Estimation of Error Constants Appearing in Non-conforming Linear

APCOM’07 in conjunction with EPMESC XI, December 3-6, 2007, Kyoto, JAPAN

Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element Xuefeng Liu1*, Fumio Kikuchi1 1

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo, 153-8914, Japan e-mail: [email protected], [email protected] Keywords : FEM, non-conforming linear triangle, a priori and a posteriori error estimates, error constants, Raviart-Thomas element Abstract As a well-known alternative to the conforming linear triangular finite element for approximation of the first-order Sobolev space, the non-conforming linear element is considered a classical discontinuous Galerkin finite element and has various interesting and attractive properties from both theoretical and practical standpoints. In particular, its a priori error analysis was performed in fairly early stage of mathematical analysis of FEM, and recently a posteriori error analysis is rapidly developing as well. For accurate error estimation of such an FEM, various error constants must be evaluated quantitatively. Based on our preceding works on the constant and conforming linear triangles, we here give some results for error constants required for analysis of the non-conforming linear triangle. More specifically, we first summarize a priori error estimation of the present non-conforming FEM, where several error constants appear. In this process, we use the lowest-order Raviart-Thomas triangular element to deal with the inter-element discontinuity of the approximate functions. Then we introduce some constants related to a reference triangle, some of which are popular in the constant and conforming linear cases. We give some theoretical results for the upper bounds of such constants. In some very special cases, exact values of constants can be obtained. In particular, a kind of maximum angle condition is required as in the case of the conforming linear triangle. Finally, we illustrate some numerical results to support the validity of such upper bounds. Our results can be effectively used in the quantitative a priori and a posteriori error estimates for the non-conforming linear triangular FEM.

REFERENCES [1] I. Babuska, A. K. Aziz, On the angle condition in the finite element method, SIAM Journal on Numerical Analysis, 13, (1976), 214-226. [2] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer (1991). [3] P.-G.. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM (2002). [4] F. Kikuchi, X. Liu, Determination of the Babuska-Aziz constant for the linear triangular finite element, Japan Journal of Industrial and Applied Mathematics, 23, (2006), 75-82. [5] F. Kikuchi, X. Liu, Estimation of interpolation error constants for the P_0 and P_1 triangular finite elements, Comput. Meths. Appl. Mech. Eng., 196, (2007), 329-336. [6] F. Kikuchi, H. Saito, Remarks on a posteriori error estimation for finite element solutions, J. Comp. Appl. Math., 199, (2007), 329-336

APCOM’07 in conjunction with EPMESC XI, December 3-6, 2007, Kyoto, JAPAN

Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element Xuefeng Liu1 *, Fumio Kikuchi1 1

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo, 153-8914, Japan e-mail: [email protected], [email protected] Abstract The non-conforming linear (P1 ) triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both theoretical and practical senses. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuˇska-Aziz maximum angle condition is required just as in the case of the conforming P1 triangle. Some applications and numerical results are also illustrated to see the validity and effectiveness of our analysis. Key Words: FEM, non-conforming linear triangle, a priori and a posteriori error estimates, error constants, Raviart-Thomas element INTRODUCTION As a well-known alternative to the conforming linear (P1 ) triangular finite element for approximation of the first-order Sobolev space (H 1 ), the non-conforming P1 element is considered a classical discontinuous Galerkin finite element [4] and has various interesting properties from both theoretical and practical standpoints [10, 22]. In particular, its a priori error analysis was performed in fairly early stage of mathematical analysis of FEM (Finite Element Method), and recently a posteriori error analysis is rapidly developing as well. For accurate error estimation of such an FEM, various error constants must be evaluated quantitatively [2, 6, 8, 17, 20, 21]. Based on our preceding works on the constant (P0 ) and the conforming P1 triangles [14, 15], we here give some results for error constants required for analysis of the non-conforming P1 triangle. More specifically, we first summarize a priori error estimation of the present non-conforming FEM, where several error constants appear. In this process, we use the lowest-order Raviart-Thomas triangular H(div) element to deal with the inter-element discontinuity of the approximate functions [9, 16]. Then we introduce some constants related to a reference triangle, some of which are popular in the P0 and the conforming P1 cases. We give some theoretical results for the upper bounds of such constants. Finally, we illustrate some numerical results to support the validity of such upper bounds. Our results can be effectively used in the quantitative a priori and a posteriori error estimates for the non-conforming P1 triangular FEM. A PRIORI ERROR ESTIMATION We here summarize a priori error estimation of the non-conforming P1 triangular FEM. Let Ω be a bounded convex polygonal domain in R2 with boundary ∂Ω, and let us consider a weak formulation of the Dirichlet boundary value problem for the Poisson equation: Given f ∈ L2 (Ω), find u ∈ H01 (Ω) s. t. (∇u, ∇v) = (f, v) ; ∀ v ∈ H01 (Ω) .

(1)

Here, L2 (Ω) and H01 (Ω) are the usual Hilbertian Sobolev spaces associated to Ω, ∇ is the gradient operator, and (·, ·) stands for the inner products of both L2 (Ω) and L2 (Ω)2 . It is well known that the solution exists uniquely in H01 (Ω) and also belongs to H 2 (Ω) for the considered Ω.

Let us consider a regular family of triangulations {T h }h>0 of Ω, to which we associate the non-conforming P1 finite element spaces {V h }h>0 . Each V h is constructed over a certain T h , and the functions in V h are linear in each K ∈ T h with continuity only at midpoints of edges, and also vanish at the midpoints on ∂Ω to approximate the homogeneous Dirichlet condition [10, 22]. Then the finite element solution uh ∈ V h is determined by, for a given f ∈ L2 (Ω), (∇h uh , ∇h vh ) = (f, vh ) ; ∀ vh ∈ V h ,

(2)

where ∇h is the “non-conforming” or discrete gradient defined as the L2 (Ω)2 -valued operator by the element-wise relations (∇h v)|K = ∇(v|K) for ∀v ∈ V h + H 1 (Ω) and ∀K ∈ T h . Eq. (2) is formally of the same form as in the conforming case, so that, for error analysis, it is natural to consider an appropriate interpolation operator Πh from H01 (Ω) (or its intersection with some other spaces) to V h . However, the situation is not so simple. That is, using the Green formula, we have ¯ XZ ∂u ¯¯ (∇h uh , ∇h vh ) = (∇u, ∇h vh ) − vh ¯ dγ ; ∀ vh ∈ V h , (3) ∂n K∈T h

∂K

∂K

¯ ∂u ¯ where ∂n denotes the trace of the derivative of u in the outward normal direction of ∂K, and dγ ∂K does the infinitesimal element of ∂K. Conventional efforts of error analysis have been focused on the estimation of the second term in the right-hand side of (3), which is absent in the conforming case. To cope with such difficulty, we introduce the lowest-order Raviart-Thomas triangular H(div) finite element space W h associated to each T h [9, 16]. Then, noting that the normal component of ∀qh ∈ W h is constant and continuous along each inter-element edge, we can derive (qh , ∇h vh ) + (div qh , vh ) = 0, and hence (∇h uh − ∇u, ∇h vh ) = (qh − ∇u, ∇h vh ) + (div qh + f, vh ) ; ∀ qh ∈ W h , ∀vh ∈ V h . Then by Lemma 6 of [12], a refinement of Strang’s second lemma [10], we have ¸2 · (qh − ∇u, ∇h wh ) + (div qh + f, wh ) 2 2 , k∇u − ∇h uh k = inf k∇u − ∇h vh k + sup k∇h wh k vh ∈V h wh ∈V h \{0}

(4)

(5)

where k · k stands for the norms of both L2 (Ω) and L2 (Ω)2 . Using the Fortin operator ΠFh : H(div; Ω) ∩ 1 H 2 +δ (Ω)2 → W h (δ > 0) (cf. [9]) and the orthogonal projection one Qh : L2 (Ω) → X h := space of step functions over T h , we obtain a priori error estimate: · ¸2 (f − Qh f, wh − Qh wh ) 2 2 F k∇u − ∇h uh k ≤ inf k∇u − ∇h vh k + k∇u − Πh ∇uk + sup , (6) k∇h wh k vh ∈V h wh ∈V h \{0} where qh in (5) is taken as ΠFh ∇u. We can obtain a more concrete error estimate in terms of the mesh parameter h∗ > 0 (h will be used in a different meaning later) by deriving estimates such as, for ∀v ∈ H01 (Ω) ∩ H 2 (Ω) and ∀g ∈ H 1 (Ω) + V h , kv − Πh vk ≤ γ0 h2∗ |v|2 , k∇v −

ΠFh ∇vk

≤ γ2 h∗ |v|2 ,

k∇v − ∇h Πh vk ≤ γ1 h∗ |v|2 , kg − Qh gk ≤ γ3 h∗ k∇h gk ,

(7)

where | · |k denotes the standard seminorm of H k (Ω) (k ∈ N) [10], and γ0 , γ1 , γ2 and γ3 are positive error constants dependent only on {T h }h>0 . Then we obtain, for the solution u ∈ H01 (Ω) ∩ H 2 (Ω), ½ h∗ {γ12 |u|22 + (γ2 |u|2 + γ3 kf k)2 }1/2 for f ∈ L2 (Ω), (8) k∇u − ∇h uh k ≤ h∗ {γ12 |u|22 + (γ2 |u|2 + γ32 h∗ |f |1 )2 }1/2 for f ∈ H 1 (Ω), where the term |u|2 can be bounded as |u|2 ≤ kf k for the present Ω. We can also use Nitsche’s trick to evaluate a priori L2 error of uh [10, 17]. That is, let us define ψ ∈ H01 (Ω) (∩H 2 (Ω)) for eh := u − uh by (∇ψ, ∇v) = (eh , v) ; ∀v ∈ H01 (Ω) .

(9)

Then, for ∀vh ∈ V h and ∀qh , q˜h ∈ W h , we have keh k2 = (˜ qh − ∇h vh , ∇h eh ) + (∇h vh − ∇ψ, ∇u − qh ) + (ψ − vh , div qh + f ) + (div q˜h + eh , eh ) . (10) Substituting vh = Πh ψ, qh = ΠFh ∇u and q˜h = ΠFh ∇ψ above, we find keh k2 = (ΠFh ∇ψ − ∇ψ + ∇ψ − ∇h Πh ψ, ∇h eh ) + (∇h Πh ψ − ∇ψ, ∇u − ΠFh ∇u) + (ψ − Πh ψ, f − Qh f ) + (eh − Qh eh , eh − Qh eh ) ,

(11)

since div qh = div ΠFh ∇u = −Qh f and div q˜h = div ΠFh ∇ψ = −Qh eh . Then we have, by (7) as well as the relations |u|2 ≤ kf k and |ψ|2 ≤ keh k, ¤ £ (12) keh k2 ≤ (γ1 + γ2 )h∗ k∇h eh k + (γ0 + γ1 γ2 )h2∗ kf k keh k + γ32 h2∗ k∇h eh k2 , where the term γ0 h2∗ kf k·keh k can be replaced with γ0 γ3 h3∗ |f |1 keh k if f ∈ H 1 (Ω). This may be considered a quadratic inequality for keh k, and solving it gives an expected order estimate ku−uh k = keh k = O(h2∗ ): µ ¶ q h∗ h 2 ke k ≤ A1 + A1 + 4A2 ; A1 = (γ1 + γ2 )k∇h eh k + (γ0 + γ1 γ2 )h∗ kf k, A2 = γ32 k∇h eh k2 . (13) 2 RELATION TO RAVIART-THOMAS MIXED FEM We have already introduced the Raviart-Thomas space W h for auxiliary purposes. But it is well known that the present non-conforming FEM is closely related to the Raviart-Thomas mixed FEM [3, 18]. Here we will summarize the implementation of such a mixed FEM by slightly modifying the original nonconforming P1 scheme described by (2). The original idea in [3, 18] is based on the enrichment by the conforming cubic bubble functions with the L2 projection into W h , but we here adopt non-conforming quadratic bubble ones to make the modification procedure a little simpler. Firstly, we replace f in (2) by Qh f . Then uh is modified to u∗h ∈ V h defined by (∇h u∗h , ∇h vh ) = (Qh f, vh ) ; ∀ vh ∈ V h .

(14)

Secondly, we introduce the space VBh of non-conforming quadratic bubble functions by defining its basis function ϕK associated to each K ∈ T h : ϕK vanishes outside K and its value at x ∈ K is given by 3

1 X (i) 1 ϕK (x) = |x − xG |2 − |x − xG |2 , 2 12 i=1

(15)

where | · | is the Euclidean norm of R2 , xG the barycenter of K, and x(i) for i = 1, 2, 3 the i-th vertex of K. It is easy to see that the line integration of ϕK for each edge e of K vanishes: Z ϕK dγ = 0 . (16) e

Now the enriched non-conforming finite element space V˜ h is defined by the following linear sum: V˜ h = V h + VBh .

(17)

By (16) and the Green formula, we find the following orthogonality relation for (∇h ·, ∇h ·): (∇h vh , ∇h βh ) = 0 ; ∀ vh ∈ V h , ∀ βh ∈ VBh .

(18)

Then the modified finite element solution u˜h ∈ V˜ h is defined by (∇h u˜h , ∇h v˜h ) = (Qh f, v˜h ) ; ∀ v˜h ∈ V˜ h .

(19)

Thanks to (18), the present u˜h can be obtained as the sum u˜h = u∗h + αh ,

(20)

where u∗h ∈ V h is the solution of (14), and αh ∈ VBh is determined by (∇h αh , ∇h βh ) = (Qh f, βh ) ; ∀βh ∈ VBh ,

(21)

i. e., completely independently of u∗h . Moreover, αh can be decided by element-by-element computations. More specifically, denoting αh |K as αK ϕK |K, Eq. (21) leads to αK (∇ϕK , ∇ϕK )K = (Qh f, ϕK )K ; ∀K ∈ T h ,

(22)

where (·, ·)K denotes the inner products of both L2 (K) and L2 (K)2 . Define {ph , uh } ∈ L2 (Ω)2 × X h by ph = ∇h u˜h , uh = Qh u˜h .

(23)

By applying the Green formula to (19), we can show that ph ∈ W h , and also that the present pair {ph , uh } satisfies the determination equations of the lowest-order Raviart-Thomas mixed FEM: ½ (ph , qh ) + (uh , div qh ) = 0 ; ∀ qh ∈ W h , (24) (div ph , v h ) = −(Qh f, v h ) ; ∀ v h ∈ X h . By the uniqueness of the solutions, {ph , uh } is nothing but the¡ unique solution of (24). ¢ R In conclusion, denoting the constant value of Qh f |K by f K = K f dx/meas(K) , we have for ∀ K ∈ T h and ∀x ∈ K that 3 ¢ 1 1 X (i) 1 ¡ G 2 ∗ ∗ αK = − f K , u˜h (x) = uh (x) + αK ϕK (x) = uh (x) − f K |x − x | − |x − xG |2 , 2 4 6 i=1 ph (x) =

∇u∗h (x)

1 − f K (x − xG ) , 2

3

uh (x) =

u∗h (xG )

¡ 1 X (i) 2 ¢ 1 |x | , − f K |xG |2 − 16 3 i=1

(25)

which coincide with those in [18] and are easy to compute by post-processing. A POSTERIORI ERROR ESTIMATION The consideration in the preceding section suggests the a posteriori error estimation based on the hypercircle method [11, 16]. Taking notice of the fact that ph ∈ W h obtained in the preceding section belongs to H(div; Ω) with div ph = −Qh f , we find that, for ∀v ∈ H01 (Ω), 1 1 k∇v − ph k2 = k∇(v − uh )k2 + k∇uh − ph k2 , k∇uh − (∇v + ph )k = k∇v − ph k , 2 2

(26)

where uh ∈ H01 (Ω) is the solution of (1) with f replaced by Qh f : (∇uh , ∇v) = (Qh f, v) ; ∀ v ∈ H01 (Ω) .

(27)

Eq. (26) implies that the three points ∇uh , ∇v and ph in L2 (Ω)2 make a hypercircle, the first having a right inscribed angle. Noting that (f − Qh f, v) = (f − Qh f, v − Qh v) for ∀v ∈ H01 (Ω) ⊂ L2 (Ω), we have by (7) that ¢ ¡ |u − uh |1 = k∇(u − uh )k ≤ γ3 h∗ kf − Qh f k ≤ γ32 h2∗ |f |1 if f ∈ H 1 (Ω) . (28) Taking an appropriate v ∈ H01 (Ω), we obtain a posteriori error estimates related to ph = ∇h u˜h :

1 1 k∇u − ph k ≤ k∇v − ph k + k∇(u − uh )k , k∇u − (∇v + ph )k ≤ k∇v − ph k + k∇(u − uh )k . (29) 2 2 h h A typical example of v is the conforming P1 finite element solution uC h ∈ VC , where VC is the conforming P1 space over T h . Another example is a function vhC ∈ VCh obtained by appropriate post-processing of uh or u∗h , such as nodal averaging or smoothing. A cheap method of constructing a nice vhC may be an interesting subject. Again we need the constant γ3 to evaluate the term k∇(u − uh )k above. If we use ∇h uh based on the original uh ∈ V h in (2), instead of u˜h ∈ V˜h , we must evaluate some additional terms. Fortunately, such evaluation can be done explicitly by γ3 and some constants related to {ϕK }K∈T h . The error k∇u − ∇h uh k thus evaluated can be also used to give a posteriori L2 estimate based on (13).

ERROR CONSTANTS To analyze the error constants in (7), let us consider their elementwise counterparts. Let h, α and θ be positive constants such that h>0, 0 0) is of the , qh2 The Fortin interpolation qh∗ = {qh1 form of qh in (48) and characterized by the conditions: Z Z Z ∗ ∗ (ˆ qh1 − qˆ1 ) ds = 0 , div(qh∗ − q) dx = 0 , (49) (qh2 − q2 ) ds = e1

e2

Tα,θ

where qˆ for q and qˆh∗ for qh∗ are defined by the relations in (47) and (48), respectively. {1,2} † † } for the same q, which is a constant , qh2 Let us now introduce another interpolation Πα,θ q = qh† = {qh1 vector function that satisfies only the former two conditions of (49). Then we have the L2 estimate sZ r kdiv qk 1+α cos θ+α2 {1,2} {1,2} kq − ΠFα,θ qk ≤ kq − Πα,θ qk + p |x|2 dx = kΠα,θ q − qk + kdiv qk. (50) 24 2 |Tα,θ | Tα,θ {1,2}

† † To bound kq − Πα,θ qk, let us evaluate kˆ q1 − qˆh1 k and kq2 − qh2 k by using C1 (α, θ) and C2 (α, θ):

Theorem 2. It holds for q = {q1 , q2 } ∈ H 1 (Tα,θ )2 that n 1 2 2 C1,α,θ + C2,α,θ + 2C1,α,θ C2,α,θ cos2 θ 2 sin θ q o1/2 2 2 + (C1,α,θ + C2,α,θ ) C1,α,θ + C2,α,θ + 2C1,α,θ C2,α,θ cos 2θ , (51)

{1,2} kq − Πα,θ qk ≤ C6 (α, θ)|q|1 ; C6 (α, θ) := √

where Ci,α,θ = Ci (α, θ) (i = 1, 2), and |q|1 =

p

|q1 |21 + |q2 |21 .

Remark 3. From (50) and (51), it is easy to derive the following estimate for the Fortin interpolation operator ΠFα,θ,h for Tα,θ,h : r 1+α cos θ+α2 kq−ΠFα,θ,h qk ≤ C6 (α, θ)h|q|1 +C7 (α, θ)hkdiv qk ; C7 (α, θ) := , ∀q ∈ H 1 (Tα,θ,h )2. (52) 24 Because of the factor sin θ in (51), the maximum angle condition applies to estimate (51), and hence to (52) [1, 5, 17]. On the other hand, the estimates for Π0α,θ,h and Π1,N α,θ,h are free from such conditions as may be seen from (43) and the comments there. GLOBAL INTERPOLATION OPERATORS F So far, we have introduced and analyzed local interpolation operators Π0α,θ,h , Π1,N α,θ,h and Πα,θ,h . For each K ∈ T h , we can find an appropriate Tα,θ,h congruent to K under a mapping ΦK : K → Tα,θ,h . Then it is natural to define the P1 non-conforming interpolation operator Πh : H01 (Ω) → V h by Πh u|K = −1 h 1 [Π1,N α,θ,h (v|K ◦ ΦK )] ◦ ΦK for ∀v ∈ H0 (Ω) and ∀K ∈ T . Similarly, the orthogonal projection operator Qh : L2 (Ω) → X h is related to Π0α,θ,h , while the global Fortin operator ΠFh is defined through ΠFα,θ,h , ΦK and the Piola transformation for 2D contravariant vector fields [9]. For each K ∈ T h , define {αK , θK , hK } as {α, θ, h} of the associated Tα,θ,h . Then, our analysis shows that the estimates in (7) can be concretely given by, for ∀v ∈ H01 (Ω) ∩ H 2 (Ω) and ∀g ∈ H 1 (Ω) + V h , h kv − Πh vk ≤ C5h h2∗ |v|2 ≤ C0h C{1,2} h2∗ |v|2 ,

k∇v − ∇h Πh vk ≤ C4h h∗ |v|2 ≤ C0h h∗ |v|2 ,

k∇v − ΠFh ∇vk ≤ C6h h∗ |v|2 + C7h h∗ k∆vk ,

kg − Qh gk ≤ C0h h∗ k∇h gk ,

(53)

where h∗ = max hK , CJh = max CJ (αK , θK ) (J = 0, 4, 5, 6, 7, {1, 2}) . K∈T h

K∈T h

(54)

Remark 4. Relations such as (20), (23) and (25) may suggest the possibility of finding interpolations for ∇u in W h better than that by the Fortin operator, which are free from the maximum angle condition [5]. However, ∇h (Πh u + αh ), for example, is not shown to belong to W h , because we cannot prove the interelement continuity of normal components unlike ∇h u˜h . Our numerical results show that such a condition is probably essential for the non-conforming P1 triangle. See also [1] for related topics. NUMERICAL RESULTS Firstly, we performed numerical computations to see the actual dependence of various constants on α and θ by adopting the conforming P1 element and a kind of discrete Kirchhoff plate bending element [13], the latter of which is used to deal with directly the 4-th order partial differential eigenvalue problems related to C4 (α, θ) and C5 (α, θ). That is, we obtained some numerical results for C4 (α) and C5 (α) (θ = π/2) together with their upper bounds. We used the uniform triangulations of the entire domain Tα : Tα is subdivided into small triangles, all being congruent to Tα,π/2,h with e. g. h = 1/20. The left-hand side of Fig. 2 illustrates the graphs of approximate C4 (α) and C0 (α) versus α ∈]0, 1], while the right-hand side does similar graphs for C5 (α) and C0 (α)C{1,2} (α). In both cases, the theoretical upper bounds based on (43) give fairly good approximations to the considered constants C4 (α) and C5 (α). Asymptotic behaviors of the constants for α ↓ 0 observed in the figures can be analyzed as in [15].

0.5

0.1 ◦ ∗

0.4

0.3

C4 (α) C0 (α)

0.08

∗ ∗ ∗ ∗ ∗ ∗ ∗◦ ∗◦ ∗◦ ◦ ◦ ◦ ◦ ◦ ∗ ∗ ∗ ∗ ◦ ∗ ◦ ∗ ◦ ◦ ∗ ◦ ◦ ◦ ◦ ◦ ◦ ∗ ◦ ∗ ◦ ∗ ∗ ∗

0.2

0.06 ∗◦

∗◦

∗◦

∗◦

∗◦

∗◦

0.04

Tα

Tα

α

0.02 1

0

0.2

0.4

0.6

∗ ◦

∗ ◦

◦ ∗

α

0.1

0

∗ ◦

∗ ◦

∗ ◦

C5 (α) C0 (α)C{1,2} (α)

1

0.8

1

0

0

0.2

0.4

0.6

α

0.8

1 α

Figure 2: Numerical results for C4 (α) & C0 (α) (left), and for C5 (α) & C0 (α)C{1,2} (α) (right) ; 0 < α ≤ 1 We also tested numerically the validity of our a priori error estimate for k∇u − ∇h uh k. That is, we choose Ω as the unit square {x = {x1 , x2 }; 0 < x1 , x2 < 1}and f as f (x1 , x2 ) = sin πx1 sin πx2 , and consider the N × N Friedrichs-Keller type uniform triangulations (N ∈ N). In such situation, 1 u(x1 , x2 ) = 2 sin πx1 sin πx2 , and all the triangles are congruent to a right isosceles triangle T1,π/2,1/N , 2π i. e., h∗ = h = 1/N . Moreover, we can use the following values or upper bounds for necessary constants: √ h (55) C0h = C0 = 1/π , C{1,2} = C{1,2} / 1/4 , C6h = C1 = C2 / 1/2 , C7h = C7 = 1/ 12 . Figure 3 illustrates the comparison of the actual k∇u − ∇h uh k and its a priori estimate based on our analysis. The difference is still large, but anyway our analysis appears to give correct upper bounds and order of errors. Probably, a posteriori estimation mentioned previously would give more realistic results.

∗ ◦

0.2

◦

0.1

∗

◦

0.05

∗

◦ 0.02 0.01

◦

||∇u − ∇h uh || a priori estimate

∗

slope= 1

∗

0.005 0.0625

0.125

0.25

0.5 h (log-scale)

Figure 3: Numerical results for k∇u − ∇h uh k and its a priori estimate versus h CONCLUDING REMARKS We have obtained some theoretical and numerical results for several error constants associated to the non-conforming P1 triangle, which we hope to be effectively used in quantitative error estimates, which are necessary for adaptive mesh refinements [7] and numerical verifications. Especially for numerical verification of partial differential equations by Nakao’s method [19], accurate bounding of various error constants is essential. Moreover, we are planning to extend our analysis to its 3D counterpart, i. e., the non-conforming P1 tetrahedron with face DOF’s.

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