c 2005 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J NUMER ANAL Vol 42, No 5, pp c 2005 Society for Industrial and Applied Mathematics ERROR ESTIMATES FOR A FINITE VOLUME ELEMENT METHOD FOR ELLIPTIC PDES IN NONCONVEX POLYGONAL DOMAINS P CHATZIPANTELIDIS AND R D LAZAROV Abstract We consider standard finite volume piecewise linear approximations for second order elliptic boundary value problems on a nonconvex polygonal domain Based on sharp shift estimates, we derive error estimations in H 1 -, L 2 - and L -norms, taking into consideration the regularity of the data Numerical experiments and counterexamples illustrate the theoretical results Key words finite volume element method, nonconvex polygons, error estimations AMS subject classifications 65N15, 65N30 DOI /S Introduction We analyze the standard finite volume element method for the discretization of second order linear elliptic PDEs on a nonconvex polygonal domain Ω R 2 Namely, for a given function f, we seek u such that (11) Lu div(a u) = f in Ω, and u = 0 on Ω with A =(a ij ) 2 i,j=1 a given symmetric matrix function with real-value entries a ij W 1,1 i, j 2 We assume that the matrix A is uniformly positive definite in Ω, ie, there exists a positive constant α 0 such that (12) ξ T A(x)ξ α 0 ξ T ξ ξ R 2, x Ω The class of finite volume methods is based on some approximation of the balance relation (13) A u nds= f dx, b b which is valid for any subdomain b Ω Here n denotes the outer unit normal vector to the boundary of b There are various approaches to the finite volume method One, the finite volume element method, uses a finite element partition of Ω, where the solution space consists of continuous piecewise linear functions, a collection of vertex-centered control volumes, and a test space of piecewise constant functions over the control volumes (cf, eg, [6, 10, 25, 28]) A second approach, usually called the finite volume difference method, uses cell-centered grids and approximates the derivatives in the balance equation by finite differences (cf, eg, [22, 29, 33]) Another approach uses mixed reformulation of the problem [12, 16] The first approach is quite close to the finite element method but nevertheless has some new properties that make it attractive for the applications [1, 20] The second approach is closer to the classical finite difference method and extends it to more general than rectangular meshes It is used mostly on perpendicular bisection or Voronoi type meshes Approximations on such rectangular Received by the editors May 5, 2003; accepted for publication (in revised form) February 6, 2004; published electronically January 20, Department of Mathematics, Texas A&M University, College Station, TX, (chatzipa@ mathtamuedu, lazarov@mathtamuedu) 1932

2 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1933 K b z z z K z z K Fig 11 Left-hand side: A sample region with dotted lines indicating the corresponding box b z Right-hand side: A triangle K partitioned into three subregions K z and triangular meshes were studied, for example, in [34] and [26], respectively The third approach is close to mixed and hybrid finite element methods and can deal, for example, with irregular quadrilateral and hexahedral cells [12, 30] Finite volume discretizations for more general convection-diffusion-reaction problems were studied by many authors For a comprehensive presentation and more references of existing results we refer to the monographs on the finite volume difference method [22] and on the finite volume element method [28], and for various applications on the special issue [21] We shall consider a finite volume element discretization of (11), in the standard conforming space of piecewise linear functions, X h = {χ C(Ω) : χ K is linear K T h and χ Ω =0} with {T h } 0<h<1 a given family of triangulations of Ω with h denoting the maximum diameter of the triangles of T h For simplicity we shall assume that T h is a quasiuniform triangulation However, this assumption is only required to show L -norm error estimates For L 2 - and H 1 -norm error estimations, nondegenerate triangulations [9, equation (4416)] are sufficient The finite volume problem will satisfy a relation similar to (13) for b in a finite collection of subregions of Ω called control volumes, the number of which will be equal to the dimension of the finite element space X h These control volumes are constructed in the following way: Let z K be the barycenter of K T h We connect z K with line segments to the midpoints of the edges of K, thus partitioning K into three quadrilaterals K z, z Z h (K), where Z h (K) are the vertices of K Then with each vertex z Z h = K Th Z h (K) we associate a control volume (also called a box) b z, which consists of the union of the subregions K z, sharing the vertex z (see Figure 11) We denote the set of interior vertices of Z h by Zh 0 The finite volume element method is then to find u h X h such that (14) (A u h ) nds= f dx, z Zh 0 b z b z Before we start our description of this work we introduce some notation We will use the standard notation for the Sobolev spaces Wp s and H s = W2 s (cf [2]) Namely, L p (V ), 1 p<, denotes the space of p-integrable real functions over V R 2, (, ) V the inner product in L 2 (V ), H s (V ) and H s (V ) the seminorm and norm, respectively, in H s (V ), W s p (V ) and Wp s (V ) the seminorm and norm, respectively,

3 1934 P CHATZIPANTELIDIS AND R D LAZAROV in Wp s (V ), p 1, and s R In addition, if V = Ω we suppress the index V, and if p = 2 and s = 0 we also suppress these indexes and denote W 0 = Further, 2 we shall denote with p the adjoint of p, ie, 1 p + 1 p =1,p>1 We begin with some comments It is well known that in the case of a polygonal Ω, if f L p,1<p<, then the solution u of (11) is not always in Wp 2 (cf, eg, [24] and section 2) However, it is always in W 2 p or in a fractional order space H 1+s for some 0 <s<1, where s and p, given in section 2, depend on both the maximal interior angle of Ω and p In short, for p large, s and p depend on the maximal interior angle, while for p close to 1, they depend on p In this paper we study the influence of the corner singularities imposed by the nonconvex polygonal domain Ω and the possible insufficient regularity of the righthand side f, say,f L p (Ω), p<2, or f H l (Ω), 0 l<1/2, on the convergence rate of the finite volume element method For domains with smooth boundary and convex polygonal domains, H 1 - and L 2 -norm error estimates were derived in [15] and [23], respectively, taking into account the regularity of f Note that we use the conservative version of the method, namely the right-hand side of the scheme is computed by the L 2 inner product of f with the characteristic functions of the finite volumes (or equivalently by the duality between H l and H l for 0 l<1/2) The reason for l<1/2 is that (14) makes sense for at least f L 1 For results concerning finite volume schemes for problems with more singular f, ie, f H 1, we refer to [19], where an approximation of f is considered b As a model for our analysis we shall consider the corresponding Galerkin finite element method, which is to find u h X h such that (15) a(u h,χ)=(f,χ), χ X h, with a(, ) the bilinear form defined by a(v, w) = A v wdx It is known that u h satisfies (cf, eg, [3, 8] and [9, Chapter 12]) Ω (16) u u h + h δ u u h H 1 Ch s+δ { u H 1+s, u W 2 p, any δ < π/ω, where s is given by (24) or (26), p by (23), and ω denotes the biggest interior angle of Ω (cf section 2) Note that the convergence rate of the finite element method (15) is optimal in the H 1 -norm and suboptimal in the L 2 -norm, since X h has the following approximation properties (cf, eg, [9, p 285]): (17) inf χ X h ( v χ + h v χ H 1) { Ch 1+s v H 1+s, v H 1+s H0 1, 0 <s<1, Ch 3 2/p v W 2 p, v Wp 2 H0 1, 1 p 2 In the literature there are various techniques for improving the convergence rate of a finite element method in nonconvex domains, eg, mesh refinement, augmenting the basis functions with appropriate singular functions (cf, eg, [8, 11]) Also, recently in [18] such a method was analyzed for some finite volume element methods Here, we are interested in the analysis of (14) in a mesh T h, which does not have any prior knowledge of the singularity imposed by the domain

4 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1935 Table 11 Theoretical convergence rate of the finite volume element method versus the finite element method in a nonconvex polygonal domain, when the exact solution u of problem (11) is in H 1+s, where s is defined by (24) or (26), and any δ<π/ω p ω =2/(2 π/ω), s 0 =1 π/ω H 1 -norm L 2 -norm L -norm p ω =2p ω/(3p ω 2) FVE FE FVE FE FVE FE p ω < 2 1 <p< p ω s + δ p ω <p ω p ω <p<p ω min(1,s+ δ) f L p p ω p 1 1 <α< p ω s s + δ s + δ s f Wα t p ω α 2 min(s + δ, 1+t) s 0 < 1/2 l<s 0 1 l f H l s 0 <l<1/2 1 l In Theorems 43 and 52, we show optimal order H 1 -norm error estimates for the finite volume element method (14), if f L p, p>1, and f H l, l (0, 1/2) Thus, the finite element (15) and finite volume element method (14) converge with the same rate in the H 1 -norm However, as in the convex case (cf, eg, [13, 27]), the situation in the L 2 -norm error estimate is quite different The convergence rate in the L 2 -norm of the finite volume element method (14) is suboptimal and lower than the corresponding finite element method In Theorem 43, for f L p, p>1, we show L 2 -norm error estimations where the order cannot be higher than 1 However, assuming additional regularity for f, namely, f Wα, t t (0, 1], α (1, 2], we are able to show, in Theorem 46, L 2 -norm error estimations that, depending on α and t, could be of the same order as the finite element method For example, this is true for α or t sufficiently close to 1 Also, in Theorem 48 we derive almost optimal order L -norm error estimates In section 5, we consider the case where f H l, l (0, 1/2) with A = I and show optimal order H 1 -norm, suboptimal L 2 -norm, and almost optimal L -norm error estimates In Theorem 52, we show again that the convergence rate of the finite volume element method (14) in the L 2 -norm is suboptimal and lower than the corresponding suboptimal rate of the finite element method In Table 11, we summarize the theoretical results concerning the convergence rate of the finite volume element method in the H 1 -, L 2 -, and L -norms obtained in sections 4 and 5 and compare them with the corresponding known results for the finite element method According to (16) the rate of the finite element method in the H 1 -norm and L 2 -norm is s and s + δ, respectively, for any δ<π/ωand s given by either (24) or (26), depending on whether f L p or f H l Note that if we assume that f Wα, t with t (0, 1] and α (1, 2], both methods give the same convergence rate, if α< p ω =2p ω /(3p ω 2) with p ω =2/(2 π/ω) Otherwise, this is determined by min(s + π/ω, 1+t) Also, in section 7 we present some numerical results for Poisson s equation on a Γ-shaped domain The particular examples we consider justify the theoretical results of Theorems 43, 46, and 48 However, these do not show the lower convergence rate in the L 2 -norm of Theorem 43, which occurs if f L p, p>1, and f / Wα, t for any α (1, 2] and t (0, 1) To show that the L 2 -norm estimates of Theorems 43 and 52 are sharp, following [27], we consider two counterexamples A short presentation of parts of this work can be found in [14] For simplicity we choose not to include convection terms in the differential equation (11) But they

5 1936 P CHATZIPANTELIDIS AND R D LAZAROV Ω ω S 0 Fig 21 A nonconvex domain Ω can be included provided they are bounded and the diffusion term is dominating A brief description of this paper is the following: In section 2 we give, in short, known sharp regularity estimates for the exact solution of problems (11) and (21), based on [24, 4] In section 3 we present the finite volume element method In sections 4 and 5, we analyze the finite volume element method (14) and derive error estimates in the H 1 -, L 2 - and L -norms The approach follows the one developed in [13] and uses known sharp regularity results for the solutions of elliptic boundary value problems (cf [24]) In section 6, we derive some auxiliary results, needed in proving Theorems 43, 46, 48, 52, 54, and 55 Finally in section 7, we present numerical examples that illustrate the theoretical results of section 4 2 Preliminaries Let us first consider the Dirichlet problem for Poisson s equation: Given f L p, p>1, find a function u :Ω R 2 such that (21) u = f, in Ω, and u =0 on Ω withωabounded, nonconvex, polygonal domain in R 2 For simplicity we assume that Ω has only one interior angle greater than π, namely ω (π, 2π) (cf Figure 21) It is well known that for such domains there exists a unique solution u H0 1 of (21) The solution u could be represented in the form u = u S + u R, where u R Wp 2 H0 1, and u S = cr λm 1 ωλm sin(λ m θ)η(re iθ ), expressed in polar coordinates (r, θ) with respect to the vertex S 0 with angle ω (cf [24]) Here c is a constant, λ m = mπ ω, m N, and η is a cutoff function which is one near S 0 and zero away from S 0 A crucial role in determining the regularity of the solution u of (11) is played by the constant p ω 2 2 π/ω According to [24, p 233], (22) if f L p,p>1, then u W 2 p, where (23) p = { p, p<pω, γ, any γ<p ω, p p ω, p ω 2 2 π/ω Using also the imbedding W 2 p H 1+s, for s =2 2/ p (cf, eg, [24, p 34]), we obtain the following: (24) if f L p,p>1, then u H 1+s, for s =2 2 p Also, for problem (21) we have (cf, eg, [4]), (25) if f H l, 0 l 1, then u H 1+s,

6 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1937 where (26) s = { 1 l, s0 <l 1, δ, any δ < π/ω, 0 l s 0, s 0 = 2 p 0 1=1 π ω For the more general problem (11), similar results hold Let S be a vertex of Ω, and denote the corresponding interior angle of Ω by ω(s) Let A and T be matrices such that A =(a ij (S)) 2 i,j=1 and T AT T = I, and let ω A (S) be the angle at the vertex TS of the transformed domain T Ω={T x : x Ω} Define ω = max ω 2 A(S) and p ω = S 2 π/ω 3 The finite volume element method In order to analyze the finite volume element method (14) we shall need to rewrite it in a variational form resembling the one for the finite element problem (15) (cf, eg, [13]) For this purpose we introduce the space Y h = {η L 2 (Ω) : η bz is constant, z Z 0 h,η bz =0if z Ω} For an arbitrary η Y h we multiply the integral relation (14) by η(z) and sum over all z Zh 0 Thus we obtain the following Petrov Galerkin formulation of the finite volume element method: Find u h X h such that (31) a h (u h,η)=(f,η), η Y h, where the bilinear form a h (, ) :X h Y h R is defined by a h (v, η) = (32) η(z) (A v) n ds, v X h,η Y h b z z Z 0 h Further, we consider the interpolation operator I h : C(Ω) Y h, defined by (33) I h v = v(z)ϕ z, z Z 0 h where ϕ z is the characteristic function of b z Then, we can rewrite (14) as a h (u h,i h χ)= (34) χ(z) f dx, χ X h b z z Z 0 h Note that for every f L p and χ X h, (f,i h χ)= (35) χ(z) fϕ z dx = z Z Ω h Thus (34) can be written equivalently in the form z Z 0 h χ(z) f dx b z (36) a h (u h,i h χ)=(f,i h χ), χ X h Existence of u h follows from the fact that a h is coercive, for h sufficiently small (cf, eg, [13] or [28, Theorem 321]), c 0 > 0: c 0 χ 2 H 1 a h(χ, I h χ), χ X h

7 1938 P CHATZIPANTELIDIS AND R D LAZAROV Then this, the local stability of I h, I h χ Lp(K) C χ L p(k), χ X h,k T h,p>1, and the Sobolev imbedding χ Lp C χ H 1, χ X h,p>1, give the stability of the finite volume scheme (36), (37) u h H 1 C f Lp, p > 1 Also, note that if A(x) is a constant matrix over each finite element K T h, then a h (χ, I h ψ)=a(χ, ψ), χ, ψ X h (cf, eg, [27]) In particular, if A = I, wehave (38) a h (χ, I h ψ)=a(χ, ψ) = χ ψdx, χ, ψ X h (cf, eg, [6]) Thus, (36) takes the form Ω (39) a(u h,χ)=(f,i h χ), χ X h In the case of general matrix A(x), the identity (38) is not valid However, following [13], we are able to rewrite a h in a form similar to a Indeed, we transform the left-hand side of (14) using integration by parts to get, for z Zh 0 and K T h, Lχ dx + A χ nds= A χ n ds, χ X h K z K z K K z b z Thus, multiplying by ψ(z), ψ X h, and summing over the triangles having z as a vertex and then over the vertices z Zh 0, we obtain (310) a h (χ, I h ψ)= K {(Lχ, I h ψ) K +(A χ n, I h ψ) K }, χ, ψ X h This is similar to a(χ, ψ) (A χ, ψ) = K {(Lχ, ψ) K +(A χ n, ψ) K }, χ, ψ X h Due to this similarity and for convenience, in what follows we shall use (310) as a definition of the bilinear form a h 4 Nonsmooth data: L p case In this section we shall derive H 1 -, L 2 -, and L -norm estimates of the error u u h for f L p, p>1 First, we shall demonstrate that the finite element method (15) and finite volume element method (36) have the same convergence rate in the H 1 -norm The L 2 -norm error estimate is quite different, and we derive two separate results First, we will show suboptimal order L 2 -norm error estimates for f L p, p>1, where the order is less than in the corresponding order for the finite element scheme (15) Next, assuming higher regularity for f, namely f W t α, t (0, 1], α (1, 2], we will show again suboptimal order L 2 -norm error estimates, but now depending on α and t, these could be of the same order as the corresponding estimates of the finite element scheme Finally, we show almost optimal L -norm estimates of the error u u h

8 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1939 For the analysis of the finite volume element method (36) we shall need to estimate the errors ε h and ε a defined by ε h (f,χ) =(f,χ) (f,i h χ), f L p,χ X h, ε a (χ, ψ) =a(χ, ψ) a h (χ, I h ψ), χ, ψ X h In section 6 we will give the proof of the following two lemmas Lemma 41 There exists a constant C such that for every χ X h, (41) (42) ε h (f,χ) Ch f Lp χ W 1 p, f L p, 1 p + 1 p =1, ε h (f,χ) Ch 1+t f W t p χ W 1 p, f W t p, 0 <t 1 Lemma 42 Assume that A W 2 Then there exists a positive constant C = C(A) such that (43) ε a (ψ, χ) Ch ψ W 1 p χ W 1 p, χ, ψ X h, ε a (u h,χ) Ch ( ) (44) (u u h ) L2 + h u W χ W, χ X 1 h 2 p p Next, we derive H 1 - and L 2 -norm error estimates for the finite volume element method (14) Theorem 43 Let u and u h be the solutions of (11) and (14), respectively, with f L p, p>1 Then, there exists a constant C, independent of h, such that (45) (46) u u h H 1 C ( h s u W 2 p + h min(1,2 2/p) f Lp ) Ch s f Lp, u u h C ( h s+δ u W 2 p + h min (1,s+δ) f Lp ), for any δ < π/ω, with p and s given by (23) and (24), respectively Remark 44 The H 1 -norm error estimation (45) is of optimal order (cf (17)) However, the L 2 -norm error estimation is not of the same order as the finite element approximation (cf (16)) for every p For example, for p sufficiently close to 1, s + δ < 1, thus, u u h = O(h s+δ ) However, for p 2, s =2 2/ p π/ω Therefore, since s + δ 2π/ω > 1, u u h = O(h) The most interesting outcome of this theorem is that the convergence rate for the L 2 -norm is suboptimal and lower than the rate of the finite element method (15) This estimate is sharp, as first demonstrated by a counterexample in [27], for convex domains Later in section 7 we give a similar example to the one in [27], which shows the sharpness of the L 2 -error estimate (46) Proof In view of (22), u W 2 p, with p defined by (23) Using the triangle inequality, (47) u u h H 1 u χ H 1 + u h χ H 1, χ X h, and the approximation properties (17) of X h, it suffices to consider the last term of (47) The positive definiteness of A, (12), gives (48) α 0 u h χ 2 H 1 a(u h χ, u h χ), χ X h

9 1940 P CHATZIPANTELIDIS AND R D LAZAROV Thus, in view of a(u h χ, u h χ) =a(u u h,u h χ)+a(u χ, u h χ) a(u u h,u h χ)+c u χ H 1 u h χ H 1, χ X h, and (48), we get for every χ X h, (49) u h χ 2 H 1 C a(u u h,u h χ) + C u χ 2 H 1 In addition, using the definitions of ε h and ε a,wehave (410) a(u u h,u h χ) =a(u, u h χ) a h (u h,i h (u h χ)) ε a (u h,u h χ) = ε h (f,u h χ) ε a (u h,u h χ), χ X h Applying then, to this relation, (41), (43), and the inverse inequality we obtain χ W 1 p Ch2/p 1 χ H 1, p > 2, χ X h, (411) a(u u h,u h χ) C(h min (1,2 2/p) f Lp + h u h H 1) u h χ H 1 Thus, for h sufficiently small, this estimate, (37) and (49) yield (412) u h χ H 1 C u χ H 1 + Ch min (1,2 2/p) f Lp, χ X h, which combined with (17) and (47) gives u u h H 1 C ( h s u W 2 p + h min(1,2 2/p) f Lp ) Using now the fact that for p<p ω, s =2 2/p and for p p ω, s<min(1, 2 2/p), we get (413) u u h H 1 Ch s( u W 2 p + f Lp ) Finally, employing the a priori regularity estimation of u, (22), we obtain the desired estimate (45) We now prove (46) by using a duality argument We consider the following auxiliary problem: Seek ϕ H0 1 such that (414) Lϕ = u u h in Ω and ϕ =0 on Ω In view of (22) and the fact that u u h L 2, we have ϕ W 2 γ, where γ<p ω, ie 2/γ =2 π/ω + ε, with arbitrary small ε>0, and satisfies the a priori estimate (415) ϕ W 2 γ C u u h, γ < p ω Now let Π h : W 2 γ H 1 0 X h denote the standard nodal interpolation operator It is well known that Π h has the following approximation property (cf, eg, [17, Theorem 316] and [2, Theorem 54]), (416) Π h v v H 1 Ch π/ω ε v W 2 γ, v W 2 γ H 1 0,

10 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1941 and Π h is bounded in W 1 q (cf, eg, [17, Theorem 316] and [2, Theorem 54]), (417) Π h v W 1 q C v W 2 γ, v W 2 γ H 1 0, q p γ =2γ/(2 γ), where p γ =2γ/(3γ 2) Using (414) and Green s formula, we easily obtain (418) u u h 2 = (u u h,lϕ)=a(u u h,ϕ) = a(u u h,ϕ Π h ϕ)+a(u u h, Π h ϕ):=i + II The first term, I, can obviously be bounded in the following way by using (413) and (416): (419) I C u u h H 1 ϕ Π h ϕ H 1 Ch s+π/ω ε( u W 2 p + f Lp ) ϕ W 2 γ Also, in view of (410), the second term, II, can be written in the form (420) II = a(u u h, Π h ϕ)=ε h (f,π h ϕ) ε a (u h, Π h ϕ) Then using (41) and (44), II can be estimated by (421) II Ch f Lp Π h ϕ W 1 + h( ) (u u h ) p L2 + h u W Πh ϕ 2 p W 1 p In order to bound Π h ϕ and Π W 1 hϕ p W 1 in (421) we consider two different cases p for p: (1) p p γ =2γ/(3γ 2), and (2) 1 <p<p γ We can easily see that p γ <p ω Thus, in view of the definition of p (cf (23)), for p p γ we also have p p γ and for 1 <p<p γ, p<p γ Let us first consider the case p p γ Then we have p p γ and p p γ so for the respective norms of Π h ϕ in (421) we can apply the estimate (417) Using also (413) we get (422) II C(h f Lp + h 1+s ( u W + f 2 p Lp )) ϕ W 2 γ C(h f Lp + h 1+s u W 2 p ) ϕ W 2 γ Combining now this estimation with (419), (415), and (418), we obtain the desired estimate (46), for p p γ In the remaining case 1 < p < p γ we cannot directly employ (417) for the estimation of II However, the inverse inequality and (417), give χ W 1 Ch 2/q 2/p γ χ W 1, q > p q γ, χ X h, p γ (423) Π h v W 1 q Ch 2/q 1+π/ω ε v W 2 γ, v W 2 γ H 1 0,q>p γ Using now this estimation in (421) and the fact that for 1 <p<p γ,2/p =2 2/p = s, weget (424) II Ch 2/p +π/ω ε ( f Lp + h u W 2 p ) ϕ W 2 γ Ch s+π/ω ε( f Lp + h s u W 2 p ) ϕ W 2 γ

11 1942 P CHATZIPANTELIDIS AND R D LAZAROV Then, combining this estimation with (419), (415), and (418), we obtain u u h Ch s+δ ( u W 2 p + f Lp ) Finally, (46) follows from the fact that for 1 <p<p γ, s =2 2 p < 2 2 p γ = 2 γ 1= 1 π ω + ε Remark 45 For the proof of Theorems 43 and 46 it is not necessary to assume a quasi-uniform mesh T h This is done in order to simplify the proof, and it is only required for the validity of the inverse inequalities that are used This assumption can be avoided by applying local inverse inequalities which hold in more general triangulations Next, we shall demonstrate that under some additional assumptions on the smoothness of the data the convergence rate in the L 2 -norm can be improved and be equal to the rate of the corresponding finite element method Theorem 46 Let u and u h be the solutions of (11) and (14), respectively Assume that f Wα, t 1 <α 2, 0 <t 1, and A W 2 Then there exists a constant C, independent of h, such that (425) u u h C(h s+δ f Lp + h 1+t+min(0,1 2/α+δ) f W t a ), for any δ < π/ω, with p =2α/(2 tα) and p and s given by (23) and (24), respectively Remark 47 For α< p ω =, we have 2/α > 1+π/ω Thus1+t+min(0, 1 2pω 3p ω 2 2/α + δ) =2+t 2/α + δ =2 2/p + δ s + δ Therefore, u u h = O(h s+δ ), ie, in this case the L 2 -norm error estimate of the finite volume element method has the same convergence rate as the corresponding finite element method If α p ω, ie, 2/α 1+δ, then the order of u u h is min(s + δ, 1+t) Proof The proof will be similar to the one for (46) First, let us note that since f Wα, t we have by imbedding (cf [2, Theorem 757]) that f L p, with p =2α/(2 tα) Thus, in view of (22), u W 2 p with p given by (23) Let again γ<p ω, such that 2/γ =2 π/ω + ε, with arbitrary small ε>0, and let ϕ Wγ 2 H0 1 be the solution of the auxiliary problem (414) Obviously, in order to show a higher order L 2 -norm error estimation of u u h, we need to derive better bounds for I and II of (418) It is obvious that the estimation of I, (419), derived in Theorem 43 is of the desired order Thus, it suffices to show a better estimate for II than the ones derived in Theorem 43 (cf (422) and (424)) Using (42) and (44) in (420), we get (426) II C(h 1+t f W t α Π h ϕ + W 1 h1+s f α Lp Π h ϕ ) W 1 p Similarly, as in Theorem 43 we need to derive bounds for Π h ϕ W 1 α and Π h ϕ W 1 p, and we will need to consider various cases for α and p with respect to p γ =2γ/(3γ 2) Since p =2α/(2 tα), we can easily see that p>α; thus we have the following three cases: (1) p>α p γ, (2) p p γ >α, and (3) p γ >p>α First, we consider the case p>α p γ For such p, according to (23), we have p >p γ Thus, using (417) in (426), and the fact that 1/2 < π/ω < 1, we get (427) II C(h 1+t f W t α + h s+π/ω ε f Lp ) ϕ W 2 γ

12 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1943 Therefore, combining this estimation, (419), (415), (418), and the fact that if α>p γ, then 2/α < 2/p γ =1+π/ω ε, we obtain the desired result, (425), in the case p>α p γ Now let p p γ >α Again we can easily see that p p γ Therefore, applying (423) and (417) in (426) and using the fact that 1/2 < π/ω < 1, we obtain (428) II C(h t+2/α +π/ω ε f W t α + h 1+s f Lp ) ϕ W 2 γ C(h 1+t+1 2/α+π/ω ε f W t α + h s+π/ω ε f Lp ) ϕ W 2 γ Therefore, combining this estimation, (419), (415), (418), and the fact that if α p γ, then 2/α > 1+π/ω ε, we obtain the desired result, (425), if p>p γ α In the remaining case p γ >p>α, we have p = p<p γ Thus, applying (423) in (426) and using the fact that 1/2 < π/ω < 1, we have (429) II C(h t+2/α +π/ω ε f W t α + h 2/p +π/ω ε+s f Lp ) ϕ W 2 γ C(h 1+t+1 2/α+π/ω ε f W t α + h s+π/ω ε f Lp ) ϕ W 2 γ Therefore, combining this estimation, (419), (415), and (418) we obtain the desired result, (425), for the remaining case p γ p>α Finally, we will show an almost optimal L -norm error estimate Theorem 48 Let u and u h be the solutions of (11) and (14), respectively, with f L p, p>1, and A W 2 Then there exists a constant C, independent of h, such that (430) u u h L Ch s log 1 h f L p Proof We split the error u u h by adding and subtracting the Galerkin finite element approximation u h (cf (15)); thus u u h = (u u h )+(u h u h ) The estimation of u u h L is well known (cf, eg, [32]) However, we shall briefly demonstrate it In view of [32, equation (08)] and the standard imbedding W 2 p C 0,2 2/ p (cf, eg, [24, Theorem 1452]), we have u u h L Ch s log 1 h u C 0,s Chs log 1 h u W 2 p, where s = 2 2/ p and C m,l is the space of m times continuously differentiable functions whose mth order derivative fulfills a uniform Hölder condition of order l Then, combining this with the elliptic regularity estimate, (431) u W 2 p C p f Lp (cf [24, Theorem 527]), we obtain (432) u u h L C p h s log 1 h f L p, p > 1 We turn now to the estimation of u h u h L Let x 0 K 0 T h such that u h u h L = (u h u h )(x 0 ) and δ x0 = δ C 0 (Ω) a regularized Dirac δ-function satisfying (δ, χ) =χ(x 0 ), χ X h

13 1944 P CHATZIPANTELIDIS AND R D LAZAROV For such a function δ (cf, eg, [9]) we have supp δ B = {x Ω: x x 0 h/2}, δ Lp Ch 2(1 p)/p, 1 <p< Ω δ =1, 0 δ Ch 2, Also let us consider the corresponding regularized Green s function G H 1 0, defined by (433) a(g, v) =(δ, v), v H 1 0 Then, we have (434) u h u h L =(δ, u h u h )=a(g, u h u h )=a(g h,u h u h ) = a(u u h,g h )=ε h (f,g h ) ε a (u h,g h ), where G h X h is the finite element approximation of G, ie, a(g, χ) =a(g h,χ), χ X h Further, using (41), (43), and the inverse inequality in (434) we obtain (435) χ W 1 q Ch 2/q 1 χ H 1, χ X h,q>2, u h u h L C { h ( ) (u u h ) L2 + h u W Gh + h f } 2 p W 1 p L p G h W 1 p C { h 2 2/ p( ) (u u h ) L2 + h u W 2 p + h min(1,2 2/p) f Lp } Gh H 1, with p>1 In addition, in view of [31, Lemma 31] we get (436) 1 G h H 1 C G L2 C (q 1) δ 1/2 L q with q 1 Choosing now q = 1 + (log 1 h ) 1 we have (437) G h H 1 ( C log h) 1 1/2 Combining now (434) (437) and Theorem 43, we obtain (438) ( u h u h L Ch 2s log h) 1 1/2 u W 2 (log + 1 ) 1/2 Chmin(1,2 2/p) f p h Lp From this, (431), and (432) we get the desired estimation (430) Remark 49 Assuming f L will not improve the convergence rate in (430), we can easily see that in this case (438) does not contribute terms of order higher than 1 However, (432) gives terms of order almost 2 2/ p, which is less than 1 Also, if we assume f Wα, t then similarly as in Theorem 46 we can show u h u h L = O(h 1+t ), but again the error u u h L will be at most of order 2 2/ p

14 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS Nonsmooth data: H l case In this section we will consider problem (21), ie, A = I, and we shall derive H 1 -, L 2 - and L -norm estimates of the error u u h for f H l, l (0, 1/2) We will show optimal H 1 -, suboptimal L 2 -, and almost optimal L -norm error estimates The H 1 - and L -norm estimations are of the same order with the corresponding estimations for the finite element scheme, whereas the L 2 -norm estimates are smaller This time for the analysis of the finite volume element method (36) we shall need in addition the following lemma, which we prove in section 6 Lemma 51 There exists a constant C such that for every χ X h, (51) ε h (f,χ) Ch 1 l f H l χ H 1, f H l, 0 <l<1/2 Theorem 52 Let u and u h be the solutions of (21) and (14), respectively, with f H l, 0 l<1/2 Then there exists a constant C, independent of h, such that (52) (53) u u h H 1 C ( h s u H 1+s + h 1 l f H l) Ch s f H l, u u h C ( h s+δ u H 1+s + h 1 l f H l), any δ < π/ω Remark 53 The convergence rate of the H 1 -norm is of optimal order (cf (17)) However, since s + δ > 1 1 l, for δ arbitrarily close to π/ω and δ < π/ω, the convergence rate in the L 2 -norm is suboptimal and lower than the rate of the corresponding finite element method (cf (15)) Later in section 7 we give an example similar to the one in [27], which shows the sharpness of the L 2 -error estimate (53) Proof The proof is similar as in Theorem 43, thus it suffices to estimate the first term of the right-hand side of (49) If f H l, with 0 <l<1/2, then in view of (25), u H 1+s, with s defined by (26) Since A = I, ε a 0 Therefore, using (51) in (410) we obtain (54) a(u u h,u h χ) Ch 1 l f H l u h χ H 1, χ X h Then, in view of the approximation property (17) of X h we get u u h H 1 C ( h s u H 1+s + h 1 l f H l) Using now the fact that for s 0 <l<1/2, s =1 l (cf (26)), and for 0 l s 0, s<1 l, and the a priori regularity estimate (25), we obtain the desired estimate (52) We now turn to (53) Using again the same arguments as in Theorem 43 it suffices to estimate term II of (418) Let again γ<p ω, such that 2/γ =2 π/ω + ε, with arbitrarily small ε>0, and let ϕ Wγ 2 H0 1 be the solution of the auxiliary problem (414) Combining (410) and (51), we have (55) II Ch 1 l f H l Π h ϕ H 1 Finally, since, p γ < 2, we can employ (417) in the estimation above, and then combining (418), (416), (52), and (415), we obtain the desired estimate (53) In Theorem 52 we demonstrated that u u h H 1 Ch s, for u H 1+s, s<π/ω In general, we know that u/ H 1+π/ω,eveniff is smooth In Theorem 54, we will show that for f H l, with l (0,s 0 ), u u h H 1 C l h π/ω, where the constant C l blows up when l s 0 This is a slight improvement of the result of Theorem 52, which in this case gives u u h H 1 Ch π/ω ε with ε>0 arbitrarily small Here we use the technique developed in [5]

15 1946 P CHATZIPANTELIDIS AND R D LAZAROV Theorem 54 Let u and u h be the solutions of (21) and (14), respectively, with f H l, 0 l<s 0 Then there exists a constant C, independent of h, such that (56) u u h H 1 C 1 s 0 l hπ/ω f H l Proof Obviously, if f H l, with 0 l<s 0, then f H l, l (s 0, 1/2) Then according to Theorem 52, we have that (57) u u h H 1 Ch 1 l ( u H 2 l + f H l) Also, since u H 2 l, wehave (58) (f,v) =a(u, v) u H 2 l v H l, v H 1 0 ; thus, f H l u H 2 l, which in view of (57) gives (59) u u h H 1 Ch 1 l u H 2 l In addition, we can easily see that if u H 2 H 1 0, (510) u u h H 1 Ch u H 2 Then, by interpolation between (59) and (510), we get (511) u u h H 1 Ch 1 s0 u X, where X =[H 2 H0 1,H 2 l H0 1 ] Here [V,W] s0/ l, θ,q, 0 θ 1, 1 q, denote the Banach spaces intermediate between V and W defined by the K-functional, which are used in interpolation theory (cf, eg, [7, Chapter 5]) Denote now with L 2,ψ the orthogonal space with respect to the L 2 -inner product to the space spanned by the function ψ = ϕ + u R, where ϕ = r π/ω sin(ϑπ/ω)η, and u R H0 1 the variational solution of u R = ϕ Then, in view of [5, Theorem 41], we have (512) u X C f Y with Y =[L 2,ψ,H l] s0/ l, ;thus (513) u u h H 1 Ch 1 s0 f Y Further, since l >s 0,[L 2,ψ,H 1 ] l,2 =[L 2,H 1 ] l,2 = H l (cf, eg, [5, equation (316)]) Therefore, in view of the reiteration theorem for the interpolation of spaces (cf, eg, [7, Chapter 5]), we get (514) Y =[L 2,ψ,H 1 ] s0, In addition, in view of [5, Theorem 31 and Remark 31], we have (515) f [L2,ψ,H 1 ] s0, C 1 s 0 l f H l, f H l Thus, combining (513) (515) we get the desired estimate

16 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1947 Finally, we will show an almost optimal L -norm error estimate Theorem 55 Let u and u h be the solutions of (21) and (14), respectively, with f H l, 0 <l<1/2 Then there exists a constant C, independent of h, such that (516) u u h L Ch s log 1 h f H l Proof The proof is similar to the one for Theorem 48 Hence, we will derive bounds for u u h L and u h u h L This time using [32, equation (08)] and the standard imbedding H 1+s C 0,s (cf, eg, [24, Theorem 1452]), we have u u h L Ch s log 1 h u C 0,s Chs log 1 h u H 1+s Then, combining this with the elliptic regularity estimate, u H 1+s C l f H l (cf [4]), we obtain (517) u u h L C l h s log 1 h f H l, 0 l<1/2 We turn now to the estimation of u h u h L Since A = I, (434) gives (518) u h u h L = ε h (f,g h ), where G h X h is the finite element approximation of the regularized Green function G (cf (433)) Then, using Lemma 51 and (437), we obtain ( u h u h L Ch 1 l log h) 1 1/2 f H l From this and (517) we get the desired estimation (516) 6 Auxiliary results In this section we shall prove Lemmas 41, 42, and 51 of the previous sections Proof of Lemma 41 We can easily see that the interpolation operator I h satisfies the property χ I h χ q L = q(k) (χ χ(z)) q dx K z (61) z Z h (K h q K χ q W 1 q (K), χ X h,q>1, with Z h (K) the set of the vertices of K Also, since in the construction of the control volumes we choose z K to be the barycenter of K, wehave (62) χdx= I h χ dx, K T h, χ X h K K In view of (61), (41) follows easily Let now f K be the mean value of f in K Thus, (63) f f K Lp(K) Ch K f W 1 p (K), f W 1 p (K), p>1

17 1948 P CHATZIPANTELIDIS AND R D LAZAROV Then, by interpolation of this estimate and f f K C f Lp(K) L p(k), we get, for f Wp(K), t p>1, and 0 <t 1 (64) f f K Lp(K) Cht K f W t p (K) Since f K is constant over K, due to (62), we have (f,χ I h χ) K =(f f K,χ I h χ) K, χ X h Thus, due to this, (64), and (61), we get for every χ X h, (f,χ I h χ) K = (f f K,χ I h χ) K Ch 1+t K f W t p (K) χ W 1 p (K), which concludes the proof of (42) We now turn to the proof of Lemma 42 For this we shall need the following auxiliary result Lemma 61 Let K be a triangle and e a side of K Then for ϕ Wp 1 (K), p>1, there exists a constant C independent of K such that ϕ(χ I h χ) ds Ch ϕ Wp 1(K) χ W 1 p (K), χ P 1(K) e Proof of Lemma 61 It is obvious that, for c constant, I h c = c and (65) I h χds= χ ds, χ X h, e E h e Thus, we have for every χ P 1 (K) and ϕ L 2 (e), ϕ(χ I h χ) ds = (ϕ c 1 )(χ c 2 I h (χ c 2 )) ds, e e e for all constants c 1,c 2 R, K T h, and e E h (K) Using now in the relation above the fact that I h χ χ L (e) L (e) and a local inverse inequality, we get for all constants c 1,c 2 R, χ P 1 (K), and ϕ Wp 1 (K), ϕ(χ I h χ) ds ϕ c 1 χ c Lp(e) 2 I h (χ c 2 ) Lp (e) (66) e h 1/p e ϕ c 1 χ c Lp(e) 2 I h (χ c 2 ) L (e) Ch 1/p e ϕ c 1 χ c Lp(e) 2 L (e) C ϕ c 1 Lp(e) χ c 2 Lp (e) with h e = e In view of the Bramble Hilbert lemma and a standard homogeneity argument, we can easily show inf c R ϕ c L p(e) Ch1 1/p e ϕ W 1 p (K), ϕ W 1 p (K), p>1 Finally, combining this with (66) we obtain the desired estimate

18 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1949 We now turn to the proof of Lemma 42 Proof of Lemma 42 First we will show (43) In view of Green s formula, we have (67) ε a (ψ, χ) = K (Lψ, χ I h χ) K + K (A ψ n, χ I h χ) K = I + II For the first term we have from (61), I C K Lψ Lp(K) χ I hχ Lp (K) C K h K ψ W 1 p (K) χ W 1 p (K) The bound for II follows at once from Lemma 61 since A ψ n W 1 p (K) C ψ W 1 p (K) We now turn to (44) Let ψ = u h in (67) and ( A) K be the average over K Then in view of (61) (63) we have for every χ X h, (Lu h,χ I h χ) K =([ A ( A) K ] u h,χ I h χ)) K Ch 2 K u h W 1 p (K) χ W 1 p (K) with p given by (23) From the estimation above we easily obtain the desired bound for I Let now E h (K) be the set of edges of K T h and Āe = A(m e ), where m e is the midpoint of the edge e We will show that for every χ X h, II = (68) ((A Āe) (u h u) n, χ I h χ) e K e E h (K) Provided that this holds, we may apply Lemma 61 and the estimate (A Āe) (u h u) W 1 p (K) C( (u u h ) + h u ) (69) L2(K) W 2 p (K) to obtain II Ch ( (u u h ) L2 + h u W 2 p ) χ W 1 p, χ X h, which gives the desired estimate for II Therefore, it remains to prove (68) We will show, for every ψ X h, (A u n, ψ I h ψ) K = (610) (Āe u n, ψ I h ψ) e =0 K K e E h (K) In the first sum we have by Green s formula for every ψ X h, (A u n, ψ) K = (A u, ψ) K (Lu, ψ) K =(A u, ψ) (Lu, ψ) =0 K K In addition, K (A u n, I hψ) K = 0 because I h ψ is piecewise constant on each interior edge e and A u n is continuous across e (in the trace sense), and I h ψ =0 on Ω Since the first sum in (610) vanishes for each smooth A and is continuous in A on L 1 ( K), the second sum is the limit of sums with a smooth A and, therefore, also vanishes Finally, since Āe u h n is constant on each e, in view of (65) we have (Āe u h n, χ I h χ) e =0, χ X h K e E h (K)

19 1950 P CHATZIPANTELIDIS AND R D LAZAROV It remains now to prove Lemma 51 Proof of Lemma 51 In view of the definition of ε h, it suffices to show χ I h χ H l Ch 1 l χ L2, 0 <l<1/2 The fractional order seminorm H l is given by w 2 H = w(x) w(y) 2 (611) dy dx; l Ω Ω x y 2(1+l) therefore, χ I h χ 2 H = 2 (χ I h χ)(x) (χ I h χ)(y) dy dx l z,w Z b z b w h x y 2(1+l) 4 χ(z) 2 2 x z dy dx 2(1+l) b z b w x y z,w Z h z w + z Z h b z b z χ(z) 2 x y 2+2(1 l) dy dx =4I + II For the estimation of II we rewrite the integral with respect to the y variable in polar coordinates (r, θ) having as center x; thus x y = r and h x y 2+2(1 l) dy C b z 0 r (1 l)p 2+1 dr = Ch 2(1 l) Therefore, b z χ 2 x y 2+2(1 l) dy dx Ch 2(1 l) χ 2 L, 2(b z) b z which gives the desired estimate for II Let us consider now z w and fix temporarily an x K z Using again polar coordinates with center x we estimate the integral with respect to y, x y 2(1+l) dy C r 1 2(1+l) dr Cr0 2l (x) =III, b w r 0(x) where r 0 (x) = dist(x, b w ) Let us assume that vertices z and w are in a different triangle and r 0 (x) >kh; therefore, III C r 0 (x) 2l Ch 2l Thus, b z χ(z) 2 x z 2 x y 2(1+l) dy dx Ch 2(1 l) χ 2 L 2(b z) b w Finally, let us consider the case that z w and are vertices of the same triangle K Then r 0 (x) could be arbitrarily small and in order for r 0 (x) 2l (x) dx < +, b z we need to assume that l<1/2 In a such case, we have (614) χ(z) 2 x z 2 x y 2(1+l) dy dx Ch 2 b w χ(z) 2 r0 2l (x) dx b z b z

20 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1951 Next we will estimate the right-hand side of the relation above For this, it suffices to bound K z χ(z) 2 r0 2l Let us denote with x 1 and x 2 the two coefficients of a point x in K z and introduce a rotation and translation of the (x 1,x 2 )-coordinate system to ( x 1, x 2 ), where x 1 -axis is the common edge K z K w We can easily see that for any point in x K z, r 0 (x) = dist(x, b w ) dist(( x 1, 0),b w )= x 1 Therefore, r0 2l (x) x 2l 1, x K z Then h h 2 χ(z) 2 r0 2l (x) dx Ch 2 χ(z) 2 K z 0 h Ch 2(1 l) χ 2 L 2(K z), 0 x 2l 1 d x 1 d x 2 assuming l<1/2 Hence, the relation above and (614) give χ(z) 2 x z 2 x y 2(1+l) dy dx Ch 2(1 l) χ 2 L 2(b z) b z b w Combining this with (612) and (613), we obtain the desired estimate 7 Numerical results In this section we will illustrate on several numerical examples the theoretical results of section 4 Our examples are similar to the ones considered in [8, 27] First, we will show that the theoretical L 2 -norm convergence rate of Theorem 46 is satisfied for the model Dirichlet boundary value problem for the Poisson equations in a Γ-shaped domain (cf Figure 71), with vertices (0, 0), (1, 0), (1, 1), ( 1, 1), ( 1, 1), and (0, 1) As in [8], we consider the following two singular functions for this Γ-shaped domain: ( ) ( ) 2 2 S 1 (r, θ) =φ(r)r 2/3 sin 3 θ, S 2 (r, θ) =φ(r)r β sin 3 θ, where β (0, 1) and φ is a cutoff function defined by 1 0 r 1/4, φ(r) = 192r r 4 440r r r , 1/4 r 3/4, 0 3/4 r (-1,1) (1,1) y 3/4 1/4 (0,0) x (1,0) (-1,-1) (0,-1) Fig 71 A Γ-shaped domain

21 1952 P CHATZIPANTELIDIS AND R D LAZAROV Table 71 Approximate theoretical convergence rate for exact solution u = S 1 + S 2 +(x x 3 )(y 2 y 4 ) β p ω =3/2, p ω =6/5 1/3 1/2 2/3 3/4 f ( u) almost in Wα t W 10/66 11/10 W 1/6 6/5 W 4/3 6/5 W 5/12 6/5 rate in H 1 -norm = s 1/3 1/2 2/3 2/3 s +2/3, (α < pω ) s =1 rate in L 2 norm min(s +2/3, 1+t), (α pω ) s = 7 6 s = 4 3 s = 4 3 rate in L -norm s 1/3 1/2 2/3 2/3 For f = (S 1 + S 2 )+6x(y 2 y 4 )+(x x 3 )(12y 2 2) the exact solution is u = S 1 + S 2 +(x x 3 )(y 2 y 4 ) We can easily see that ( ) ( ) 2 2 (S 1 + S 2 )=φ(r)(β 2 (2/3) 2 )r β 2 sin 3 θ +(2β +1)φ (r)r β 1 sin 3 θ ( ) 2 + φ (r)r β sin 3 θ + 7 ( ) ( ) φ (r)r 1/3 sin 3 θ + φ (r)r 2/3 sin 3 θ Since φ is a smooth cutoff function and 6x(y 2 y 4 )+(x x 3 )(12y 2 2) is a polynomial, the nonsmoothness of f results from (S 1 + S 2 ) and for β (0, 1) this is dictated from the term r β 2, except in the case β =2/3, where the leading term is r 1/3 According to [24, Theorem 1453], if a function g can be written as g = r γ ϕ(ϑ), in polar coordinates, where ϕ is smooth function, then g Wα, t with t>0 and α>1, for γ>t 2/α Thus, applying this to f, we have that f is almost in Wα β 2+2/α, with α (1, 2/(2 β)), for β 2/3, and f Wα 1/3+2/α, with α (1, 6), for β =2/3 In addition, in view of the imbedding L p Wa, t with p =2α/(2 tα), for β 2/3, then f L p, with p =2/(2 β), and for β =2/3, f L 6 Since we have considered a Γ-shaped domain, the largest interior angle is 3π/2; therefore, p ω =2/(2 (π/ 3π 2 ))=3/2 Thus, in view of (22), the solution u of the Poisson problem is almost in W 2 p, with p = min(2/(2 β), 3/2), or else u is almost in H 1+s with s = min(β,2/3) For example, we consider β =1/3, 1/2, 2/3, and 3/4 Then f is almost in the Sobolev spaces W 10/66 11/10, W 1/6 6/5, W 4/3 5/12 6/5, and W6/5 for β =1/3, 1/2, 2/3, and 3/4, respectively In Table 71 we present the theoretical and in Tables 72 and 73 the computed rates of convergence of the finite volume element method which illustrate the results of Theorem 46 The computation is done in the following way: For a given triangulation with number of nodes N and stepsize 2h, we compute the finite volume solution and the norms of the errors u u 2h T, where T = H 1,L 2,L Then we split each triangle into four similar triangles and compute the solution u h and the corresponding norms of the errors, u u h T Then the computed rates are given u u by log 2h T 2 u u h This procedure is repeated up to seven levels of refinement The T integrals in the finite volume formulation were approximated with a 13-point Gaussian quadrature For the solution of the corresponding linear system we used a multigrid preconditioner One may argue that the suboptimal order of the L 2 -norm error estimates in Theorems 43 and 54 of the finite volume element method might be an artifact of the proof and expect the same rate as in the finite element method However, this is not correct In what follows, we consider a counterexample which is based on a

22 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1953 Table 72 Experimental convergence rate for β = 1 3 and β = 1 2 β =1/3 β =1/2 # of nodes H 1 L 2 L H 1 L 2 L Theoretical Table 73 Experimental convergence rate for β = 2 3 and β = 3 4 β =2/3 β =3/4 # of nodes H 1 L 2 L H 1 L 2 L Theoretical similar argument given in [27] The following arguments can easily be modified and apply to a model problem in a convex domain This can then be used to illustrate the theoretical convergence rates derived in [14, 23] First we will show that the L 2 -norm estimate in Theorem 43 is sharp We consider the model problem (71) u = f in Ω, and u =0 on Ω, where f L 2 and Ω is the Γ-shaped domain with vertices (0, 0), (2, 0), (2, 2), ( 2, 2), ( 2, 2), and (0, 2) Since π/ω =2/3, according to Theorem 43 we know that u u h H 1 Ch s f L2 with s =2/3 ε, ε>0 arbitrarily small Let us assume then that (46) is not true and the finite volume and finite element methods converge in the L 2 -norm with the same rate, ie, u u h L2 Ch 2s f L2 Obviously, (u u h,φ) u u h L2 = sup φ L 2\{0} φ L2 Hence, our assumption leads to (72) (u u h,φ) Ch 2s φ L2 f L2

23 1954 P CHATZIPANTELIDIS AND R D LAZAROV Fig 72 An example of a triangulation The successive uniform refinement occurs by splitting the triangles into four Next, let us denote ψ H 1+s H 1 0 the solution of the auxiliary problem (73) ψ = φ in Ω, and ψ =0 on Ω Thus, (74) (u u h,φ)=a(u u h,ψ)=a(u u h,ψ Π h ψ)+a(u u h, Π h ψ), where Π h ψ is the interpolant of ψ in X h Obviously, then (75) a(u u h, Π h ψ)=(f,π h ψ I h Π h ψ) We can easily see that Thus combining (72) (75), we get a(u u h,ψ Π h ψ) Ch 2s φ L2 f L2 (f,π h ψ I h Π h ψ) Ch 2s φ L2 f L2 Since f is an arbitrary function of L 2, this leads to Hence, Π h ψ I h Π h ψ L2 Ch 2s φ L2 ψ I h Π h ψ L2 Ch 2s φ L2 Then, since φ is also an arbitrary function, this should be true for any function ψ H 1+s H 1 0 Therefore, let us consider a function ψ H 1+s H 1 0 such that (76) ψ(x 1,x 2 )=x 1 (1 x 1 ), (x 1,x 2 ) Ω 1 =[1/2, 3/2] [1/2, 3/2] For this ψ we should get (77) ψ I h Π h ψ L2(Ω 1) Ch2s

24 ERROR ESTIMATES FOR FVEM IN POLYGONAL DOMAINS 1955 ( 1 i 1 j n, 2 + ) n ( 1 i+1 1 j n, 2 + n ) 1 b ij 2 b ij 1 i 1 ( j 2 + n, 2 + n ) 1 i+1 1 j ( 2 + n, 2 + n ) Fig 73 A sample square K ij The two regions b 1 ij and b2 ij are separated with the dashed line We discretize Ω 1 into n 2 equal size squares with length h =1/n, and each square is divided further into two right triangles in the same direction Next, we construct the relative control volumes by connecting the barycenter of its triangle with the middle of the edges Let us denote z ij the vertices (1/2+i/n, 1/2+j/n), i, j =0,,n 1 Also, let K ij be the square [1/2+i/n, 1/2+(i +1)/n] [1/2+j/n, 1/2+(j +1)/n], i, j =1,,n, and b 1 ij = K ij (b ij b i(j+1) ) and b 2 ij = K ij (b (i+1)j b (i+1)(j+1) ) (cf Figure 73) Then, since ψ depends only on x, I h ψ I has the same value on the control volumes b ij, j =0,,n 1, for every i =0,,n 1 For this reason, on the square K ij, I h ψ = ψ(z ij )onb 1 ij and I hψ = ψ(z (i+1)j )onb 2 ij Then we have ψ I h Π h ψ 2 L = 2(Ω 1) ψ Ω 2 dx 1 dx 2 + (I h Π h ψ) 2 dx 1 dx 2 1 Ω 1 2 ψi h Π h ψdx 1 dx 2 Ω 1 Finally, we have = 3/2 1/2 2 n 1 x 2 1(1 x 1 ) 2 dx 1 + (ψ 2 (z ij ) b 1 ij + ψ 2 (z (i+1)j ) b 2 ij ) n 1 i,j=0 = 10 81n n 4 i,j=0 ( ψ(z ij ) x 1 (1 x 1 ) dx 1 dx 2 b 1 ij + ψ(z (i+1)j ) ψ I h Π h ψ L2(Ω 1) = 10 9 b 2 ij 1 n + o x 1 (1 x 1 ) dx 1 dx 2 ) ( ) 1 = O(h) n Combining this with (77) we get a contradiction, since 2s 4/3 Similar arguments can be used in order to show now the sharpness of the L 2 -norm error estimate in Theorem 52 Thus, let us consider this time the model problem (71), with f H 1/3

25 1956 P CHATZIPANTELIDIS AND R D LAZAROV According to Theorem 52 we have that u u h H 1 Ch s f H 1/3, with s =2/3 ε, ε>0, arbitrarily small Let us assume that the L 2 -norm error estimate in Theorem 54 does not hold and the finite volume and finite element methods converge in L 2 -norm with the same rate, ie, u u h L2 Ch 2s f H 1/3 Repeating similar arguments as in the previous counterexample, the function ψ H 1+s H 1 0 that satisfies (76) and (73) should also satisfy (78) Π h ψ I h Π h ψ H 1/3 Ch 2s φ L2 We discretize again Ω 1 in the same way as before, into n 2 equal size squares with length h =1/n, and each square is divided further into two right triangles in the same direction We construct the control volumes b ij in the same manner as before and denote z ij the vertices (1/2+i/n, 1/2+j/n), i, j =0,,n 1 Then, using the definition of H l (611), we can estimate Π h ψ I h Π h ψ H 1/3 (Ω 1) from below by (79) n 1 Π h ψ I h Π h ψ 2 H 1/3 (Ω 1) i,j=1 b zij bzij Π h ψ(z ij ) (x y) 2 x y 2(1+1/3) dy dx Also, let x =(x 1,x 2 ), y =(y 1,y 2 ), and K ij =[1/2+i/n, 1/2+i/n +1/3n] [1/2+ j/n, 1/2+j/n+1/3n], and since ψ is invariant in the x 2 -direction and x y 2/3n, (79) gives (710) Π h ψ I h Π h ψ 2 H 1/3 (Ω 1) n 1 Π h ψ(z ij ) (x y) 2 dy dx i,j=1 K ij K ij x y 2(1+1/3) ( ) 2(1+1/3) n 1 3n 2 (n 1) Π h ψ(z i1 ) 2 x 1 (x 1 y 1 ) 2 dy dx i=1 K i1 K i1 Next, we can easily see that Π hψ(z i1) x 1 =(2i +1)/n and (x 1 y 1 ) 2 1 dy dx = K i n 6 Thus, n 1 i=1 K i1 K i1 K i1 2 n 1 Π h ψ(z i1 ) x 1 (x 1 y 1 ) 2 1 dy dx n 8 i 2 = Finally, employing this in (710) we get Π h ψ I h Π h ψ H 1/3 (Ω C ( ) 1 1) n + o = O(h 2/3 ) 2/3 n 2/3 Combining this with (78) we get a contradiction, since 2s 4/3 i=1 n(n 1)(2n 1) n 8