with the gradient-based monitors can be even larger than those of standard finite element methods with uniform meshes. The reason is that the gradients are not necessarily large around the free boundary. It is clear that a suitable monitor should be able to capture the free boundary efficiently. A posteriori error estimators of residual type have been widely used in adaptive mesh refinement, though they have not been used in moving mesh methods yet. In the literature, the following a posteriori error estimator of residual was derived in Chen and Nochetto [9] and Liu and Yan [26], for the finite element approximation of the obstacle problem with zero obstacle (2.5), where K hg (0)={vh ¥ V hg : vh \ 0} with ¯ ): g ¥ C 2(W gˆ 2=gˆ 21 +g 22 +g 23 ,

(3.1)

with gˆ 21 =C h 2y F (f+div(A Nu h )) 2, y

y

g 22 =

C

l 5 “W=”

g 23 =F

“W

h l F [(A Nu h ) · n] 2, l

|(A Nu h ) · n| |g − p h g|,

where p h is the standard Lagrange interpolation operator, u h is the solution of (2.6), h l is the size of an edge l, and [(A Nu h ) · n] denotes the jump of A-normal derivative on an interior edge l, defined by [(A Nu h ) · n]l =(A Nu h |y 1l − A Nu h |y 2l ) · n,

l=y¯ 1l 5 ¯y 2l ,

where n is the outer normal vector of y 1l . When g is smooth, g 23 is of higher order, and normally not needed in computation. For the case g=0, this a posteriori error estimator has been shown to be reliable in [9], and further extended to the non-zero obstacle case. As a matter of fact, the estimator gˆ 2 may be applicable to the finite element approximation of more general variational inequalities, see [26]. The main disadvantage of this estimator is that it only provides an upper bound for

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Li, Liu, and Ma

the error. It is clear that in general the estimator cannot be sharp since one could change the value of f in the coincidence set provided f [ 0 without changing that of the solution. Thus the value of f inside the coincidence set should take little effect in sharper error estimates. In this section, we derive sharp error estimates for the finite element approximation of (2.5). Relevant discussions can be found in [25]. Here we adopt a simpler and more direct approach. By exploiting the fact that the solution u of (2.5) satisfies the following nonlinear equation: a(u, v)+(fqu , v)=(f, v)

-v ¥ H 10 (W),

(3.2)

where

˛

qu (x)=

1

u(x)=0

0

u(x) > 0

when, for example, the solution u ¥ H 2(W) and the free boundary has Lebesgue measure zero, we will show that sharper a posteriori error estimates can indeed be obtained, and only g 21 needs to be improved. What we shall do is to replace f with f|Sh , where Sh is a small neighborhood of the non coincidence set, to be described below. When the free boundary consists of a finite numbers of piecewise smooth curves, meas(Sh 0 W+) Q 0 as the maximum diameter of the finite elements tends to zero. 3.1. Upper Bounds In this section, we derive upper bounds of the approximation error. We define W+ y: y … W+, y ¥ T h}, h ={1 ¯ − W bh =W 0 (W+ h 2 W h ),

W h− ={1 ¯y: y … W −, y ¥ T h}, + b W+b h =W h 2 W h .

For a given set Q, we use the notations: a ={1 ¯y: ¯y 5 Q 2 a ] ”, y ¥ T h}, Q

a 0 “Q a. 2 =Q 2 2 Q

We denote by qQ the characteristic function of Q. Let “T h be the set of all edges not on “W h: “T h={l ¥ “y : y ¥ T h, l 5 “W=”}. We need the following lemmas in deriving error estimates of residual type.


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Lemma 3.1 [10]. Let p h be the standard Lagrange interpolation operator. Then for m=0, 1, and 1 < q [ ., ||v − p h v||m, q, y [ Ch 2y − m |v|2, q, y ,

-v ¥ W 2, q(W).

(3.3)

Lemma 3.2 [20]. For all v ¥ W 1, q(W), 1 [ q [ ., ||v||0, q, “y [ C(h y−1/q ||v||0, q, y +h 1y − 1/q |v|1, q, y ).

(3.4)

We also need the following positivity-preserving and H 1-stable interpolation operator Ph : L 1(W) Q V h0 defined by J

(Ph v)(x)= C i=1

1 1 meas(B F v 2 j (x), ) i

i

Bi

(3.5)

where ji (x) is the canonical basis function associated with the interior nodes {Pi } Ji=1 of mesh T h, i.e., ji (Pj )=dij , Bi is the largest ball inside of Di ={1 ¯y: Pi is a vertex of y}. The operator Ph was introduced in [9], whose basic properties are provided below. Lemma 3.3 [9]. If v ¥ H 10 (W), then ||v − Ph v||0, y [ Ch y |v|1, y˜ ,

y ¥ T h,

(3.6)

||v − Ph v||0, l [ Ch 1/2 |v|1, ˜l , l

l ¥ “T h.

(3.7)

Lemma 3.4 [9]. Let vh ¥ V h0 , y ¥ T h, and l ¥ “T h. Then ||vh − Ph vh ||0, y [ C C h 3/2 ||[(Nvh ) · n]||0, l , l

(3.8)

l … y˜

||vh − Ph vh ||0, l [ C C h lŒ ||[(Nvh ) · n]||0, lŒ .

(3.9)

lŒ … ˜l

With the above lemmas, we are able to prove the following upper bounds. Theorem 3.1. Assume that A is a constant matrix, f ¥ L 2(W), g — 0. Let u and u h be the solutions of (2.5) and (2.6) respectively. Then |u − u h | 21, W [ Cg 2,

(3.10)

where g 2 :=g 21 +g 22 with g 21 = C h 2y F f 2qW2 +b , h y ¥ Th

y

g 22 = C h l F [(A Nu h ) · n] 2. l ¥ “T h

l

(3.11)

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Li, Liu, and Ma

Proof. Let e=u − u h . For any vh ¥ K h0 (0), it follows from (2.5) and (2.6) that c |e| 21, W [ a(u, u − u h ) − a(u h , u − u h ) [ (f, u − u h ) − a(u h , u − u h )+a(u h , vh − u h ) − (f, vh − u h ) =(f, u − vh )+a(u h , vh − u). ¯ + and supp Ph u … W 2¯ +b By taking vh =Ph u and noting that supp u … W h , we have (f, u − Ph u)=(fqW2 +b , u − Ph u)=(fqW2 +b , (I − Ph )(u − u h )+(I − Ph ) u h ). h h It follows from (3.6) with v=u − u h and (3.8) with vh =u h that ||(I − Ph )(u − u h )||0, y [ Ch y |u − u h |1, y˜ , ||[(Nu h ) · n]||0, l , ||(I − Ph ) u h ||0, y [ C C h 3/2 l l … y˜

as in [9]. Thus, by the positivity of the matrix A (2.1) c (f, u − Ph u) [ C(g 21 +g 22 )+ |e| 21, W . 2

(3.12)

Next, it follows from (3.7), (3.9), and the Green’s formula that a(u h , vh − u) = C F (A Nu h ) N(vh − u) y ¥ Th

y

= C F [(A Nu h ) · n](vh − u) l ¥ “T h

l

[ C ||[(A Nu h ) · n]||0, l (||(I − Ph )(u − u h )||0, l +||(I − Ph ) u h ||0, l ) l ¥ “T h

1

[ C ||[(A Nu h ) · n]||0, l h 1/2 |u − u h |1, ˜l + C h lŒ ||[(A Nu h ) · n]||0, lŒ l l ¥ “T h

2

lŒ … ˜l

c [ Cg 22 + |e| 21, W . 4 Then the desired result follows from (3.12) and (3.13).

(3.13) i


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Remark 3.1. If A is a non-constant smooth matrix, then it can be seen from the above proof that one simply needs to replace g 21 in (3.11) by g 21 = C h 2y F (f+div(A Nu h )) 2 qW2 +b . h y ¥ Th

y

3.2. Lower Bounds To this end, we need the following lemmas for the bubble functions, the proof of which can be found in [2, 30]. Lemma 3.5. Let y 1l , y 2l be two elements in T h with a common edge l=y¯ 1l 5 ¯y 2l . For any constants By and Dl , there exist polynomials wy in H 10 (y) and wl in H 10 (y 1l 2 y 2l ) such that for m=0, 1, F By wy =h 2y F B 2y ,

− m)+2 |wy | 2m, y [ Ch 2(1 F B 2y , y

F Dl wl =h l F D 2l ,

− m)+1 |wl | 2m, y 1l 2 y 2l [ Ch 2(1 F D 2l . l

y

y

l

y

y

l

In the following we show the above estimator is equivalent on discretization error if u satisfies a(u, v)=(fqW+, v),

-v ¥ H 10 (W).

(3.14)

This assumption holds in many practical applications. For instance it holds when u ¥ H 2(W) and the free boundary consists of a finite number of piecewise smooth curves, see [26]. Theorem 3.2. Assume that all the conditions stated in Theorem 3.1. Then

1

c g 2 − C h 2y F (f − f¯ ) 2 qW+h − C h 2y F f 2qW2 +b h y ¥ Th

y

y ¥ Th

y

2 [ |u − u |

2 h 1, W

[ Cg 2, (3.15)

where g is defined in Theorem 3.1, f¯ |y => f/|y| and C is a positive constant independent of h y and h l . Proof. For any y ¥ W+ h , let wy be defined as in Lemma 3.5 with By =f¯ |y . By (2.5),

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Li, Liu, and Ma

h 2y F f 2 [ 2h 2y F (f¯ 2+(f − f¯ ) 2) y

y

=2 F (f+f¯ − f) wy +2h 2y F (f − f¯ ) 2 y

y

=2 F (A N(u − u h )) · Nwy +2 F (f¯ − f) wy +2h 2y F (f − f¯ ) 2 y

y

y

[ C |u − u h | 21, y +d(|wy | 21, y +h y−2 ||wy || 20, y )+Ch 2y F (f − f¯ ) 2 y

[ C |u − u h | 21, y +Cdh 2y F f 2+Ch 2y ||f − f¯ || 20, y . y

The last inequality implies h 2y ||f|| 20, y [ C |u − u h | 21, y +Ch 2y ||f − f¯ || 20, y ,

if y … W+ h .

(3.16)

To estimate gl , we define wl as in Lemma 3.5 with Dl =[(A Nu h ) · n]l . It follows from (3.14) and the Green’s formula that, for any l ¥ “T h, h l F [(A Nu h ) · n] 2 l

=F [(A Nu h ) · n] wl l

=F 1

(A Nu h ) · Nwl

=F 1

A N(u h − u) · Nwl +F 1

y l 2 y 2l

y l 2 y 2l

y l 2 y 2l

fqW+wl

[ C |u − u h | 21, y 1l 2 y 2l +d(|wl | 21, y 1l 2 y 2l +h l−2 ||wl || 20, y 1l 2 y 2l )+Ch 2l ||fqW+|| 20, y 1l 2 y 2l [ C |u − u h | 21, y 1l 2 y 2l +Cdh l F [(A Nu h ) · n] 2+Ch 2l ||fqW+|| 20, y 1l 2 y 2l . l

Thus we obtain that, for any l ¥ “T h, h l F [(A Nu h ) · n] 2 l

[ C |u − u h | 21, y 1l 2 y 2l +Ch 2l ||f − f¯ || 20, (y 1l 2 y 2l ) 5 W+h +Ch 2l ||f|| 20, (y 1l 2 y 2l ) 5 W+. (3.17)


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Therefore, for any l ¥ “T h, (g 21 )y 1l +(g 21 )y 2l +(g 22 )l + ). [ C(|u − u h | 21, y 1l 2 y 2l +h 2l ||f − f¯ || 20, (y 1l 2 y 2l ) 5 W+h +h 2l ||f|| 20, (y 1l 2 y 2l ) 5 (W˜+b h 0 Wh ) (3.18)

i

Then the desired result follows from (3.16) and (3.18).

Although a sharper error estimator is obtained, it cannot be directly is uncomapplied in mesh refinement since the characteristic function qW2 +b h putable—we usually do not know the location of the free boundary. Nevertheless one can substitute it with some a posteriori quantities, thus obtain some a posteriori error indicators, which can then be used in adaptive finite element method. In some applications, one may roughly know the location of the free ˜ +b boundary, For example, one may be able to estimate a set S such that W h . +b Then one can replace qW2 h with qS . In any case, one can always take S=W and this just give us the estimator gˆ 2. Also one may utilize this error estimator via an iterative procedure, e.g., using the numerical non-coincidence ˜ +b sets {u h > 0} to approximate W h , although a special care has to be taken to prevent from possible numerical instability. Another idea is to approxby the following function q hW+: For a > 0, let imate qW2 +b h u q hW+= a h . h +u h Then, in practical computations we replace g 21 by 2 g 21 = C h 2y F f 2q hW+. y ¥ Th

y

Let us show that this approximator may be more reasonable. To demonstrate this, we divide W into three parts: W h− ,

W a/2 :={x ¥ W+ : u h < h a/2},

and

a/2 W+b . h 0W

Then, for y … W h− , u −u > > h +u −u

||q hW+ − qW+b ||0, ., y = h

h

a

h

[ min{1, h −a ||u h − u||0, ., y }, 0, ., y

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Li, Liu, and Ma

a/2 and for y … W+b , h 0W

h > > h +u a

||q hW+ − qW+b ||0, ., y = h

a

h

[ h a/2. 0, ., y

Therefore, if the error ||u h − u||0, ., y for y … W h− (where u — 0) is of the order h b with b > a, then the difference caused by this approximation is a higha/2 order small local quantity for all y … W h− 2 (W+b ). It can be shown h 0W a/2 that if ||u h − u||0, ., W Q 0, then meas(W ) Q 0, as h Q 0. The most suitable selection of h may not be always easy to know—it may even depend on the mesh adaptivity techniques to be used. One may have to follow some heuristic rules. In the moving mesh method, for example, one can just use h=max y h y . It has been found that the resulting meshes are not sensitive to the selection of h. For example, we tested h between 0.1 and 1.0 for our test problems, and similar computational meshes were always obtained. This is probably due to the fact that on W+, g 22 is in fact dominant. Since h is normally small, the larger a is, the more mesh refinements likely happen. On the other hand, efficiency may be poor if a > 2, as usually b [ 2. In this paper, we take a=1.1. Again it was found that the computational results were not sensitive to the selection of a, as expected. The case f ] 0 can be dealt with by using the transformation u g=u − f provided f ¥ H 2(W). 4. NUMERICAL METHODS We construct the error-estimator-based monitor by using the equivalent a posteriori error estimator derived in the last section: M|on y =`c+g 21, y + C

g 22, l ,

(4.1)

l: edge of y

where c is a positive constant, g1, y is the integration in every element g 21, y =h 2y F (f(1 − q huh )) 2, y

and g2 is the integration on every side g 22, l =h l F [(A Du h ) · n] 2. l

The reason for adding a positive constant c in the monitor (4.1) is that the monitor is used as a metric and therefore can not be zero at any point. The


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magnitude of c will decide the density of the nodes on the coincidence set. If c is too large, then there will have too many elements inside the set, which are not needed, as main approximation error is concentrated around the free boundary. On the other hand, if c is too small, there will have a large jump of density of elements over the free boundary as the monitor is relatively large just outside the coincidence set and near zero inside the set. This, as observed in our numerical experiments, leads to very singular meshes and higher approximation error around the free boundary. We overcome this difficulty by smoothing the monitors around free boundary by using one of the following two strategies: Monitor smoothing strategy I. In order to make the monitor more smooth, we first interpolate M from L 2(W) into H1, h (W), from piecewise constant to piecewise linear ; M|on y |y| (p h M)|at P = y: P is vertex of y ; y: P is vertex of y |y| and then project it back into L 2(W): ˜ |on y =1 M 3

C

(p h M)at P .

P is vertex of y

Monitor smoothing strategy II. An alternative approach for smoothing the monitor is the following: Given d > 0. If My* =0, let M gy* =max y ¥ T gh My /2, where T gh ={y: dist(y, y g ) [ d}. Then we replace My* by M gy* . Denote Xi as nodes and F j as basis functions, such that F j(Xi )=dij . Let H ij=F NF i · NF j dx,

q i=F fF i dx,

W

W

Yi =f(Xi ),

(4.2)

where f and f are given data in (1.1). The finite element approximation is u h =; Ui F i. In all the numerical experiments, A in the variational inequality (2.2) is chosen as the unit matrix. Thus it follows from the equivalence between (2.2) and the minimization problem (2.3) that the variational inequality (2.4) can be approximated by the following optimization problem: min 12 U THU+q TU s.t. U \ Y.

(4.3)

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Li, Liu, and Ma

In order to solve the problem (4.3) with adaptive mesh, we use the following algorithm: Algorithm 0. (i) solve the optimization problem (4.3) on the current mesh; (ii) move the mesh to a new location and update the solution on new meshes, as described in [23, 24]; (iii) judge if the error between the initial computational mesh and the solution of (2.7) is sufficiently small. if not, go to (i). In part (i), we use a projection gradient method introduced in [17] to solve the optimization problem (4.3). We briefly describe the method here. For given H and q, if the projection operator is denoted by proj(•), then • (1) g=HU+q; • (2) e=U − proj(U − g); • (3) error=||e||2 , if e < E then exit; ||e||

2

• (4) r=||e|| 2 +e2THe ; 2

• (5) U=U − re, go to (1). 5. NUMERICAL EXPERIMENTS In order to illustrate the foregoing numerical schemes, we consider several test problems. Our special attention is to capture the free boundary and decrease the approximation errors using minimum number of elements. Example 5.1. In the first example, we choose a square physical domain and a constant f: W=[ − 1.5, 1.5] × [ − 1.5, 1.5], f=−2. The obstacle function f is zero. The exact solution is

˛

u=

r2 2

0

− ln(r)+ln(1) − 12

if r \ 1 if r < 1

where r=`x 2+y 2. The free boundary is u(x, y)=0 along the circle r=1. In Fig. 1, we display the numerical results obtained by using the moving mesh methods with 30 × 30 elements. The constant c in the monitor function (4.1) is set to be 10 −3. For comparison, we also solve this problem using a uniform grid and a moving mesh based on the non-equivalent error estimator (3.1). It is observed from Fig. 1 that in these two cases large


47

Fig. 1. Mesh and the solution error for Example 5.1 with 30 × 30 nodes: (a) moving mesh method based on the equivalent error estimator; (b) uniform mesh solution; (c) moving mesh method based on the non-equivalent error estimator (3.1). The maximum errors are approximately 0.7 × 10 −3, 1.9 × 10 −3, and 1.7 × 10 −3, respectively.

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Li, Liu, and Ma

errors occur on the free boundary. On the other hand, the moving mesh method based on the equivalent a posteriori error estimator removes the leading errors from the free boundary and as a result the overall error level is decreased significantly. More importantly, it is observed that the moving mesh approximation with the equivalent error estimator provides a satisfactory identification for the free boundary, as also seen in Fig. 1. Example 5.2. This example is as same as Example 5.1, except that f is given by

˛

f=

2 r

−4

4

0.95

if r \ 0.5 2 0.95 r

−( )

if r < 0.5.

Note that f above is not well defined at r=0, but the data of f in the neighborhood of the origin does not affect the solution of (2.4). The exact solution is u=(r − 1) 2 for r \ 1 and 0 for r < 1. Again, we compare the performance of the computations on moving meshes and uniform meshes in Fig. 2, with 30 × 30 nodes used. Observations similar to Example 5.1 are made from Fig. 2. It is clear from Figs. 1 and 2 that the mesh refinement scheme based on the non-equivalent a posteriori error estimates refines the meshes almost uniformly everywhere in the computational domain, and as a result the coincidence set can not be captured. On the other hand, the moving mesh scheme based on the equivalent a posteriori error estimates only refines the domain outside the coincidence set and therefore is more efficient. To demonstrate the convergence of our moving mesh method, we compute Example 5.2 with 20 × 20, 30 × 30, and 40 × 40 nodes, and display the adaptive meshes and the solution errors in Fig. 3. It is observed that the moving mesh methods with the sharper error estimators increase the accuracy significantly. The leading errors at the free boundary are eliminated in the moving mesh solutions. Example 5.3. In this example, we choose W=[ − 1, 1] × [ − 1, 1], f=1. Let K={u ¥ H 10 (W) : u [ f} with f(x)=dist(x, “W). Note that the above problem is a variation of type (2.4) and the obstacle function is piecewise linear (and non-zero). This problem was studies by Kornhuber [19]. For this nonzero-constraint problem, we modify the characteristic function q hu h by letting ha q hu h = , u h − fh +h a where a is some positive constant.


49

Fig. 2. Mesh and the solution error for Example 5.1 with 30 × 30 nodes: (a) moving mesh method based on the equivalent error estimator; (b) uniform mesh solution; (c) moving mesh method based on the non-equivalent error estimator (3.1). The maximum errors are approximately 1.0 × 10 −3, 2.0 × 10 −3, and 1.5 × 10 −3, respectively.

50

Li, Liu, and Ma

Fig. 3. Adaptive meshes and solution errors with 20 × 20, 30 × 30, and 40 × 40 nodes (from top to bottom), respectively, for Example 5.2.


51

Fig. 4. Adaptive meshes and numerical solution with 20 × 20, 30 × 30, and 40 × 40 nodes, respectively, for Example 5.3.

52

Li, Liu, and Ma

Fig. 5. Adaptive meshes and numerical solutions with 10 × 20, 20 × 40, and 30 × 60 nodes (from top to bottom), respectively, for Example 5.4. The solution domain is [0, 5] × [0, 10].


53

To demonstrate the convergence of our moving mesh methods, we compute Example 5.3 with 20 × 20, 30 × 30, and 40 × 40 nodes. Figure 4 shows the adaptive meshes and the numerical solutions. Although the exact solution is unknown, the convergence is observed clearly from this figure. Example 5.4. In the last example, the obstacle f=0, f=−1 and

˛

u|“W =

0.5h 21 − qx

0 [ x [ l, y=0

0.5(h 1 − y)

2 +

x=l, 0 < y [ h 1

0.5(h 2 − y)

2 +

x=0, 0 < y [ h 1

0

0 < x < l, y=h 1

where h 1 =10, h 2 =2, l=5, q=(h 21 − h 22 )/(2l), and (h − y)+=max(h − y, 0). The problem is defined in the domain W=[0, l] × [0, h 1 ]. This problem arises in the modeling of the seepage of an incompressible, inviscid fluid through an unsaturated rectangular dam. The true solution of this problem is not readily accessible. This problem was studied numerically by Ainsworth et al. [3] who derived local a posteriori error estimator for assessing the error in a finite element approximation. The adaptive meshes and the numerical solutions are shown in Fig. 5. The numerical solutions as well as the curve of free boundary seem convergent as the number of elements is sufficiently large. ACKNOWLEDGMENTS We thank Dr. B. S. He of Nanjing University for introducing us the optimization code used in this work. The research of the first author was supported by Peking University, Hong Kong Baptist University Joint Research Institute for Applied Mathematics. This research was supported by Special Funds for Major State Basic Research Projects of China and UK EPSRC (Grant GR/L67387). REFERENCES 1. Ainsworth, M., and Oden, J. T. (1993). A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65, 23–50. 2. Ainsworth, M., and Oden, J. T. (1997). A posteiori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 142, 1–88. 3. Ainsworth, M., Oden, J. T., and Lee, C. Y. (1993). Local a posteriori error estimators for variational inequalities. Numer. Methods Partial Differential Equations 9, 22–33. 4. Azarenok, B. N. (2002). Variational barrier method of adaptive grid generation in hyperbolic problems of gas dynamics. SIAM J. Numer. Anal. 40, 651–682.

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5. Baines, M. J. (1994). Moving Finite Elements, Oxford University Press. 6. Beckett, G., Mackenzie, J. A., Ramage, A., and Sloan, D. M. (2002). Computational solution of two-dimensional unsteady PDEs using moving mesh methods. J. Comput. Phys. 182, 478–495. 7. Brackbill, J. U., and Saltzman, J. S. (1982). Adaptive zoning for singular problems in two dimensions. J. Comput. Phys. 46, 342–368. 8. Ceniceros, H. D., and Hou, T. Y. (2001). An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 172, 609–639. 9. Chen, Z. M., and Nochetto, R. H. (2000). Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84, 527–548. 10. Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam. 11. Davis, S. F., and Flaherty, J. E. (1982). An adaptive finite element method for initialboundary value problems for partial differential equations. SIAM J. Sci. Stat. Comp. 3, 6–27. 12. Dvinsky, A. S. (1991). Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comput. Phys. 95, 450–476. 13. Elliott, C. M., and Ockendon, J. R. (1982). Weak and Variational Methods for Moving Boundary Problems, Research Notes in Mathematics, Vol. 59, Pitman, Boston. 14. French, D. A., Larsson, S., and Nochetto, R. H. (2001). A posteriori error estimates for a finite element approximation of the obstacle problem in L .. Comput. Methods Appl. Math. 1, 18–38. 15. Friedman, A. (1982). Variational Principles and Free-Boundary Problems, Academic Press, New York. 16. Glowinski, R., Lions, J. L., and Tremolieres, R. (1976). Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam. 17. He, B. S. (1994). Solving a class of linear projection equations. Numer. Math. 68, 71–80. 18. Kinderlehrer, D., and Stampacchia, G. (1980). An Introduction to Variational Inequalities and Their Applications, Academic Press, New York. 19. Kornhuber, R. (1996). A posteriori error estimates for elliptic variational inequalities. Comput. Math. Appl. 31, 49–60. 20. Kufner, A., John, O., and Fucik, S. (1977). Function Spaces, Nordhoff, Leyden, The Netherlands. 21. Li, S., and Petzold, L. (1997). Moving mesh methods with upwinding schemes for timedependent PDEs. J. Comput. Phys. 131, 368–377. 22. Li, R., Liu, W. B., Ma, H. P., and Tang, T. (2002). Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41, 1321–1349. 23. Li, R., Tang, T., and Zhang, P. (2001). Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 170, 562–588. 24. Li, R., Tang, T., and Zhang, P. (2002). A moving mesh finite element algorithm for singular problems in two and three space dimensions. J. Comput. Phys. 177, 365–393. 25. Liu, W. B., Ma, H. P., and Tang, T. (2001). Mixed error estimates for elliptic obstacle problems. Adv. Comput. Math. 15, 261–283. 26. Liu, W. B., and Yan, N. (2000). A posteriori error estimates for a class of variational inequalities. J. Sci. Comput. 35, 361–393. 27. Miller, K., and Miller, R. N. (1981). Moving finite element methods I. SIAM J. Numer. Anal. 18, 1019–1032. 28. Ren, Y., and Russell, R. D. (1992). Moving mesh techniques based upon equidistribution, and their stability. SIAM J. Sci. Stat. Comput. 13, 1265–1286.


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Moving Mesh Method with Error-Estimator-Based Monitor and Its ...

Moving Mesh Method with Error-Estimator-Based Monitor and Its ...

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