Error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions

Transcription

1 IMA Journal of Numerical Analysis 3) 3, 5 47 Error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions WEIZHU BAO Department of Computational Science, National University of Singapore, Singapore, 7543 Received on 6 September ; revised on April In this paper we design high-order non)local artificial boundary conditions ABCs) which are different from those proposed by Han, H. & Bao, W. 997 Numer. Math., 77, ) for incompressible materials, and present error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions. The finite-element approximation especially its corresponding stiff matrix) becomes much simpler sparser) when it is formulated in a bounded computational domain using the new non)local approximate ABCs. Our error bounds indicate how the errors of the finite-element approximations depend on the mesh size, terms used in the approximate ABCs and the location of the artificial boundary. Numerical examples of the exterior Stokes equations outside a circle in the plane are presented. Numerical results demonstrate the performance of our error bounds. Keywords: exterior Stokes equations; finite-element approximation; artificial boundary; non)local artificial boundary condition ABC); error bounds.. Introduction Let Γ i be a smooth bounded simple closed curve in R and Ω be the exterior domain with the boundary Γ i see Fig. ). We consider the following boundary value problem: P) Find u, p) such that ν u + grad p = f div u = in Ω,.) u = on Γ i, u is bounded, p, when r = x x + x +..) Here x = x, x ) is the Cartesian coordinate system, polar coordinate system is r,θ), u = u, u ) T is the velocity, p is the pressure, ν > isthe viscosity constant and f = f, f ) T is a given function with compact support. The well-posedness of the exterior Stokes problem P) is treated in Girault & Sequeira 99). In fact, an equivalent form of P) is also used to model incompressible materials Brezzi & Fortin, 99), i.e. ν div εu) + grad p = f, div u = in Ω,.3) u = on Γi, σ nu, p) = νεu)n pn = g, on Γi,.4) u is bounded p, when r = x x + x + ;.5) bao@cz3.nus.edu.sg c The Institute of Mathematics and its Applications 3

2 6 W. BAO Ω Ω s Ω e Ω i Q Γ i Γ e FIG.. Set-up of the domain and artificial boundary. where n is the unit outward normal vector, g = g, g ) T is a given function, Γ i = Γi Γi with Γi Γi =, and εu) = ε ij u)) i, j= is the strain tensor corresponding to the displacement u, which is given by ε ij u) = ui + u ) j, i, j..6) x j x i In finding numerical solutions of the problem P) or.3).5), one of the main numerical difficulties is the unboundedness of the domain Ω which means that the classical finiteelement FEM) or finite-difference method FDM) can not be used directly. In the last two decades, several methods have been proposed to solve boundary value problems in unbounded domains Givoli, 99); for example, the boundary element method BEM) Kohr, 997), the infinite-element method Ying, 995), coupling of BEM and FEM Antonio & Meddahi, ; Meddahi & Sayas, ), etc. One of the most popular methods is to introduce an artificial boundary and set up ABCs on it. Then the original problem is reduced to a boundary value problem in a bounded computational domain. Thus a numerical approximation of the original problem can be obtained by solving the reduced problem. In recent years, many authors have worked on this subject for various problems by different techniques, see Engquist & Majda 977); Feng 984); Halpern & Schatzman 989); Bao 998, 997, ); Bao & Han 996); Bao & Xin ); Han et al. 994) and references therein. In the above works, two types of ABC were designed: nonlocal and local ABCs. Each type has its own advantages and disadvantages. For nonlocal ABCs, it is very easy to design high-order approximate boundary conditions, but the stiff matrix of the finiteelement approximation for the original problem by using a nonlocal ABC becomes much denser than that of a local ABC. On the other hand, it is usually not easy to implement high-order local boundary conditions because the high-order derivatives usually higher than second-order) will appear in the conditions see 3.4)). Furthermore, several authors

3 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 7 also gave error estimates for the numerical solution, see Han & Wu 99) and Givoli et al. 997). But their error estimates only depend on the mesh size and the approximate ABCs. How the error depends on the location of the artificial boundary is unknown. But this is a very interesting problem for engineers. Recently we got new error bounds which depend not only on the mesh size and terms used in the approximate ABCs but also on the location of the artificial boundary for the finite-element approximation of elliptic equations Han & Bao, ; Bao & Han, ) and the linear elastic system Han & Bao, ) in an unbounded domain. The key idea was to use an equivalent norm in the error analysis. In.4), if Γi =, the variational formulation of the Stokes problem must be based on the partial differential equations PDEs).3) in order to cope with the normal stress boundary condition.4). In this case, to design the ABCs at a given artificial boundary Γ e should be based on the continuity of the displacement and normal stress related to.3), i.e. u Γe, σ n u, p) Γe = νεu)n pn Γe..7) This kind of ABC for incompressible materials was studied by Han & Bao 997) and error bounds for the finite-element approximation were also given there Han & Bao, 997). In contrast, if Γi =, i.e. pure Dirichlet boundary condition.) is posed on the inside boundary Γ i,the variational formulation can be obtained from the PDEs.) instead of.3). Thus the variational formulation itself and especially its corresponding stiff matrix will become much simpler and sparser, respectively. In this case, to design ABCs at a given artificial boundary Γ e must be based on the continuity of the velocity and normal stress related to.), i.e. u Γe, σ n u, p) Γe = ν u n pn..8) Γe The purpose of this paper is to design high-order non)local ABCs for the exterior Stokes problem P) based on.8), which are different from those proposed by Han & Bao 997) for incompressible materials, and provide error estimates for the finite-element approximation of P). The advantage of the new ABCs is such that the stiff matrix of the finite-element approximation of P) becomes much sparser and thus could save computational time in solving the corresponding linear system. Our error estimates depend on not only the mesh size and the approximate ABCs but also the location of the artificial boundary. They can be used to determine how to choose the mesh size, terms used in the ABCs and the location of the artificial boundary for practical computations. The layout of this paper is as follows. In Section we design a family of high-order nonlocal ABCs at a given artificial boundary for the problem P). In Section 3 we propose a family of high-order local ABCs. In Section 4 we introduce the finite-element formulation of the problem P) in a bounded computational domain using an approximate nonlocal ABC and prove error bounds for the finite-element approximation. In Section 5 similar results for using high-order local ABCs are presented. In Section 6 we report on some numerical experiments. Finally, in Section 7 we draw some conclusions.

4 8 W. BAO. High-order nonlocal ABCs We introduce a circle Γ e with radius R such that supp f B R ) := {x R : x < R}, then Ω is divided into two parts: the unbounded part Ω e := Ω \ B R ) and the bounded part Ω i := Ω \ Ω e see Fig. ). The restriction of the solution u, p) of the problem P) to the unbounded domain Ω e is then the solution of the following problem: P e ) Find u, p) such that ν u + grad p = div u =, in Ω e,.) u Γe = ur,θ) u is bounded, p, when r = x x + x +..) The general solution of P e )issee Han & Bao, 997, pp for details) u i r,θ)= r R W r,θ) ) + G i r,θ), x i R r < +, θ π, i =,,.3) where G, G and W are harmonic functions satisfying G i r,θ)= ai ) R a + n i n cos nθ + bi n sin nθ r R r < +, θ π, i =,,.4) W r,θ)= with a i n = π π n= n n p n cos nθ + q n sin nθ Rn r n, R r < +, θ π; G i R,θ)cos nθ dθ = π π.5) u i R,θ)cos nθ dθ, i =,, n,.6) b i n = π π G i R,θ)sin nθ dθ = π π u i R,θ)sin nθ dθ, i =,, n,.7) p n = an b n, q n = bn + a n, n.8) Combining.3).5) and.) on noting the boundary condition at infinity for p in.), a computation shows pr,θ)= ν n ) p n cos nθ + q n sin nθ Rn r n, R r < +, θ π. n=.9)

5 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 9 Applying the standard method for elliptic problems Givoli, 99; Han & Bao, ), Helmholtz-type equations Goldstein, 98), the linear elastic system Han & Wu, 99; Han & Bao, ) etc., we now design the nonlocal ABCs for P) at Γ e with the transmission conditions u r=r + = u r=r, σ n u, p) r=r + = σ n u, p) r=r, on Γ e,.) where σ n = σ n,σ n ) T is the normal stress defined in.8) and related to.) with n = cos θ,sin θ) T the unit outward normal vector at Γ e.in.) ± refer to the limits from Ω e and Ω i, respectively. The transmission condition.) was also used in Halpern 996, ) for numerical solutions of the exterior Stokes problem defined in the whole plane by using the spectral method. Combining.8),.), noting.3),.4),.9),.6) and.7), one obtains σ n = cos θ ν u ) p + ν sin θ u x x Γe = ν cos θ cos θ u r sin θ ) u p + ν sin θ sin θ u r θ r + cos θ ) u r θ Γ e = ν u r p cos θ G = ν + r W + r R ) W p cos θ Γe r x r x Γe = ν G + rν cos θ W r r sin θ ) W 4νr cos θ W r θ r Γe = ν G W ν x r r + x ) W..) r θ Γ e Plugging.5) into.), noting.8), one has σ n = ν G n r ν R cos θ p n cos nθ + q n sin nθ) Γe R n= n sin θ p n sin nθ + q n cos nθ) R n= = ν G n r ν r=r R p n+ cos nθ + q n+ sin nθ)= ν G + ν G + ν G r r R θ r=r = ν G + ν G r R θ = ν u ν π R,θ) n u R,φ)cos nφ θ)dφ r=r R θ π R = ν π n u R,φ)cos nφ θ) u R,φ)sin nφ θ) dφ π R = ν π cos nφ θ) u R,φ) sin nφ θ) u R,φ) dφ π R θ n φ n φ T u)..)

6 3 W. BAO Similarly, we obtain σ n = ν π cos nφ θ) u R,φ) sin nφ θ) u R,φ) + dφ π R θ n φ n φ T u)..3) It is easy to see that the boundary conditions.),.3) are different from those designed in Han & Bao 997). In fact,.).3) are the exact boundary conditions at Γ e for the problem P). Thus the restriction of the solution u, p) of the problem P) to the bounded domain Ω i is the solution of the following problem: P i ) Find u, p) such that ν u + grad p = f div u =, in Ω i,.4) u = on Γ i, σ n u, p) = T u) T u), T u)) T, on Γ e..5) In the exact boundary conditions.).3), there are infinite terms in the right-hand sides. When we solve the problem numerically, we need to truncate the right-hand sides of.).3) into finite terms. Let Ti N u) = ν π R N π θ cos nφ θ) u i R,φ) i sin nφ θ) u + ) R,φ) i+ ) i dφ n φ n φ Ti N u), i =,..6) Then we derive a series of approximate ABCs at Γ e : σ n u, p) = T N u) T N u), T N u) ) T, on Γe, N =,,,...,.7) where T u) =, ) T is the stress free boundary condition which is often used in engineering literature. Then the original problem P) can be reduced to the following problem defined in the bounded domain Ω i approximately for N =,,,...: P N ) Find u N, p N ) such that ν u N + grad p N = f div u N =, in Ω i,.8) u N = on Γ i, σ n u N, p N ) = T N u N ), on Γ e..9) 3. High-order local ABCs In this section we design high-order local ABCs at Γ e for the problem P). We consider a solution u, p) of P), which consists of the first N harmonics at Γ e. Thus we assume u i R,θ)= ai N ) a + n i cos nθ + bi n sin nθ, i =, ; 3.)

7 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 3 TABLE The coefficients α N) m in the first five local ABCs α N) α N) α N) 3 α N) 4 α N) 5 N = N = 7/6 /6 N = 3 74/6 5/6 /6 N = 4 533/4 43/44 /36 /8 N = 5 388/378 4/643 7/78 3/648 /59 where the an, b n, a n and b n are constants Fourier coefficients, see.6) and.7)). Substituting 3.) into.8), we obtain σ ni u, p)= ν R N ) n a n i cos nθ + bi n sin nθ + ) i ν u i+ ) i R,θ), i =,. R θ 3.) Following the standard method used for elliptic problems Bao & Han, ) and the linear elastic system Han & Bao, ), we design high-order local ABCs at Γ e for the problem P): where T i N u) = ν R σ n u, p) = T N u) N ) m α m N) m u i R,θ) θ m m= T N u), T N u) ) T, N =,,... ; 3.3) + ) i ν R u i+ ) i θ R,θ), i =, ; 3.4) and the coefficients α m N) are listed in Table for the first five local ABCs. Then the original problem P) can be reduced to the following problem defined in the bounded domain Ω i approximately for N =,,...: P N ) Find ũ N, p N ) such that ν ũ N + grad p N = f div ũ N =, in Ω i, 3.5) ũ N = on Γ i, σ n ũ N, p N ) = T N ũ N ), on Γ e. 3.6) 4. Error bounds for the case of using nonlocal ABCs In this section, we present error bounds for the finite-element approximations of problems P N ). These error bounds depend on not only the mesh size, h, and terms used in the

8 3 W. BAO approximate ABCs, N see.6)), but also the location of the artificial boundary, R. This kind of error bound is very useful in engineering applications. Let H m Ω i ) and H s Γ e ) be the usual Sobolev spaces on the domain Ω i and the boundary Γ e with integer m and real number s. Furthermore, m,ωi and m,ωi denote the usual norm and semi-norm on H m Ω i ), respectively Adams, 975). Suppose { } V = v = v,v ) T H Ω i ) : v Γi =, W = L Ω i ). Then the boundary value problem P i )isequivalent to the following variational problem: VP) Find u, p) V W such that where au, v) = ν au, v) + a u, v) + bp, v) = f v), v V, 4.) bq, u) =, q W ; 4.) Ω i i, j= u i v i dx ν u : vdx, u, v V, 4.3) x j x j Ω i a u, v) = T u) v ds = ν π π n u i R,φ)v i R,θ)cos nφ θ) Γ e π i= + ) i u i R,φ)v R,θ)sin nφ θ) i+ ) i dφ dθ = ν π π u ir,φ) v i R,θ) cos nφ θ) π φ θ n i= + ) i u ir,φ) v R,θ) i+ ) i sin nφ θ) dθ dφ, u, v V,4.4) φ θ n bq, v) = q div v dx, v V, q W, 4.5) Ω i f v) = f v dx, v V. 4.6) Ω i Notice that, in the bilinear functional au, v), the integrand is now u : vinstead of εu) : εv) which is used in Han & Bao 997) for incompressible materials. Thus, the stiff matrix of the corresponding finite-element approximation becomes much sparser and could save computational time in solving the corresponding linear system. Furthermore, let a N u, v) = ν π N π π i= + ) i u ir,φ) φ u ir,φ) φ v i+ ) i R,θ) θ v i R,θ) cos nφ θ) θ n sin nφ θ) n dθ dφ, u, v V.4.7) Then the boundary value problem P N )isequivalent to the following variational problem:

9 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 33 VP N ) Find u N, p N ) V W such that au N, v) + a N un, v) + bp N, v) = f v), v V, 4.8) bq, u N ) =, q W. 4.9) If we replace V and W by their conforming finite-element subspaces V h V and W h W in which h represents the mesh size Ciarlet, 978), then the finite-element approximation of the problem VP N )is VP h N ) Find uh,n, p h,n ) V h W h such that au h,n, v h ) + a N uh,n, v h ) + bp h,n, v h ) = f v h ), v h V h, 4.) bq h, u h,n ) =, q h W h. 4.) We note that the symmetric bilinear form a, ) is bounded and coercive on V V from the Poincaré inequality Adams, 975), i.e. there exist two positive constants M, M such that au, v) M u V v V, u, v V, 4.) M v V av, v), v V. 4.3) Thus we can define an equivalent norm on the space V : Therefore, we have that v =av, v) / = ν v,,ωi, v V. 4.4) au, v) u v, u, v V, 4.5) v av, v), v V. 4.6) For the bilinear forms a, ) and a N, ), wehave the following lemma. LEMMA 4. The following inequality holds: a N v, v) a v, v) 3av, v) 3 v, v V, N, 4.7) a u, v) 3 u v, a N u, v) 3 u v, u, v V, N, 4.8) where a u, v) and a N u, v) are defined in 4.4) and 4.7), respectively. Proof. Follow the proof for elliptic problems in Han & Bao ) and notice the inequality 3.4) on page of Han & Bao, ). For the bilinear form bq, v),wehave the following lemma. LEMMA 4. There exists a generic constant β > independent of R, such that bq, v) sup β q W, q W. 4.9) v V \{} v Proof. Follow the proof in Han & Bao 997).

10 34 W. BAO In order to derive the well-posedness of the problem VP h N ) and an error bound of the finite-element approximation, we suppose that the following discrete inf sup condition between V h and W h holds: bq h, v h ) sup v h V h \{} v h β qh W, q h W h, 4.) where β is a constant independent of h, N and R. It follows immediately from 4.5) 4.) and Theorem 4. in Chapter I of Girault & Raviart 986) that the variational problems VP), VP N ) and VP h N ) are well-posed; that is, for f V,the dual of V, there exists a unique u, p) V W solving VP), a unique u N, p N ) V W solving VP N ), a unique u h,n, p h,n ) V h W h solving VP h N ), and u + u N + u h,n + p W + p N W + p h,n W C f V. 4.) Note that the well-posedness of VP) implies immediately the well-posedness of the original problem P). Let R = max{ x : x supp f Γ i }, Γ = {R,θ) : θ π} and Ω ={x Ω i : x < R } and Γ r ={r,θ) : θ π}. Werecall an equivalent definition of Sobolev space H s Γ r ) for any real number s Adams, 975): and Thus we use w H s Γ r ) wr,θ)= p + p m cos mθ + q m sin mθ) πp m= + π + m ) s pm + q m )<. m= / w s,γr = πm s pm + q m ) 4.) m= as a semi-norm of the space H s Γ r ). Then we have the following estimate. LEMMA 4.3 Suppose that u, p) V W is the solution of the exterior Stokes problem P) and there exists an integer k such that u Γ H k+ Γ ). Then we have that a u, v) a N u, v) C N + ) k R where C is a generic constant independent of u, N, h and R. R ) max{,n } u k+,γ v, v V, 4.3)

11 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 35 Proof. Assume that u i R,θ)= pi + pn i cos nθ + qi n sin nθ), i =,, 4.4) where v i R,θ)= ci + cn i cos nθ + di n sin nθ), i =, ; 4.5) p i n = π π u i R,θ)cos nθ dθ, q i n = π π u i R,θ)sin nθ dθ, i =,, n ; 4.6) c i n = π π v i R,θ)cos nθ dθ, d i n = π π v i R,θ)sin nθ dθ, i =,, n. 4.7) Noting that u, p) satisfies the homogeneous exterior Stokes equations say.) with f = )inthe domain {x : x > R },bythe method of separation of variables, we obtain u r,θ)= r R + p R n n ) pn q n ) cos nθ + q n + p n ) sin nθ r n n=3 + pn cos nθ + q n sin nθ)rn r n, R r, θ π, 4.8) u r,θ)= r R + p R n n ) qn + p n ) cos nθ + p n q n ) sin nθ r n n=3 + pn cos nθ + q n sin nθ)rn r n, R r, θ π. 4.9) Setting r = R in 4.8) and 4.9), we obtain where u i R,θ)= ai + an i cos nθ + bi n sin nθ), i =, ; 4.3) ) { n an = R p n, n =,,, R pn + n )R R ) p R n q n ), n 3; 4.3)

12 36 W. BAO ) { n bn = R q n, n =,, R qn + n )R R ) q R n + p n ), n 3; 4.3) ) { n an = R p n, n =,,, R pn n )R R ) q R n + p n ), n 3; 4.33) ) { n bn = R q n, n =,, R qn + n )R R ) p R n q n ), n ) Inserting 4.3), 4.5) into 4.4) and 4.7), using the orthogonality of the cosines and sines, noting 4.3) 4.34), 4.6), 4.8), 4.7), 4.), 4.4) and 4.5), we obtain a u, v) a N u, v) = νπ n bn c n a n d n b n c n + a n d n + i= N νπ n bn c n a n d n b n c n + a n d n + = νπ 3νπ C + n=n+ i= ) an i ci n + bi n di n i= ) an i ci n + bi n di n ) n bn c n a n d n b n c n + a n d n + an i ci n + bi n di n n n=n+ n=n+ n / an i ) + bn i )) i= i= n 3 n=max{,n } R C N + ) k R n n=n+ / cn i ) + dn i )) pn i ) + qn i )) R n R n pn i ) + qn i )) R n / R i= n v ) max{,n } u k+,γ v. 4.35) Combining Lemmas , we obtain the following error bound. THEOREM 4. Let u, p) be the solution of the problem P) and u h,n, p h,n ) be the solution of the problem VP h N ). Suppose the discrete inf sup condition 4.) holds, f i=

13 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 37 L Ω i ) and u Γ H k+ Γ ) k ). Then we have the following error bound: u u h,n + p p h,n W C inf u vh + inf p qh W v h V h q h W h ) max{,n } R u R k+,γ + N + ) k. 4.36) Proof. The proof of this theorem is by using a standard technique of the mixed finiteelement method Girault & Raviart, 986) and noting the estimate 4.3). The details are omitted here. Suppose that u H k+ Ω i ), p H k Ω i ), u Γ H k+ Γ ) and the interpolation errors of V h to V and W h to W Ciarlet, 978; Girault & Raviart, 986) satisfy inf u vh V + inf p ph W C h k ) u k+,ωi + p k,ωi. 4.37) v h V h q h W h Then combining 4.37) and 4.36), noting 4.3), 4.4), and the Poincaré inequality, we obtain u u h,n,ω + p p h,n,ω C u u h,n,ω + p p h,n,ω ) C u u h,n,ωi + p p h,n,ωi ) C u u h,n + p p h,n W ) C h k u k+,ωi + p k,ωi ) + N + ) k ) max{,n } u R k+,γ. 4.38) R For example, for the Taylor Hood P/P) element which satisfies 4.) Girault & Raviart, 986), the error bound 4.38) holds for k =. 5. Error bounds for the case of using local ABCs In this section, we present the finite-element formulation of the problem P N ) and provide an error bound for the finite-element approximation. To cope with the high-order local ABCs 3.3), we define Let Ṽ ={v H Ω i ) : v Γe H N Γ e ), v Γi = }. ã N u, v) = Γ e v T N u) ds = ν + N m= α N) m i= π m u i R,θ) θ m v R,θ) u θ R,θ) v R,θ) u θ R,θ) m v i R,θ) θ m dθ, u, v Ṽ 5.)

14 38 W. BAO Then the weak form of the problem P N ) is: ṼP N ) Find ũ N, p N ) Ṽ W such that aũ N, v) +ã N ũn, v) + b p N, v) = f v), v Ṽ, 5.) bq, ũ N ) =, q W. 5.3) If we replace Ṽ and W by their conforming finite-dimensional subspaces, Ṽ h Ṽ and W h W in which h is the mesh size Ciarlet, 978), then the finite-element approximation of the problem ṼP N ) is: ṼP h N ) Find ũh,n, p h,n ) Ṽ h W h such that aũ h,n, v h ) +ã N ũh,n, v h ) + b p h,n, v h ) = f v h ), v h Ṽ h, 5.4) bq h, ũ h,n ) =, q h W h. 5.5) From 4.5), 4.6) and 4.9), the well-posedness of the problem ṼP N ) depends on the property of the bilinear form ã N u, v). Forany u, v Ṽ,wecan also formally expand u Γe = ur,θ)and v Γe = vr,θ)in Fourier series see 4.3) and 4.5)). Substituting 4.3) and 4.5) into 5.) and using the orthogonality of the cosines and sines, we obtain ) ã N u, v) = νπ n bn c n a n d n b n c n + a n d n where +γ N) n γ N) n = i= N m= a i n ci n + bi n di n), u, v Ṽ, 5.6) n m α m N), n N. 5.7) Thus the property of ã N N) u, v) depends on the property of γ n.infact, we have the following lemma. LEMMA 5. For any odd N 9, there exist two generic positive constants C ) and N C ) depending only on N such that N ã N u, v) C ) u N N,Γ R v N,ΓR, u, v Ṽ, 5.8) C ) N v N,Γ R ã N v, v), v Ṽ. 5.9) Proof. For any odd N 9, noting 5.7) and Lemma 3. in Bao & Han ), we know that there exist positive constants C ) and C ) such that N N γ n N) n, C ) N nn γ n N) C ) N nn, n =,, 3,... 5.) If N is even, then γ n N) γ n N) <, when n is sufficient large, lim n + n N = αn) N <. 5.)

15 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 39 From 5.) and 5.6), noting 4.) with r = R, 4.3) and 4.5), we have that ã N u, v) C) N n bn c n a n d n b n c n + ) a n d n + n N an i ci n + bi n di n C ) N n N i= a i n ) + b i n )) / n N i= i= d i n ) + c i n )) / C ) N u N,Γ R v N,ΓR, u, v Ṽ. 5.) Furthermore, from 5.) and 5.6) with u = v, noting 4.) with r = R and 4.5), we obtain ã N v, v) νπ +νπ νπ C ) N γ N) n cn i ) + dn i )) i= n dn c n c n d n + cn i ) + dn i )) γ N) n i= n N i= i= c i n ) + d i n )) + νπ ) ) n dn c n + cn d n c i n ) + d i n )) = C ) N v N,Γ R, v Ṽ. 5.3) Thus the desired inequalities 5.8) and 5.9) are proved. REMARK 5. For any even N, noting 5.) and 5.6), the inequality 5.9) does not hold. In fact, it is easy to construct a function v Ṽ such that ã N v, v) <foran even N. Thus the bilinear form au, v) +ã N u, v) is not coercive on Ṽ. Therefore one cannot prove the well-posedness of the problem ṼP N ). This phenomenon was also observed in numerical simulations of elliptic equations Bao & Han, ; Givoli et al., 997), and the linear elastic system Han & Bao, ) in an unbounded domain: the errors of the finite-element solution when choosing N =, 4,... in high-order local ABCs are much larger than those when choosing N =,, 3,... From the discussion above, noting the Poincaré inequality, we assign the following norm on Ṽ: / / v := av, v) + v N,Γ R v + v N,Γ R, v Ṽ. 5.4) For the well-posedness of the problem ṼP h N ), we assume that the following discrete inf sup condition between Ṽ h and W h holds: bq h, v h ) sup v h Ṽ h v \{} h β qh W, q h W h, 5.5)

16 4 W. BAO where β is a constant independent of h, N and R. It follows immediately from 4.5), 4.6), 4.9), 5.5), 5.8) and 5.9) that the variational problems ṼP N ) and ṼP h N ) are well-posed in the case of odd N 9 or N = ; that is, for f Ṽ, the dual of Ṽ,there exists a unique ũ N, p N ) Ṽ W solving ṼP N ), a unique ũ h,n, p h,n ) Ṽ h W h solving ṼP h N ), and ũ N + ũ h,n + p N W + p h,n W M N f Ṽ, odd N 9, 5.6) where M N is a constant. When N =, then ã N u, v) = a N u, v). We have dealt with this case in the previous section. From now on, we always assume that N 9 is an odd integer. For the bilinear form ã N u, v),wehave the following estimate. LEMMA 5. Suppose that u, p) V W is the solution of the exterior problem P) and u Γ H N+ Γ ). Then we have the following estimate for any odd N 9: a u, v) ã N u, v) C N) R where C N) is a generic constant independent of u, R and h. R ) max{,n } u N+,Γ v N,ΓR, v Ṽ, 5.7) Proof. Inserting 4.3), 4.5) into 4.4), using the orthogonality of the cosines and sines, noting 5.6), 5.), 4.3) 4.34), 5.7) and 4.), we obtain for any odd N 9 a u, v) ã N u, v) = νπ C N) C N) n=n+ C N) R R γ n N) n n N n=n+ i= ) i= n N+ n=max{,n } i= ) an i ci n + bi n di n C N) a i n ) + b i n )) / n N n=n+ p i n ) + q i n )) R n R n n N n=n+ i= i= / v N,ΓR an i ci n + bi n di n c i n ) + d i n )) / ) max{,n } u v N+,Γ N,ΓR, v Ṽ. 5.8) Combining Lemmas 5. and 5., we obtain the following error bound. THEOREM 5. Let u, p) be the solution of the problem P) and ũ h,n, p h,n ) be the solution of the problem ṼP h N ). Suppose the discrete inf sup condition 5.5) holds, f L Ω i ) and u Γ H N+ Γ ). Then we have the following error bound for any odd

17 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 4 N 9: u ũ h,n + p p h,n W C N) inf u v h + inf p qh W v h Ṽ h q h W h ) max{,n } R + u N+,Γ. 5.9) R Proof. The proof of this theorem is also by using a standard technique of the mixed finiteelement method Girault & Raviart, 986) and noting v v for all v Ṽ and the estimate 5.7). The details are also omitted here. Suppose u H k+ Ω i ), p H k Ω i ), u ΓR H k+n Γ R ) and the interpolation errors of Ṽ h to Ṽ and W h to W Ciarlet, 978; Girault & Raviart, 986; Givoli et al., 997) satisfy: inf u v h + inf p qh W C h k u k+,ωi + p k,ωi + u k+n,γr. v h Ṽ h q h W h 5.) Then combining 5.) and 5.9), noting the Poincaré inequality, 4.) and 4.4), we obtain for any odd N 9 u ũ h,n,ω + p p h,n,ω ) ) C u ũ h,n,ω + p p h,n,ω C u ũ h,n,ωi + p p h,n,ωi ) C u ũ h N + p p h,n W C N) h k ) ) max{,n } R u k+,ωi + p k,ωi + u k+n,γr + u N+,Γ. 5.) R 6. Numerical results In this section we present numerical results which demonstrate the performance of the error bounds 4.38) and 5.). We consider the numerical implementation for the finiteelement approximation by using a non)local ABC at Γ e. When dealing with high-order local ABCs, we only consider the case of N = inthis section. In this case, the usual finite-element subspace V h of V proposed in Girault & Raviart 986) can be used as Ṽ h directly. When dealing with the case of N > inlocal ABCs, special subspace Ṽ h which has a higher regularity at Γ e should be used. A family of this kind of spaces was introduced in Givoli et al. 997). In our computations, the Taylor Hood element i.e. P/P) which satisfies the discrete inf sup condition 4.) Girault & Raviart, 986) was used to construct the finite-element subspaces V h and W h. That is to say, k = inthe interpolation errors 4.37) and 5.) Ciarlet, 978). The integrations on the circle in 5.) and 4.7) are evaluated on each element numerically by a Gaussian quadrature. EXAMPLE An exterior Stokes problem. We consider the exterior Stokes equations in the planar domain outside a circular obstacle of radius a = 5 see Fig. ). The problem is

18 4 W. BAO governed by the following boundary value problem: ν u + grad p = f in Ω ={x : 5 < x }, div u = in Ω, 6.) u 5,θ)= gθ) g θ), g θ)) T, on Γ i = Ω, 6.) u is bounded, p, when r = x + x + ; 6.3) where x 5 48ν)x + 48ν)x + 4ν x ), 5 x <, f x) =, x ; x + 48ν)x ν)x 4ν x ), 5 x <, f x) =, x ; g θ) = cos θ sin θ 4ν 565 sin θ 5 sin θ ln 5 + sin θ + 7 sin θ, θ π; 8 g θ) = cos θsin θ 5) 4ν 565 sin θ) 7 cos θ, θ π. 8 This problem has an exact solution: u x) = 4ν 4ν x x + x 5) x x + x + 5) ln x + x 5) x + x + 5) x x +x 5) x x +x 5) ln x +x 5) x +x + 5) + x x ) 3, 5 x <,, x ; u x) = 4ν 4ν x x 5) x +x 5) x x + 5) x +x x + 5) x ) 3, 5 x <, x x 5) x +x 5) x x + 5) x +x, x ; + 5) px) = x x +x 5) x x +x + x + 5) x x ), 5 x <, x x +x 5) x x +x, x + 5) In this example, we take ν = and the unbounded domain Ω ={x R which is the exterior domain outside a circle Γ i ={x R : x = 5}. : 5 < x }

19 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 43 TABLE The effect of the mesh size h using nonlocal ABCs Mesh h = 7368 h = 3684 h = 84 h = 9 max u u h,n E 6 748E E E 4 max p p h,n u u h,n,ω 4 754E 4 69E E 4 384E 4 u u h,n,ω p p h,n,ω 3 834E 6 9E 76E 4 9E 3 _ 3 Errors _ 4 _ 5 _ 6 _ 7 _ ln h FIG.. The effect of the mesh size h by using nonlocal ABCs. + +: ln u u h,n,ω,**:ln u u h,n,ω, oo:ln p p h,n,ω. First we test the effect of the mesh size h in the error bound 4.38), we introduce a circular artificial boundary Γ e = Γ of radius R = R =. On Γ we apply the nonlocal ABC.7) with a very large N say N = 5). In this case, the numerical error comes mainly from the finite-element discretization because the error from the approximate ABC is negligible. In the annular computational domain Ω,weuse four meshes respectively. The first mesh consists of one radial layer of elements, with 6 triangular elements in the layer. We denote it as 6. The other three meshes are 3, 4 64 and 8 8. Table shows the maximum errors of u u h,n, p p h,n over the mesh points, and u u h,n,ω, u u h,n,ω, p p h,n,ω for large N say N = 5).

20 44 W. BAO 4 Errors _ 6 _ 8 _ 4 _ lnn+) FIG.3. The effect of N by using nonlocal ABCs. + +: ln u h, u h,n,ω,**:ln u h, u h,n,ω,oo: ln p h, p h,n,ω. Figure shows the errors u u h,n,ω, u u h,n,ω and p p h,n,ω for large N say N = 5). Second we test the effect of N in the error bound 4.38). In order to do so, we choose Γ e = Γ with R = R so that the effect of R in the error bound 4.38) disappears. Let u h,, p h, ) denote the finite-element approximation of the problem on the domain Ω with the mesh size h when N is very large say N = 5). Figure 3 shows the errors E N := u h, u h,n k,ω k =, ) and p h, p h,n,ω on the mesh 8 8 for N =,, 3, 5, 7, 9. Third we test the effect of the location of the artificial boundary Γ e. Let Ω R ={x : 5 < x < R} denote the bounded computational domain with the artificial boundary Γ R. We choose R =, 5,, 5, 3 respectively. The corresponding meshes we used were 4 64, 8 64, 64, 6 64 and 64. That is to say, each computational domain has a mesh with the fixed mesh size h = 84. Let u R,N, p R,N ) denote the finiteelement approximation of the problem on the domain Ω R with the corresponding mesh by using the nonlocal ABCs.7) on the artificial boundary Γ R, u R,, p R, ) correspond to the solution when N is very large say N = 5) and ũ R,N, p R,N ) correspond to the solution by using the high-order local ABC 3.3) at Γ R with N =. Figures 4 and 5 show the errors E R := u R, u R,N,Ω and E R := p R, p R,N,Ω for R =, 5,, 5, 3. Figure 6 shows the errors E R := u R, ũ R,N,Ω and p R, p R,N,Ω for R =, 5,, 5, 3. From Table and Fig., one can see that the convergent rates of u u h,n,ω and p p h,n,ω with respect to h are approximately when using nonlocal ABCs with avery large N, which are consistent with the error bound 4.38) with k =. Figure 3 confirms the effect of N in the error bound 4.38). Furthermore, Figs 4 and 5 confirm the

21 ERROR BOUNDS FOR THE FINITE-ELEMENT APPROXIMATION 45 _ ln u R, u R, N,Ω 3 _ 4 _ 5 _ 6 _ 7 N= N= N=3 N=5 8 _ 6 _ 4 _ lnr /R) FIG.4. The effect of R with respect to u by using nonlocal ABCs. 4 ln p R, p R, N,Ω _ 6 _ 8 _ 4 N= N= N=3 N=5 8 _ 6 _ 4 _ lnr /R) FIG.5. The effect of R with respect to p by using nonlocal ABCs. effect of R in the error bound 4.38) and Fig. 6 confirms the effect of R in the error bound 5.). The minor discrepancies when N = 5inFigs 4 and 5 are due to round-off errors because in this case the errors themselves are very small.

22 46 W. BAO Errors _ 3 _ 4 _ 5 _ 6 8 _ 6 _ 4 _ lnr /R) FIG.6. The effect of R by using a local ABC N = ). + +: ln u R, u R,N,Ω,**:ln u R, u R,N,Ω, oo:ln p R, p R,N,Ω. 7. Conclusions Afamily of high-order non)local ABCs for numerical simulations of the exterior Stokes equations in an unbounded domain is designed. The original problem is then reduced to a problem defined in a bounded computational domain by imposing a non)local ABC at a circular artificial boundary. The finite-element formulation is presented. Error bounds for the case of using non)local ABCs are obtained. This kind of error bounds depends on not only the mesh size, terms used in the ABCs, but also the location of the artificial boundary. They can be used to choose the mesh size, terms used in the ABCs and the location of the artificial boundary for practical computations. Numerical results demonstrate the performance of our error bounds. 8. Acknowledgements The author acknowledges support by the National University of Singapore and the referees for their valuable comments which led to the improvement of this paper. REFERENCES ADAMS, R.A.975) Sobolev Spaces. New York: Academic Press. ANTONIO, M.& MEDDAHI, S.) New implementation techniques for the exterior Stokes problem in the plane. J. Comput. Phys., 7, BAO, W.997) Artificial boundary conditions for two-dimensional incompressible viscous flows around an obstacle. Comput. Methods Appl. Mech. Engrg., 47, BAO, W.998) The approximations of the exact boundary condition at an artificial boundary for linearized incompressible viscous flows. J. Comput. Math., 6,

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24 48 W. BAO MEDDAHI, S.& SAYAS, F. J.) A fully discrete BEM-FEM for the exterior Stokes problem in the plane. SIAM J. Numer. Anal., 37, 8. YING, L.A.995) Infinite Element Methods. Braunschweig: Friedr. Vieweg & Sohn.