Curved domain approximation in Dirichlet's problem

CURVED DOMAIN APPROXIMATION IN DIRICHLET’S PROBLEM

Miyoung Lee, Sang Mok Choo, and Sang Kwon Chung

Abstract. The purpose of this paper is to investigate the piece- wise polynomial approximation for the curved boundary. We ana- lyze the error of an approximated solution due to this approxima- tion and then compare the approximation errors for the cases of polygonal and piecewise polynomial approximations for the curved boundary. Based on the results of analysis, p-version numerical methods for solving Dirichlet’s problems are applied to any smooth curved domain.

1. Introduction

We consider the Poisson equation

(1.1) −∆u = F in Ω,

with the Dirichlet boundary condition u = 0 on ∂Ω,

where F ∈ C ^∞ (Ω). Here Ω is a bounded domain in the plane with an infinitely differentiable curved boundary ∂Ω which may be neither polygonal nor polynomial.

It is usually assumed that the boundary ∂Ω of Ω is either polygo- nal structures or polynomials when some numerical methods are imple- mented and corresponding numerical estimations are obtained(see [1, 2, 3, 4]). Thus if a curved boundary is neither polygonal nor polynomial, these results may not be held.

Received February 28, 2003.

2000 Mathematics Subject Classification: Primary 65M06, 65M12, 65M15.

Key words and phrases: curved domain, polynomial boundary approximation,

p-version, Poisson equation.

Strang and Berger [5] and Thom´ee [6] obtained the error estimates for the following equation

(1.2) −∆u h = F in Ω h , u _h = 0 on ∂Ω h ,

where a polygonal domain Ω _h approximates a curved domain Ω and h is the maximum length of the edges of ∂Ω h . Due to these theoretical re- sults, the implementation of the numerical h-version method to a curved domain is meaningful.

To obtain more accurate results for the numerical solution on domains with curved boundary, we should apply a p-version method to the curved boundary. But we are aware of no studies that have examined p-version boundary approximation.

The purpose of this paper is to generalize the polygonal boundary approximation using polynomial approximation. We will approximate the curved boundary by a piecewise polynomial boundary and analyze the errors of the approximated solution due to this polynomial bound- ary approximation. In section 2, the error estimates in the maximum norm are discussed. In section 3, the L ² -error estimates for gradients of numerical solutions are discussed.

2. Error estimates in the maximum-norm

We assume that the boundary ∂Ω is a union of subboundaries (f _i (s), g _i (s)) such that

∂Ω = ∪ 1≤i≤k {(f _i (s), g _i (s)) : 0 ≤ s ≤ 1}

with

(f i (1), g i (1)) = (f i+1 (0), g i+1 (0)), (f k (1), g k (1)) = (f 1 (0), g 1 (0)), where f i , g _i ∈ C ^∞ [0, 1]. Let Ω p be the curved domain with boundary

∂Ω p = ∪ 1≤i≤k {(Π i (s), Ψ i (s)) : 0 ≤ s ≤ 1},

where Π _i and Ψ _i be the Lagrange interpolating polynomials of degree p _i satisfying

Π i ( ℓ

p _i ) = f i ( ℓ

p _i ), Ψ i ( ℓ

p _i ) = g i ( ℓ

p _i ), ℓ = 0, . . . , p i .

Then the corresponding polynomial approximation problem for (1.1) becomes

(2.1) −∆u _p = F in Ω _p , u _p = 0 on ∂Ω _p .

Let | · | Ω denote the sup-norm in Ω and k · k Ω denote L 2 -norm over Ω, |u| m,Ω = max |α|≤m |D ^α u| Ω and |x| ² = |(x 1 , x ₂ )| ² = x ² ₁ + x ² ₂ .

Using the maximum principle, we obtain the following theorem.

Theorem 2.1. Let F ∈ C ^∞ ( ˜ Ω) where ˜ Ω ) Ω. Then there exists a constant C which depends only on Ω such that

|u − u p | _Ω∩Ω

≤ C max

1≤i≤k {|f _i ^(p

⁺¹⁾ | [0,1]

1 p _i + 1 ( 1

p _i ) ^p

⁺¹ ,

|g _i ^(p

⁺¹⁾ | _[0,1] 1 p _i + 1 ( 1

p _i ) ^p

⁺¹ }|F | Ω ˜ ,

where Ω ∩ Ω _p = (Ω ∩ Ω _p ) ∪ ∂(Ω ∩ Ω _p ) and f _i ^(p

⁺¹⁾ is the (p _i + 1)st derivative of f _i .

Proof. Let d(x, Ω) be the distance between Ω and x. Let Ω ^ǫ = ∪ 1≤i≤k {x : d(x, Ω i ) < ǫ i },

where

∂Ω _i = {(f _i (s), g _i (s)) : 0 ≤ s ≤ 1}, ǫ _i = max{|f _i ^(p

⁺¹⁾ | _[0,1] 1

p _i + 1 ( 1

p _i ) ^p

⁺¹ , |g ^(p _i

⁺¹⁾ | _[0,1] 1 p _i + 1 ( 1

p _i ) ^p

⁺¹ } so that Ω ∪ Ω _p ⊂ Ω ^ǫ ⊂ ˜ Ω for large p.

Let u ^ǫ be the solution of

−∆u ^ǫ = F in Ω ^ǫ , u ^ǫ = 0 on ∂Ω ^ǫ .

Note that ∂Ω ^ǫ ∈ C ^∞ except finite number of points. Then for x ∈

∂Ω ∪ ∂Ω p , there exists y x ∈ ∂Ω ^ǫ with d(x, ∂Ω ^ǫ ) = d(x, y x ). Since u ^ǫ − u is harmonic in Ω, we obtain for some C

|u ^ǫ − u| Ω ≤ |u ^ǫ − u| ∂Ω

= sup

x∈∂Ω

|u ^ǫ (y _x ) − u ^ǫ (x)|

≤ sup

x∈∂Ω

|y _x − x||u ^ǫ | _{1, Ω}

≤ max ǫ i |u ^ǫ | 1, Ω

≤ C max ǫ _i |∆u ^ǫ | Ω

≤ C max ǫ _i |F | Ω ˜ .

On the other hand

|u p − u ^ǫ | Ω

≤ |u p − u ^ǫ | ∂Ω

= sup

x∈∂Ω

|u ^ǫ (y _x ) − u ^ǫ (x)|

≤ sup

x∈∂Ω

|y x − x||u ^ǫ | 1,Ω

≤ 2 max ǫ _i |u ^ǫ | _1,Ω

≤ C max ǫ i |∆u ^ǫ | Ω

≤ C max ǫ _i |F | Ω ˜ .

Using triangle inequality, we obtain the desired result. ¤ Remark 2.1. Let {P j } ^N ₀ be a sequence of points on ∂Ω with P 0 = P _N and Ω _h be a polygonally approximated domain with the maximum length h of edges in a triangulation for Ω _h . Thom´ee [5] obtained for convex domain Ω

|u − u h | Ω

≤ Ch ² . Thus the errors of h-version and p-version are

|u − u _h | Ω

∩Ω

= O( 1 N ² ),

|u − u _p | _Ω

_∩Ω

= O( 1

(N − 1) ^{N +1} ), respectively, whenever | −−−−→

P _i P _i+1 | = | −−−−−−→

P _i+1 P _i+2 |, f ^(q) and g ^(q) are bounded for q = 0, . . . , N . Therefore we can see that a curved boundary approx- imation is more accurate than a polygonal boundary approximation.

In most numerical studies on a curved domain, the curved boundary is polynomial. In this case, Ω = Ω _p for some p. Thus the errors of h-version and p-version are, respectively,

|u − u h | Ω

= O( 1 N ² ),

|u − u _p | Ω

= O( 1

(N − 1) ^N+1 ).

3. L 2 -error estimates for gradients

Without loss of generality, we may assume that the origin belongs to Ω and there exists a positive constant µ such that

x · η(x) ≥ µ > 0, x ∈ ∂Ω,

where η(x) is the outward unit normal vector to the boundary ∂Ω at a point x. For example, a convex domain satisfies the above condition.

Let s(x) be the unit tangent vector to the boundary ∂Ω at a point x.

Using the Green’s identity, we obtain the following lemma.

Lemma 3.1. There exists a constant C such that for a harmonic function v in Ω ∩ Ω _p

k ∂v

∂η k _{∂(Ω∩Ω}

₎ ≤ C| ∂v

∂s | _{∂(Ω∩Ω}

₎ ,

where _∂η ^∂ and _∂s ^∂ denote differentiations in the direction of normal and tangent vector, respectively.

Proof. Integrating the following identity

2

X

j,k=1

· ∂

∂x _k {x _k ( ∂v

∂x _j ) ² } − 2 ∂

∂x _j (x _k ∂v

∂x _j

∂v

∂x _k )

¸ + 2

2

X

k=1

x _k ∂v

∂x _k ∆v = 0 over Ω∩Ω _p and using Green’s identity, we obtain for a harmonic function v

(3.1)

Z

∂(Ω∩Ω

)

· x · η

2

X

j=1

( ∂v

∂x _j ) ² − 2 ∂v

∂n

∂v

∂r

¸ ds = 0,

where ^∂v _∂r denotes the derivative in direction of x = (x 1 , x 2 ). Note that for unit vectors η = (η 1 , η ₂ ) and s = (−η 2 , η ₁ ), it holds that

(3.2)

2

X

j=1

( ∂v

∂x _j ) ² = ( ∂v

∂η ) ² + ( ∂v

∂s ) ² . Setting

x · η = kxkx n , x · s = kxkx s , ∂v

∂r = (x n

∂v

∂η + x s

∂v

∂s )kxk,

and using (3.2), we may rewrite (3.1) as Z

∂(Ω∩Ω

)

· x _n ( ∂v

∂s ) ² − x n ( ∂v

∂η ) ² − 2x s

∂v

∂s

∂v

∂η

¸

kxkds = 0.

Thus we obtain for ǫ > 0

(3.3)

Z

∂(Ω∩Ω

)

x _n ( ∂v

∂η ) ² kxkds

= Z

∂(Ω∩Ω

)

· x _n ( ∂v

∂s ) ² − 2x s

∂v

∂s

∂v

∂η

¸ kxkds

≤ Z

∂(Ω∩Ω

)

·

(x _n + 2|x _s | 1 ǫ )( ∂v

∂s ) ² + 2|x _s |ǫ( ∂v

∂η ) ²

¸ kxkds.

Since there exists a positive constant µ such that x · η ≥ µ > 0, x ∈ ∂Ω,

and ∂Ω _p is any kind of polynomial approximation of ∂Ω, there exists a positive constant λ such that for sufficiently large p

x · η ≥ λ > 0, x ∈ ∂Ω p . Therefore we obtain

x · η ≥ min{µ, λ} > 0, x ∈ ∂(Ω ∩ Ω p ).

Since there exist constants r 0 and r 1 such that for all x ∈ ∂(Ω ∩ Ω p ), 0 < r 0 ≤ kxk ≤ r 1 , we can choose ǫ in (3.3) such that

x _n kxk − 2|x _s |ǫkxk > min{µ, λ} − 2r 1 ǫ > 0, x ∈ ∂(Ω ∩ Ω _p ).

It completes the desired proof. ¤

Using the Green’s formula, we obtain the following theorem.

Theorem 3.1. There exists a constant C depending on u such that k∇u − ∇u p k Ω∩Ω

≤ C max

1≤i≤k {|f _i ^(p

⁺¹⁾ | [0,1] ( 1

p _i ) ^p

⁺

, |g _i ^(p

⁺¹⁾ | [0,1] ( 1

p _i ) ^p

⁺

}.

Proof. Since e _p = u − u _p is a harmonic function in Ω ∩ Ω _p , using Lemma 3.1, we obtain for some constant C

k∇e p k ² _Ω∩Ω

= Z

∂(Ω∩Ω

)

e _p ∂e _p

∂η ds (3.4)

≤ ke p k _{∂(Ω∩Ω}

₎ k ∂e _p

∂n k _{∂(Ω∩Ω}

₎

≤ Cke _p k _{∂(Ω∩Ω}

₎ | ∂e _p

∂s | _{∂(Ω∩Ω}

₎ . Note that

(3.5) | ∂e _p

∂s | _{∂(Ω∩Ω}

₎ = | ∂e _p

∂s | _{∂(Ω∩Ω}

_)∩∂Ω + | ∂e _p

∂s | _{∂(Ω∩Ω}

_)∩∂Ω

and the second term in the righthand side of (3.5) is

| ∂e _p

∂s | _{∂(Ω∩Ω}

_)∩∂Ω

= sup

x

∈

|∇u(x p ) · s(x p )|.

Let x p = (Π i (s 0 ), Ψ i (s 0 )) and ˜ x = (f i (s 0 ), g i (s 0 )). Then we obtain

|∇u(x _p ) · s(x _p )| ≤ |∇u(˜ x) · s(x _p )| + Ckx _p − ˜ xk (3.6)

≤ C{| cos( π

2 ± θ)| + kx _p − ˜ xk}

≤ C{sin θ + kx _p − ˜ xk},

where θ is the acute angle between the tangential vectors s(x p ) on ∂Ω p

and s(˜ x) on ∂Ω.

Using the smoothness of functions f i , g _i , Π i , Ψ i , the inequality (3.6) becomes

|∇u(x _p ) · s(x _p )|

≤ C{sin θ + kx p − ˜ xk}

≤ C{

q

(f _i ^′ (s 0 ) − Π ^′ _i (s 0 )) ² + (g _i ^′ (s 0 ) − Ψ ^′ _i (s 0 )) ² + kx p − ˜ xk}.

Thus we obtain (3.7)

| ∂e _p

∂s | _{∂(Ω∩Ω}

_)∩∂Ω

≤ C

·

1≤i≤k max {|f _i ^(p

⁺¹⁾ | _[0,1] ( 1

p _i ) ^p

⁺¹ , |g _i ^(p

⁺¹⁾ | _[0,1] ( 1 p _i ) ^p

⁺¹ } + max

1≤i≤k {|f _i ^(p

⁺¹⁾ | [0,1]

1 p _i + 1 ( 1

p _i ) ^p

⁺¹ , |g _i ^(p

⁺¹⁾ | [0,1]

1 p _i + 1 ( 1

p _i ) ^p

⁺¹ }

¸

≤ C max

1≤i≤k {|f _i ^(p

⁺¹⁾ | _[0,1] ( 1

p _i ) ^p

⁺¹ , |g ^(p _i

⁺¹⁾ | _[0,1] ( 1

p _i ) ^p

⁺¹ }.

Similarly we obtain the inequality (3.7) for | ^∂e _∂s

| _{∂(Ω∩Ω}

_)∩∂Ω . Applying (3.7) and Theorem 2.1 to (3.4), we obtain the desired results. ¤

Remark 3.1. Thom´ee [5] obtained for a convex domain Ω

|∇u − ∇u h | Ω

≤ Ch

,

where h is the maximum length of the edges in a triangulation for Ω _h . Thus we obtain similar results as in Remark 2.1.

4. Concluding remark

We approximated a curved boundary by polynomials of variable de- grees and obtained the corresponding error estimates. Since a piecewise polynomial boundary approximation for a curved boundary is more ac- curate than that a polygonal boundary approximation, we were able to show that the numerical solution by p-version methods is more accurate than that obtained by using the usual boundary approximation for a curved boundary.

Acknowledgment. The authors are indebted to V. Thom´ee for encouraging and constant interest during the investigation. Also we thank to F. Milner for his stimulating encouragement to realize the value of this study.

References

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[2] M. Lee and F. Milner, Mixed finite element methods for nonlinear elliptic prob- lems: The p-Version, Numer. Methods Partial Differential Equations 12 (1996), 729–741.

[3] J. T. Oden and L. Demkowicz, Mixed finite element methods for nonlinear elliptic problems: The p-Version, Comput. Methods Appl. Math. 89 (1991), 11–40.

[4] M. Suri, On the stability and convergence of higher order mixed finite element methods for second order elliptic problems, Math. Comp. 54 (1990), 1–19.

[5] G. Strang and A. E. Berger, The Change in solution due to change in domain, Proceedings of the AMS summer institute on partial differential equations at Berkeley (1971), 199–205.

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Math. Appl. 11 (1973), 33–44.

Miyoung Lee

Department of Management Information System Konkuk University

Seoul 143-701, Korea

E-mail: [email protected] Sang Mok Choo

School of Mathematics and Physics Ulsan University

Ulsan 680-749, Korea

E-mail: [email protected] Sang Kwon Chung

Department of Mathematics Education Seoul National University

Seoul 151-742, Korea

E-mail: [email protected]