Logarithm cannot be removed in maximum norm error estimates for linear finite elements in 3D
aa r X i v : . [ m a t h . NA ] J un LOGARITHM CANNOT BE REMOVED IN MAXIMUM NORMERROR ESTIMATES FOR LINEAR FINITE ELEMENTS IN 3D
NATALIA KOPTEVA
Abstract.
For linear finite element discretizations of the Laplace equationin three dimensions, we give an example of a tetrahedral mesh in the cubicdomain for which the logarithmic factor cannot be removed from the standardupper bounds on the error in the maximum norm. Introduction
Consider the problem(1.1) − △ u = f in Ω ⊂ R d , u = 0 on ∂ Ω , where d ∈ { , } , and △ is the Laplace operator (defined by △ = ∂ x + ∂ y + ∂ z for d = 3). It is well known that a standard linear finite element approximation u h for (1.1) on a quasi-uniform mesh of diameter h is quasi-optimal in the sense that(1.2) k u − u h k L ∞ (Ω) . h | ln h | k u k W ∞ (Ω) , see, e.g., [8, Theorem 2], [6, Theorem 3.1] and [9, Theorem 5.1], for, respectively,polygonal, convex polyhedral and smooth domains. It was also shown in [3, 2, 1]that when d = 2, the logarithmic factor cannot be removed from (1.1). Surprisingly,it appears that this is still an open question for d = 3. In this note, we address thisby giving an example of a tetrahedral mesh in the cubic domain Ω = (0 , suchthat, under certain conditions on f , one has k u − u h k L ∞ (Ω) ≥ C ∗ h | ln h | , where apositive constant C ∗ depends on f , but not on h .Our mesh is constructed using the two-dimensional triangulation used in a lessknown paper [1]. Furthermore, the general approach here also follows [1] in that,on a particular mesh, we estimate the error of a finite element method using itsexplicit finite difference representation.The paper is organized as follows. In § § Notation.
We write a . b when a ≤ Cb with a generic constant C depending onΩ and f , but not on h . For any domain D in R or R , the notation C k,α ( ¯ D ) isused for the standard H¨older space consisting of functions whose k th order partialderivatives are uniformly H¨older continuous with exponent α ∈ (0 , L ∞ ( D ) and the related Sobolev space W ∞ ( D ). Mathematics Subject Classification.
Primary 65N15, 65N30; Secondary 65N06.
Key words and phrases. linear finite elements, maximum norm error estimate, logarithmicfactor. Mesh description. Main results
Consider the standard linear finite element method with lumped-mass quadra-ture (for the case without quadrature, see Corollary 2.2). With the standard finiteelement space S h of continuous piecewise-linear functions vanishing on ∂ Ω, thecomputed solution u h ∈ S h is required to satisfy(2.1) Z Ω ∇ u h · ∇ v h = Z Ω ( f v h ) I ∀ v h ∈ S h , where v I ∈ S h , for any v ∈ C ( ¯Ω), denotes its standard piecewise-linear Lagrangeinterpolant.Let Ω := (0 , and define a tetrahedral mesh in Ω as follows. With an eveninteger N , set h := N − . Starting with the uniform rectangular grid { ( x i , y j ) =( ih, jh ) } Ni,j =0 , define a triangulation of (0 , by drawing diagonals as on Fig. 1(left) [1]. Note that each interior node is shared by 6 triangles, except for ( , ),which is shared by 4 triangles.Next, partition Ω into triangular prisms by constructing a tensor product of thetwo-dimensional triangulation in the ( x, y )-plane and the uniform grid { z k = kh } in the z -direction. Finally, divide each triangular prism into three tetrahedra ason Fig. 1 (centre, right) using method A or method B. Note that for the resultingtetrahedral mesh to be well-defined, no prisms of the same type can share a verticalface. Such a well-defined mesh is generated if, for example, the shadowed triangleson Fig. 1 (left) correspond to method B of prism partition. Note also that it will beconvenient to evaluate the local contributions to (2.1) associated with triangularprisms, the set of which (rather than of all tetrahedra) will be denoted T .Introduce the notation U ijk := u h ( x i , y j , z k ) for nodal values of u h and thestandard finite difference operators defined, for t = x, y, z , by δ t U ijk := u h ( P ijk + h i t ) − u h ( P ijk ) + u h ( P ijk − h i t ) h , P ijk := ( x i , y j , z k ) , where i t is the unit vector in the t -direction. Now, we claim that the finite elementmethod (2.1) can be rewritten, for i, j, k = 1 , . . . , N −
1, as L h U ijk := − ( δ x + δ y + γ ij δ z ) U ijk = γ ij f ( x i , y j , z k ) , (2.2a) where γ ij := (cid:26) , if i = j = N , , otherwise , (2.2b)subject to U ijk = 0 for any ( x i , y j , z k ) ∈ ∂ Ω.Indeed, for a particular prism T ∈ T , using the notation E lm for the edgeconnecting vertices l and m (see Fig. 1), and assuming that E is parallel to the x -axis, a calculation shows that Z T ∇ u h ·∇ v h = | T | n ∂ x u h ∂ x v h ) (cid:12)(cid:12) E + ( ∂ x u h ∂ x v h ) (cid:12)(cid:12) E ′ ′ (2.3) + ( ∂ y u h ∂ y v h ) (cid:12)(cid:12) E + 2( ∂ y u h ∂ y v h ) (cid:12)(cid:12) E ′ ′ + X l =1 ( ∂ z u h ∂ z v h ) (cid:12)(cid:12) E ll ′ o . Here we used the fact that each tetrahedron has an edge parallel to each of thecoordinate axes. Also, within the prism T , each of such edges belongs to exactly1 tetrahedron, except for E and E ′ ′ , while each of the latter is shared by 2tetrahedra. OGARITHM IN MAXIMUM NORM ERROR BOUNDS FOR 3D LINEAR ELEMENTS 3 xy
01 1 x yz ′ ′
13 2 ′ ′ ′ ′ Figure 1.
Two-dimensional triangulation in (0 , for N = 6(left); partition of a triangular prism into 3 tetrahedra usingmethod A (centre) and method B (right).Next, set v h := φ ijk in (2.1), where φ ijk ∈ S h equals 1 at ( x i , y j , z k ) and vanishesat all other mesh nodes. With this v h , adding the contributions of (2.3) to the left-hand side of (2.1), one gets | T | n − δ x U ijk − δ y U ijk − γ ij δ z U ijk o = | T | n γ ij f ( x i , y j , z k ) o . For the right-hand side here, we used the observation that each node ( x i , y j , z k ) isshared by 24 γ ij tetrahedra. Clearly, the above relation immediately implies (2.2a).Now that our finite element method (2.1) is represented as a finite differencescheme (2.2), note that if γ ij were equal to 1 for all i, j , we would immediatelyget the standard finite-difference error bound k u − u h k L ∞ (Ω) . h [7]. However, γ N , N = = 1 results in a slightly worse convergence rate, consistent with (1.2). Theorem 2.1.
Let
Ω = (0 , and f := F ( x, y ) sin( πz ) in (1.1) , with any F ∈ C ,α ([0 , ) subject to F = 0 at the corners of (0 , , and F ( , ) = k F k ∞ > (where k · k ∞ is used for the norm in L ∞ ((0 , ) ). Then u ∈ C ,α ( ¯Ω) ⊂ W ∞ (Ω) ,and there exists a positive constant C ∗ depending on F , but not on h , such that forthe finite element approximation u h obtained using (2.1) on the above tetrahedralmesh with a sufficiently small h , one has k u − u h k L ∞ (Ω) ≥ C ∗ h | ln h | . Corollary 2.2.
The result of Theorem 2.1 remains valid for a version of (2.1) without quadrature.Proof.
Let ¯ u h be the finite element solution obtained using linear finite elementswithout quadrature. Then | u h − ¯ u h | . h | ln h | / k f k W ∞ (Ω) ; see [4, final 3 lines inAppendix A]. The desired assertion follows. (cid:3) Remark . Un-der the conditions of Theorem 2.1, for any m ∈ N , one can choose F ∈ C m,α ([0 , )subject to F = 0 in small neighbourhoods of the corners of (0 , . Then the the-orem remains valid, while now u ∈ C m +2 ,α ( ¯Ω) (as an inspection of the proof ofthis theorem reveals; see, in particular, Lemma 3.1 and Remark 3.2). Thus, ad-ditional smoothness of the exact solution would not improve the accuracy of thefinite element method. NATALIA KOPTEVA Proof of Theorem 2.1
We split the proof into a number of lemmas, which involve an auxiliary function w ( x, y ), as well as its finite difference approximation W ij , defined by M w := − ( ∂ x + ∂ y ) w + π w = F in (0 , , (3.1) M h W ij := − ( δ x + δ y ) W ij + γ ij π W ij = γ ij F ( x i , y j ) , (3.2)the latter for i, j = 1 , . . . , N −
1, subject to w = 0 and W ij = 0 on the boundaryof (0 , . Lemma 3.1.
Under the conditions of Theorem 2.1 on f , the solution of prob-lem (1.1) is u = w ( x, y ) sin( πz ) , where w ∈ C ,α ([0 , ) is a unique solution of (3.1) and | ∂ x w | + | ∂ y w | . in [0 , . Furthermore, (3.3) e F ( , ) ≥ k F k ∞ / cosh( π ) , where e F := F − π w. Proof.
The regularity of w and the bounds on its pure fourth partial derivativesfollow from [11, Remark 4 in § F ( , ) = k F k ∞ , it sufficesto show that k F k ∞ − π w ( x, y ) ≥ B ( x ) := k F k ∞ cosh( π ( x − ))cosh( π ) . The latter is obtained by an application of the maximum principle for the operator M as M ( k F k ∞ − π w ) ≥ M B , while k F k ∞ − π w = k F k ∞ ≥ B on ∂ Ω. (cid:3) Remark . In Lemma 3.1, we have w ∈ C ,α ([0 , ) rather than w ∈ C ,α ([0 , ),as the latter requires additional corner compatibility conditions on F [11, Theo-rem 3.1], while bounded fourth pure partial derivatives of w are sufficient for thefinite-difference-flavoured analysis that yields the crucial bound (3.4) below. Lemma 3.3.
For U ijk of (2.2) and W ij of (3.2) one has | U ijk − W ij sin( πz k ) | . h .Proof. First, note that − δ z [sin( πz k )] = λ h sin( πz k ) where λ h := h sin ( πh ) = π + O ( h ) [7, § II.3.2]. Combining this with (2.2a) and (3.2) yields L h [ W ij sin( πz k )] = (cid:2) − ( δ x + δ y ) W ij + γ ij λ h W ij (cid:3) sin( πz k )= γ ij F ( x i , y j ) sin( πz k ) | {z } = L h U ijk + O ( h ) , where we also used | W ij | ≤ π − k F k ∞ . The desired result follows by an applicationof the discrete maximum principle for the operator L h . (cid:3) Lemma 3.4 ([1]) . Let w solve (3.1) , and f W ij satisfy − ( δ x + δ y ) f W ij = γ ij e F ( x i , y j ) ,subject to f W ij = 0 at the boundary nodes. Then there exists a constant C , inde-pendent of h and e F , such that (3.4) w ( , ) − f W N , N ≥ C h | ln h | e F ( , ) − O ( h ) . Proof.
Recalling that e F = F − π w , rewrite (3.1) as − ( ∂ x + ∂ y ) w = e F . Now f W ij may be considered a finite difference approximation of w , for which (3.4) isobtained in [1]. Note that C is independent of h and e F (as it is related to thediscrete Green’s function for the operator − ( δ x + δ y ); see also Remark 3.5). (cid:3) OGARITHM IN MAXIMUM NORM ERROR BOUNDS FOR 3D LINEAR ELEMENTS 5
Remark C in (3.4) [1]) . It is noted in [1] that f W ij of Lemma 3.4 allowsthe representation f W ij = ˚ W ij − h G ij e F ( , ), where − ( δ x + δ y ) ˚ W ij = e F ( x i , y j ),subject to ˚ W ij = 0 at the boundary nodes, while G ij is the discrete Green’s functionfor the operator − ( δ x + δ y ) associated with the node ( N , N ). To be more precise, − ( δ x + δ y ) G ij equals h − if i = j = N and 0 otherwise, subject to G ij = 0 at theboundary nodes. Furthermore, there is a constant C > h (as wellas of e F ) such that G N , N ≥ C | ln h | . (For the latter, Andreev [1] uses [5, (16)];see also [10, Lemma 6] for a similar result in the finite element context.) Finally,for ˚ W ij , one has a standard finite-difference error bound | ˚ W ij − w ( x i , y j ) | . h .The above observations yield (3.4). Remark . It was also pointed out in [1] that f W ij = w h ( x i , y j ), where w h isa linear finite element solution for − ( ∂ x + ∂ y ) w = e F obtained using the two-dimensional triangulation of (0 , shown on Fig. 1 (left). Lemma 3.7.
Under the conditions of Theorem 2.1 on F , for the solutions of (3.1) and (3.2) with a sufficiently small h , one has max i,j =0 ,...,N | w ( x i , y j ) − W ij | ≥ C h | ln h | ,with a positive constant C that depends on F , but not on h .Proof. Set C := 4 π − C k F k ∞ / cosh( π ) and e ij := f W ij − W ij , where C and f W ij are from Lemma 3.4. Note that − ( δ x + δ y ) e ij = γ ij π [ W ij − w ( x i , y j )] (in view of e F = F − π w ). Also, for the auxiliary B ij := π C h | ln h | (cid:8) − ( x i − ) (cid:9) , note that − ( δ x + δ y ) B ij = π C h | ln h | . We now prove the desired bound by contradiction.Assume that max i,j | w ( x i , y j ) − W ij | < C h | ln h | . Then − ( δ x + δ y )[ B ij ± e ij ] > B ij ± e ij ≥
0. So | e ij | ≤ B ij , so | f W ij − W ij | = | e ij | ≤ π C h | ln h | . Combining this with (3.4), oneconcludes that w ( , ) − W N , N ≥ (cid:8) C e F ( , ) − π C (cid:9)| {z } ≥ π C h | ln h | − O ( h ) ≥ C h | ln h | for a sufficiently small h , where we also used C e F ( , ) ≥ π C (in view of (3.3)).The above contradicts our assumption that max i,j | w ( x i , y j ) − W ij | < C h | ln h | . (cid:3) Proof of Theorem 2.1.
It now suffices to combine the findings of Lemmas 3.1, 3.3and 3.7. In particular, u ( x i , y j , ) − U i,j, N = w ( x i , y j ) − W i,j + O ( h ), by Lemmas 3.1and 3.3. So, in view of Lemma 3.7, one gets k u − u h k L ∞ (Ω) ≥ C ∗ h | ln h | with anyfixed positive constant C ∗ < C . (cid:3) References [1] V. B. Andreev,
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Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
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