Weak discrete maximum principle of finite element methods in convex polyhedra
aa r X i v : . [ m a t h . NA ] M a y WEAK DISCRETE MAXIMUM PRINCIPLE OFFINITE ELEMENT METHODS IN CONVEX POLYHEDRA
DMITRIY LEYKEKHMAN AND BUYANG LI
Abstract.
We prove that the Galerkin finite element solution u h of theLaplace equation in a convex polyhedron Ω , with a quasi-uniform tetra-hedral partition of the domain and with finite elements of polynomialdegree r >
1, satisfies the following weak maximum principle: k u h k L ∞ ( Ω ) C k u h k L ∞ ( ∂Ω ) , with a constant C independent of the mesh size h . By using this result,we show that the Ritz projection operator R h is stable in L ∞ normuniformly in h for r ≥
2, i.e. k R h u k L ∞ ( Ω ) C k u k L ∞ ( Ω ) . Thus we remove a logarithmic factor appearing in the previous resultsfor convex polyhedral domains.
1. Introduction
Let S h be a finite element space of Lagrange elements of degree r > T of a convex polyhedron Ω ⊂ R , where h denotes the mesh size of the tetrahedral partition, and quasi-uniformity meansthat ρ τ > ch ∀ τ ∈ T , with ρ τ denoting the radius of the largest ball inscribed in the tetrahedron τ ∈ T .Let ˚ S h be the subspace of S h consisting of functions with zero boundary values.A function u h ∈ S h is called a discrete harmonic if it satisfies( ∇ u h , ∇ χ h ) = 0 ∀ χ h ∈ ˚ S h . (1.1)In this article, we establish the following result, which we call weak maximumprinciple of finite element methods (for higher order equations it is often calledAgmon–Miranda maximum principle). Theorem 1.1.
A discrete harmonic function u h satisfies the following estimate: k u h k L ∞ ( Ω ) C k u h k L ∞ ( ∂Ω ) , (1.2) where the constant C is independent of the mesh size h . As an application of the weak maximum principle, we show that the Ritz pro-jection R h : H ( Ω ) → ˚ S h defined by( ∇ ( u − R h u ) , ∇ v h ) = 0 ∀ v h ∈ ˚ S h This work is partially supported by NSF DMS-1913133 and a Hong Kong RGC grant (projectno. 15300519). is stable in L ∞ norm for finite elements of degree r >
2, i.e. k R h u k L ∞ ( Ω ) C k u k L ∞ ( Ω ) ∀ u ∈ H ( Ω ) ∩ L ∞ ( Ω ) . Although this result is well-known for smooth domains [27, 29], for convex poly-hedral domains the result was available only with an additional logarithmic fac-tor [20, Theorem 12].In the finite element literature, the “strict” discrete maximum principle k u h k L ∞ ( Ω ) k u h k L ∞ ( ∂Ω ) i.e., with C = 1 in (1.2), has attracted a lot of attention; see [7, 8, 25, 31, 32], tomention a few. However, the sufficient conditions for the strict discrete maximumprinciple often put serious restrictions on the geometry of the mesh. For piecewiselinear elements in two-dimensions, the strict discrete maximum principle generallyrequires the angles of the triangles to be less than π/
2, or the sum of oppositeangles of the triangles that share an edge to be less than π (for example, see [32, § L ∞ and W , ∞ norms for two-dimensional polygons. Such L ∞ - and W , ∞ -stability results have a wide range of applications, for exampleto pointwise error estimates of finite element methods for parabolic problems [17,21, 22], Stokes systems [3], nonlinear problems [11, 12, 23], obstacle problems [6],optimal control problems [1, 2], to name a few. As far as we know, [26] is the onlypaper that establishes weak maximum principle and L ∞ stability estimate (withoutthe logarithmic factor) for the Ritz projection on nonsmooth domains.In three dimensions the situation is less satisfactory. The stability of the Ritzprojection in L ∞ and W , ∞ norms is available on smooth domains [27, 29] andconvex polyhedral domains [14,20]. However, on convex polyhedral domains in [20],the L ∞ -stability constant depends logarithmically on the mesh size h , and it is notobvious how the logarithmic factor can be removed there. There are no resultson the weak maximum principles in three dimensions even on smooth domainsor convex polyhedra. The objective of this paper is to close this gap for convexpolyhedral domains. In order to obtain the result, we have to modify the argumentin [26] by extending the arguments to L p norm for some 1 < p <
2. This constitutesthe main technical difficulty in the analysis of the paper. The mere adaptation ofthe L -norm based argument used in [26] for convex polyhedral domains, wouldyield a logarithmic factor. Unfortunately, the current analysis does not allow usto extend the results to nonconvex polyhedral domains or graded meshes. Thesewould be the subject of future research.The paper is organized as follows. In section 2 we state some preliminary resultsthat we use later in our arguments. In section 3, we reduce the proof of the weakdiscrete maximum principle to a specific error estimate. Section 4 is devoted tothe proof of this estimate, which constitutes the main technical part of the paper. Finally, section 5, gives an application of the weak discrete maximum principle toshowing the stability of the Ritz projection in L ∞ norm uniformly in h for higherorder elements.In the rest of this article, we denote by C a generic positive constant, which maybe different at different occurrences but will be independent of the mesh size h .
2. Preliminary results
In this section, we present several well-known results that are used in our anal-ysis. First result concerns global regularity of the weak solution v ∈ H ( Ω ) to theproblem ( ∇ v, ∇ χ ) = ( f, χ ) ∀ χ ∈ H ( Ω ) . (2.3)On the general convex domains we naturally have the H regularity (cf. [13]).However, on convex polyhedral domains, we have the following sharper W ,p ( Ω )regularity result (cf. [9, Corollary 3.12]). Lemma 2.1.
Let Ω be a convex polyhedron. Then there exists a constant p > depending on Ω such that for any < p < p and f ∈ L p ( Ω ) , the solution v of (2.3) is in W ,p ( Ω ) and k v k W ,p ( Ω ) C k f k L p ( Ω ) . For any point x ∗ ∈ Ω we denote S d ( x ∗ ) = { x ∈ Ω : | x − x ∗ | < d } . The followingresult, which is a version of the Poincare inequality, is an extension of Lemma 1.1in [26], which was established in two dimensions for p = 2. Lemma 2.2.
Let < p < ∞ . If χ ∈ W ,p ′ ( Ω ) and x ∗ ∈ ∂Ω , then k χ k L p ( S d ∗ ( x ∗ )) Cd ∗ k∇ χ k L p ( Ω ) . Proof.
Similarly to [26, Lemma 1.1], we consider χ ∈ C ∞ ( Ω ) and extend χ byzero outside Ω . By denoting x ∗ = ( x ∗ , x ∗ , x ∗ ) and using the spherical coordinatescentered at x ∗ , we define e χ ( ρ, ϕ, θ ) = χ ( x ∗ + ρ sin( ϕ ) cos( θ ) , x ∗ + ρ sin( ϕ ) sin( θ ) , x ∗ + ρ cos( ϕ )) , for 0 ρ d ∗ , and some ϕ ∈ [0 , π ] and θ ∈ [0 , π ]. Since χ = 0 on ∂Ω , there exists θ ∗ ∈ [0 , π ] such that e χ ( ρ, ϕ, θ ∗ ) = 0. Therefore, | e χ ( ρ, ϕ, θ ) | = (cid:12)(cid:12)(cid:12)(cid:12) Z θθ ∗ ∂ θ e χ ( ρ, ϕ, θ ′ ) d θ ′ (cid:12)(cid:12)(cid:12)(cid:12) Z π | ∂ θ e χ ( ρ, ϕ, θ ′ ) | d θ ′ . From the chain rule, we have ∂ θ e χ ( ρ, ϕ, θ ) = − ∂ x e χ ( ρ, ϕ, θ ) ρ sin( ϕ ) sin( θ ) + ∂ x e χ ( ρ, ϕ, θ ) ρ sin( ϕ ) cos( θ ) . As a result, by H¨older’s inequality, we obtain | e χ ( ρ, ϕ, θ ) | p C Z π | ∂ θ e χ ( ρ, ϕ, θ ′ ) | p d θ ′ C Z π ρ p |∇ e χ ( ρ, ϕ, θ ′ ) | p d θ ′ . Therefore, Z S d ∗ ( x ∗ ) | χ ( x ) | p d x = Z d ∗ Z π Z π | e χ ( ρ, ϕ, θ ) | p ρ sin( ϕ ) d θ d ϕ d ρ C Z d ∗ Z π Z π (cid:18) Z π ρ p |∇ e χ ( ρ, ϕ, θ ′ ) | p d θ ′ (cid:19) ρ sin( ϕ ) d θ d ϕ d ρ Cd p ∗ Z d ∗ Z π Z π |∇ e χ ( ρ, ϕ, θ ′ ) | p ρ sin( ϕ ) d θ ′ d ϕ d ρ = Cd p ∗ Z S d ∗ ( x ∗ ) |∇ χ ( x ) | p d x. This proves the desired result. (cid:3)
The next result addresses the problem (2.3) when the source function f is sup-ported in some part of Ω . It establishes the stability of the solution in W ,p normand traces the dependence of the stability constant on the diameter of the support.The corresponding result in [26] is the equation (1.6) therein, which was establishedfor p = 2 in two dimensions. In our situation we need it for larger range of p . Lemma 2.3.
For any bounded Lipschitz domain Ω , there exist positive constants α ∈ (0 , ) and C (depending on Ω ) such that for − α p α and f ∈ L p ( Ω ) with supp( f ) ⊂ S d ∗ ( x ) and dist( x , ∂Ω ) d ∗ , the solution of (2.3) satisfies k v k W ,p ( Ω ) Cd ∗ k f k L p ( Ω ) . Proof.
If dist( x , ∂Ω ) d ∗ , then S d ∗ ( x ) ⊂ S d ∗ (¯ x ) for some ¯ x ∈ ∂Ω . For any χ ∈ W ,p ′ ( Ω ), there holds | ( ∇ v, ∇ χ ) | = | ( f, χ ) | k f k L p ( S d ∗ ( x )) k χ k L p ′ ( S d ∗ ( x )) k f k L p ( S d ∗ ( x )) k χ k L p ′ ( S d ∗ (¯ x )) Cd ∗ k f k L p ( Ω ) k∇ χ k L p ′ ( Ω ) , where in the last step we have used Lemma 2.2.For ~w ∈ C ∞ ( Ω ) , we let χ ∈ H ( Ω ) ∩ H ( Ω ) ֒ → W ,p ′ ( Ω ) (for − α p α )be the solution of (cid:26) ∆χ = ∇ · ~w in Ωχ = 0 on ∂Ω. The solution χ defined above satisfies ∇ · ( ~w − ∇ χ ) = 0 , and, according to [16, Theorem B], there exists a constant α ∈ (0 , ) such that k∇ χ k L p ′ ( Ω ) C k ~w k L p ′ ( Ω ) for − α p α. By using these properties, we have | ( ∇ v, ~w ) | = | ( ∇ v, ∇ χ ) | Cd ∗ k f k L p ( Ω ) k∇ χ k L p ′ ( Ω ) Cd ∗ k f k L p ( Ω ) k ~w k L p ′ ( Ω ) . Since C ∞ ( Ω ) is dense in L p ′ ( Ω ) and the estimate above holds for all ~w ∈ C ∞ ( Ω ) ,the duality pairing between L p ( Ω ) and L p ′ ( Ω ) implies the desired result. (cid:3) The next lemma concerns basic properties of harmonic functions on convex do-mains. The result is essentially the same as in [28, Lemma 8.3].
Lemma 2.4.
Let D and D d be two subdomains satisfying D ⊂ D d ⊂ Ω , with D d = { x ∈ Ω : dist ( x, D ) d } , where d is a positive constant. If v ∈ H ( Ω ) and v is harmonic on D d , i.e. ( ∇ v, ∇ w ) = 0 , ∀ w ∈ H ( D d ) , then the following estimates hold: | v | H ( D ) Cd − k v k H ( D d ) , (2.4a) k v k H ( D ) Cd − k v k L ( D d ) . (2.4b)Finally, we need the best approximation property of the Ritz projection in W ,p norm. In [14], the best approximation property of the Ritz projection in W , ∞ norm was established on convex polyhedral domains. Together with the standardbest approximation property in H norm we obtain(2.5) k v − R h v k W ,p ( Ω ) C min χ ∈ ˚ S h k v − χ k W ,p ( Ω ) ∀ v ∈ H ( Ω ) ∩ W ,p ( Ω ) , for any 2 p ∞ . Extension of the above result to 1 < p ∞ follows by duality(cf. [5, § Lemma 2.5.
On a convex polyhedron Ω , the following estimate holds for any fixed p ∈ (1 , ∞ ] : k v − R h v k W ,p ( Ω ) Ch k v k W ,p ( Ω ) ∀ v ∈ H ( Ω ) ∩ W ,p ( Ω ) . In sections 3–4, we would use several results from [24, 26, 27]. Some of theseresults were stated therein for sufficiently small mesh size h under certain hypothesison the triangulation. Since, we concentrate on the Lagrange elements, all thehypotheses in [27] are trivially satisfied and we assume that our mesh size h issufficiently small, say h h for some constant h , so these results hold.
3. Basic estimates
In this section, we derive some estimates we require to establish one of our keyresults, Theorem 1.1. This part of the argument up to (3.12) is analogous to thefirst part of the proof of [26, Theorem 1] up to equation (3.10). The dyadic decom-position part is also similar. The essential difference lies in the duality argumentin section 4, after the equation (4.21).In [27, Corollary 5.1], the following interior error estimate was established k u − u h k L ∞ ( Ω ) Ch l | ln h | ¯ r | u | W l, ∞ ( Ω ) + Cd − /q − p k u − u h k W − p,q ( Ω ) , for 0 l r , where ¯ r = 1 for r = 1, ¯ r = 0 for r > Ω ⊂⊂ Ω ⊂⊂ Ω , withdist( Ω , ∂Ω ) > d > kh and dist( Ω , ∂Ω ) > d > kh . Choosing u = 0, p = 0 and q = 2 in the above estimate, we obtain that there exists a constant C independentof h such that k u h k L ∞ ( Ω ) Cd − k u h k L ( Ω ) . (3.1)Let x ∈ Ω be a point satisfying | u h ( x ) | = k u h k L ∞ ( Ω ) with d = dist( x , ∂Ω ) . If d > kh then we can choose Ω = S d/ ( x ) and Ω = S d ( x ). In this case, thefollowing interior L ∞ estimate holds (cf. [27, Corollary 5.1] and [26, Lemma 2.1(ii)]): | u h ( x ) | Cd − k u h k L ( S d ( x )) . Otherwise, we have d kh . In this case, the inverse inequality of finite elementfunctions (cf. [5, Ch. 4.5]) implies | u h ( x ) | = k u h k L ∞ ( S h ( x )) Ch − k u h k L ( S h ( x )) . Hence, either for d > kh or d kh , the following estimate holds: | u h ( x ) | Cρ − k u h k L ( S ρ ( x )) , with ρ = d + 2 kh. (3.2)To estimate the term k u h k L ( S ρ ( x )) on the right hand side of the inequalityabove, we use the following duality property: k u h k L ( S ρ ( x )) = sup supp( ϕ ) ⊂ S ρ ( x ) k ϕ k L Sρ ( x | ( u h , ϕ ) | , which implies the existence of a function ϕ ∈ C ∞ ( Ω ) with the following properties:supp( ϕ ) ⊂ S ρ ( x ) , k ϕ k L ( S ρ ( x )) k u h k L ( S ρ ( x )) | ( u h , ϕ ) | . (3.4)For this function ϕ , we define v ∈ H ( Ω ) to be the solution of( ∇ v, ∇ χ ) = ( ϕ, χ ) ∀ χ ∈ H ( Ω ) , (3.5)and let v h ∈ ˚ S h be the finite element solution of( ∇ v h , ∇ χ h ) = ( ϕ, χ h ) ∀ χ h ∈ ˚ S h . Thus, v h is the Ritz projection of v and satisfies( ∇ ( v − v h ) , ∇ χ h ) = 0 ∀ χ h ∈ ˚ S h . (3.6)Let u be the solution of the problem (in weak form) ( ( ∇ u, ∇ χ ) = 0 ∀ χ ∈ H ( Ω ) ,u = u h on ∂Ω. (3.7)Then the continuous maximum principle of (3.7) implies k u k L ∞ ( Ω ) k u h k L ∞ ( ∂Ω ) . (3.8)Notice, that u h is the Ritz projection of u , i.e. ( ( ∇ ( u − u h ) , ∇ χ h ) = 0 ∀ χ h ∈ ˚ S h ,u − u h = 0 on ∂Ω. Therefore, we have k u h k L ( S ρ ( x )) | ( u h , ϕ ) | (here we used (3.4))= 2 | ( u h − u, ϕ ) + ( u, ϕ ) | = 2 | ( ∇ ( u h − u ) , ∇ v ) + ( u, ϕ ) | (here we used (3.5))= 2 | ( ∇ u h , ∇ v ) + ( u, ϕ ) | (here we used (3.7)) | ( ∇ u h , ∇ v ) | + 2 k u k L ∞ ( Ω ) k ϕ k L ( Ω ) | ( ∇ u h , ∇ v ) | + Cρ k u h k L ∞ ( ∂Ω ) k ϕ k L ( S ρ ( x )) , (3.9)where we have used (3.8) and the H¨older inequality in deriving the last inequality.To estimate | ( ∇ u h , ∇ v ) | , we note that( ∇ u h , ∇ v ) = ( ∇ u h , ∇ ( v − v h )) (here we use (1.1) and v h ∈ ˚ S h )= ( ∇ ( u h − χ h ) , ∇ ( v − v h )) ∀ χ h ∈ ˚ S h . (here we use (3.6)) . We simply choose χ h to be equal to u h at interior nodes and χ h = 0 on ∂Ω ; thus u h ( x ) − χ h ( x ) is zero when dist( x, ∂Ω ) > h , and for any r ≥ k u h − χ h k L ∞ ( Ω ) C k u h k L ∞ ( ∂Ω ) . If we define Λ h = { x ∈ Ω : dist( x, ∂Ω ) h } , then using the inverse inequality, | ( ∇ u h , ∇ v ) | k∇ ( u h − χ h ) k L ∞ ( Λ h ) k∇ ( v − v h ) k L ( Λ h ) Ch − k u h − χ h k L ∞ ( Ω ) k∇ ( v − v h ) k L ( Λ h ) Ch − k u h k L ∞ ( ∂Ω ) k∇ ( v − v h ) k L ( Λ h ) . (3.10)Then, substituting (3.9) and (3.10) into (3.2), we obtain k u h k L ∞ ( Ω ) C (cid:0) ρ − h − k∇ ( v − v h ) k L ( Λ h ) + 1) k u h k L ∞ ( ∂Ω ) . (3.11)The proof of Theorem 1.1 will be completed if we establish ρ − h − k∇ ( v − v h ) k L ( Λ h ) C, (3.12)which will be accomplished in the next section.
4. Estimate of ρ − h − k∇ ( v − v h ) k L ( Λ h ) Let R = diam( Ω ) and d j = R − j for j = 0 , , , . . . We define a sequence ofsubdomains A j = { x ∈ Ω : d j +1 | x − x | d j } , j = 0 , , , . . . For each j we denote A lj to be a subdomain slightly larger than A j , defined by A lj = A j − l ∪ · · · ∪ A j ∪ A j +1 ∪ · · · ∪ A j + l l = 1 , , . . . Let J = [ln ( R / ρ )] + 1, with [ln ( R / ρ )] denoting the greatest integer not ex-ceeding ln ( R / ρ ). Then 2 ρ d J +1 ρ and measure( A j ∩ Λ h ) ≤ Chd j . (4.13)By using these subdomains defined above, we have ρ − h − k∇ ( v − v h ) k L ( Λ h ) ρ − h − (cid:18) J X j =0 k∇ ( v − v h ) k L ( Λ h ∩ A j ) + k∇ ( v − v h ) k L ( Λ h ∩ S ρ ( x )) (cid:19) Cρ − h − J X j =0 h d j k∇ ( v − v h ) k L ( Λ h ∩ A j ) + Cρ − h − k∇ ( v − v h ) k L ( Λ h ∩ S ρ ( x )) , (4.14)where the H¨older inequality and (4.13) were used in deriving the last inequality.Using global error estimate in H norm, Lemma 2.1 with p = 2 and (3.3), weobtain ρ − h − k∇ ( v − v h ) k L ( Λ h ∩ S ρ ( x )) Cρ − h − h k v k H ( Ω ) Cρ − h − h k ϕ k L ( Ω ) C, where we have used ρ > h and k ϕ k L ( Ω ) ρ − h − k∇ ( v − v h ) k L ( Λ h ) Cρ − h − J X j =0 d j k∇ ( v − v h ) k L ( A j ) + C. (4.15)Now, we use the following interior energy error estimate (proved in [24, Theorem5.1], also see [26, Lemma 2.1 (i)]): k∇ ( v − v h ) k L ( A j ) C k∇ ( v − I h v ) k L ( A j ) + Cd − j k v − I h v k L ( A j ) + Cd − j k v − v h k L ( A j ) , (4.16)where I h denotes the nodal interpolant. Using the approximation theory, we obtain k∇ ( v − v h ) k L ( A j ) (cid:0) Ch + Ch d − j (cid:1) k v k H ( A j ) + Cd − j k v − v h k L ( A j ) Chd − p j k v k W ,p ( A j ) + Cd − j k v − v h k L ( A j ) for < p < , (4.17)where we have used d j > h and the following inequality in deriving the last inequal-ity: k v k H ( A j ) Cd − p j k v k W ,p ( A j ) for < p < . (4.18)The inequality above follows from Lemma 2.4, the H¨older inequality and Sobolevembedding, i.e. k v k H ( A j ) Cd − j k v k L ( A j ) Cd − − q j k v k L q ( A j ) if q > Cd − p j k v k W ,p ( A j ) for q = p − < p < q > . This proves that (4.16) holds for < p < p = , we obtain k∇ ( v − v h ) k L ( A j ) Chd − j ρ k ϕ k L ( S ρ ( x )) + Cd − j k v − v h k L ( A j ) Chd − j ρ + Cd − j k v − v h k L ( A j ) , (4.19)where the last inequality is due to the following H¨older inequality: k ϕ k L ( S ρ ( x )) Cρ k ϕ k L ( S ρ ( x )) with k ϕ k L ( S ρ ( x )) . From (4.19) we see that d j k∇ ( v − v h ) k L ( A j ) Cρ h (cid:18) hd j (cid:19) + C k v − v h k L ( A j ) . (4.20)Then, substituting (4.20) into (4.15), we have ρ − h − k∇ ( v − v h ) k L ( Λ h ) C J X j =0 (cid:18) hd j (cid:19) + Cρ − h − J X j =0 k v − v h k L ( A j ) C + Cρ − h − J X j =0 k v − v h k L ( A j ) . (4.21)It remains to estimate P Jj =0 k v − v h k L ( A j ) . To this end, we let χ be a smoothcut-off function satisfying χ = 1 on A j and χ = 0 outside A j . Then k v − v h k L ( A j ) k χ ( v − v h ) k L ( Ω ) k χ ( v − v h ) k H ( Ω ) (Sobolev embedding H ( Ω ) ֒ → L ( Ω )) k∇ ( v − v h ) k L ( A j ) + Cd − j k v − v h k L ( A j ) . (4.22)By using (4.22) and the interpolation inequality (for 1 < p < k v − v h k L ( A j ) k v − v h k − θL p ( A j ) k v − v h k θL ( A j ) with 12 = 1 − θp + θ , (4.23)we obtain k v − v h k L ( A j ) k v − v h k − θL p ( A j ) (cid:0) k∇ ( v − v h ) k L ( A j ) + Cd − j k v − v h k L ( A j ) (cid:1) θ = ( ε − θ − θ k v − v h k L p ( A j ) ) − θ (cid:0) ε k∇ ( v − v h ) k L ( A j ) + Cεd − j k v − v h k L ( A j ) (cid:1) θ ε − θ − θ k v − v h k L p ( A j ) + ε k∇ ( v − v h ) k L ( A j ) + Cεd − j k v − v h k L ( A j ) , where ε can be an arbitrary positive number. By choosing ε = d j ( ρ/d j ) σ with σ ∈ (0 , k v − v h k L ( A j ) (cid:18) ρd j (cid:19) − θσ − θ d − θ − θ j k v − v h k L p ( A j ) (4.24) + (cid:18) ρd j (cid:19) σ (cid:0) d j k∇ ( v − v h ) k L ( A j ) + C k v − v h k L ( A j ) (cid:1) . (4.25)Hence, ρ − h − J X j =0 k v − v h k L ( A j ) Cρ − h − J X j =0 (cid:18) ρd j (cid:19) − θσ − θ d − θ − θ j k v − v h k L p ( A j ) + Cρ − h − J X j =0 (cid:18) ρd j (cid:19) σ (cid:0) d j k∇ ( v − v h ) k L ( A j ) + C k v − v h k L ( A j ) (cid:1) Cρ − h − J X j =0 (cid:18) ρd j (cid:19) − θσ − θ d − θ − θ j k v − v h k L p ( A j ) + Cρ − h − J X j =0 (cid:18) ρd j (cid:19) σ k v − v h k L ( A j ) , (4.26) where we have used (4.20) in deriving the last inequality. Note that J X j =0 (cid:18) ρd j (cid:19) σ k v − v h k L ( A j ) C (cid:18) ρd j (cid:19) σ k v − v h k L ( S ρ ( x )) + 2 J X j =0 (cid:18) ρd j (cid:19) σ k v − v h k L ( A j ) . Combining the last two estimates, we obtain ρ − h − J X j =0 k v − v h k L ( A j ) Cρ − h − J X j =0 (cid:18) ρd j (cid:19) − θσ − θ d − θ − θ j k v − v h k L p ( A j ) + Cρ − h − (cid:18) ρd j (cid:19) σ k v − v h k L ( S ρ ( x )) + Cρ − h − J X j =0 (cid:18) ρd j (cid:19) σ k v − v h k L ( A j ) . If d j > κρ for sufficiently large constant κ , then the last term can be absorbed bythe left side. Hence, we have J X j =0 ρ − h − k v − v h k L ( A j ) J X j =0 Cρ − h − (cid:18) ρd j (cid:19) − θσ − θ d − θ − θ j k v − v h k L p ( A j ) + Cρ − h − (cid:18) ρd j (cid:19) σ k v − v h k L ( S ρ ( x )) . (4.27)It remains to estimate k v − v h k L p ( A j ) and k v − v h k L ( S ρ ( x )) . To this end, welet ψ ∈ C ∞ ( A j ) be a function satisfying k v − v h k L p ( A j ) v − v h , ψ ) and k ψ k L q ( A j ) , with 1 p + 1 q = 1 . (4.28)Let w ∈ H ( Ω ) be the solution of (cid:26) − ∆w = ψ in Ω,w = 0 on ∂Ω,
Then using Lemma 2.5 and Lemma 2.1, we obtain( v − v h , ψ ) = ( ∇ ( v − v h ) , ∇ w )= ( ∇ ( v − v h ) , ∇ ( w − I h w )) k∇ ( v − v h ) k L p ( Ω ) k∇ ( w − I h w ) k L q ( Ω ) Ch k v k W ,p ( Ω ) k w k W ,q ( Ω ) Ch k ϕ k L p ( Ω ) k ψ k L q ( Ω ) Ch k ϕ k L p ( S ρ ( x )) Ch ρ p − k ϕ k L ( S ρ ( x )) k ψ k L q ( A j ) Ch ρ p − , where we have used k ϕ k L ( S ρ ( x )) k ψ k L q ( A j ) k v − v h k L p ( A j ) Ch ρ p − and k v − v h k L ( S ρ ( x )) Ch . (4.29)By substituting these estimates into (4.27), we obtain J X j =0 ρ − h − k v − v h k L ( A j ) J X j =0 C (cid:18) hρ (cid:19) (cid:18) ρd j (cid:19) p − − θσ − θ + C. (4.30)Since p <
2, by choosing sufficiently small σ we have p − − θσ − θ > J X j =0 ρ − h − k v − v h k L ( A j ) J X j =0 C (cid:18) hρ (cid:19) (cid:18) ρd j (cid:19) p − − θσ − θ + C C. (4.31)Then, substituting this into (4.21), we obtain ρ − h − k∇ ( v − v h ) k L ( Λ h ) C. (4.32)This proves the desired result for sufficiently small mesh size h h , as explainedin the end of section 2.For h > h , we denote by e g h ∈ S h the finite element function satisfying e g h = u h on ∂Ω and e g h = 0 at the interior nodes of the domain Ω . Naturally, k e g h k L ∞ ( Ω ) k u h k L ∞ ( ∂Ω ) . Since χ h = u h − e g h ∈ ˚ S h , from (1.1), we have( ∇ u h , ( ∇ ( u h − e g h )) = 0and as a result k∇ ( u h − e g h ) k L ( Ω ) = ( ∇ ( u h − e g h ) , ∇ ( u h − e g h )) = − ( ∇ e g h , ∇ ( u h − e g h )) C k∇ e g h k L ( Ω ) k∇ ( u h − e g h ) k L ( Ω ) . Thus, using the inverse inequality and that h > h , we have k∇ ( u h − e g h ) k L ( Ω ) C k∇ e g h k L ( Ω ) Ch − k e g h k L ( Ω ) Ch − k e g h k L ∞ ( Ω ) Ch − k u h k L ∞ ( ∂Ω ) . By using the inverse inequality and the above estimate, we also have k u h − e g h k L ∞ ( Ω ) Ch − k u h − e g h k L ( Ω ) Ch − k∇ ( u h − e g h ) k L ( Ω ) Ch − k u h k L ∞ ( ∂Ω ) . By the triangle inequality, this proves k u h k L ∞ ( Ω ) k e g h k + k u h − e g h k L ∞ ( Ω ) C k u h k L ∞ ( ∂Ω ) for h > h .Combining the two cases h h and h > h , we obtain the desired result ofTheorem 1.1.
5. Application to the Ritz projection
In this section, we adopt Schatz’s argument to prove the maximum-norm stabilityof the Ritz projection. This argument uses the weak maximum principle establishedabove to remove a logarithmic factor for finite elements of degree r > Ω can be extended to a larger convex domain e Ω quasi-uniformly, with Ω ⊂⊂ e Ω .The logarithmic factor has been removed in previous articles only for r ≥ u ∈ H ( Ω ), we denote by R h u ∈ ˚ S h the Ritz projection of u ,defined by(5.33) ( ∇ ( u − R h u ) , ∇ χ h ) = 0 ∀ χ h ∈ ˚ S h . Theorem 5.1.
Under assumption (A) , for finite elements of degree r ≥ the Ritzprojection satisfies (5.34) k R h u k L ∞ ( Ω ) C k u k L ∞ ( Ω ) ∀ u ∈ H ( Ω ) ∩ C ( Ω ) . Proof.
Let e u be the zero extension of u to the larger domain e Ω . Let ˚ S h ( e Ω ) be thefinite element space subject to the tetrahedral partition of e Ω (with zero boundaryvalues), and let e u h be the Ritz projection of e u in the domain e Ω , i.e.(5.35) Z e Ω ∇ ( e u − e u h ) · ∇ χ h d x = 0 ∀ χ h ∈ ˚ S h ( e Ω ) . Since u = e u on Ω , it follows that k u − u h k L ∞ ( Ω ) = k e u − u h k L ∞ ( Ω ) (5.36) k e u − e u h k L ∞ ( Ω ) + k e u h − u h k L ∞ ( Ω ) (5.37) : = E + E . (5.38)By using [27, Theorem 5.1] (which requires r ≥ h sufficiently small, say h h ∗ ), we have E C k e u − I h e u k L ∞ ( Ω ′ ) + C k e u − e u h k L ( Ω ′ ) , (5.39)where Ω ′ is some intermediate domain satisfying Ω ⊂⊂ Ω ′ ⊂⊂ e Ω . Since theLagrange interpolation operator I h is stable in the L ∞ norm on C ( Ω ), it followsthat k e u − I h e u k L ∞ ( Ω ′ ) C k e u k L ∞ ( Ω ) = C k u k L ∞ ( Ω ) . (5.40)To estimate k e u − e u h k L ( Ω ′ ) , we use a duality argument. Thus, k e u − e u h k L ( Ω ′ ) k e u − e u h k L ( e Ω ) = sup e ϕ ∈ C ∞ e Ω ) k e ϕ k L e Ω ) Z e Ω ( e u − e u h ) e ϕ d x. In particular, there exists a e ϕ ∈ C ∞ ( e Ω ) satisfying k e ϕ k L ( e Ω ) k e u − e u h k L ( e Ω ) Z e Ω ( e u − e u h ) e ϕ d x. (5.41) For this e ϕ we define e ψ ∈ H ( Ω ) to be the weak solution of(5.42) ( − ∆ e ψ = e ϕ in e Ω, e ψ = 0 on ∂ e Ω, and denote by e ψ h ∈ ˚ S h ( e Ω ) the Ritz projection of e ψ in e Ω , i.e.(5.43) Z e Ω ∇ ( e ψ − e ψ h ) · ∇ e χ h d x = 0 ∀ e χ h ∈ ˚ S h ( e Ω ) . If we denote by e T the set of tetrahedra in the partition of e Ω , then testing (5.42) by e u − e u h yields Z e Ω ( e u − e u h ) e ϕ d x = Z e Ω ∇ ( e u − e u h ) · ∇ e ψ d x (here we use integration by parts)= Z e Ω ∇ ( e u − e u h ) · ∇ ( e ψ − e ψ h ) d x (here we use (5.35))= Z e Ω ∇ e u · ∇ ( e ψ − e ψ h ) d x (here we use (5.43))= X τ ∈ e T Z τ ∇ e u · ∇ ( e ψ − e ψ h ) d x = − X τ ∈ e T Z τ e u ∆ ( e ψ − e ψ h ) d x + Z ∂τ e u ∂ n ( e ψ − e ψ h ) d s C k e u k L ∞ ( e Ω ) X τ ∈ e T (cid:16) k ∆ ( e ψ − e ψ h ) k L ( τ ) + k ∂ n ( e ψ − e ψ h ) k L ( ∂τ ) (cid:17) C k u k L ∞ ( Ω ) (cid:18) h − k∇ ( e ψ − e ψ h ) k L ( e Ω ) + X τ ∈ e T k e ψ − e ψ h k W , ( τ ) (cid:19) , (5.44)where in the last step we have used k e u k L ∞ ( e Ω ) = k u k L ∞ ( Ω ) and the trace inequality k ∂ n ( e ψ − e ψ h ) k L ( ∂τ ) Ch − k∇ ( e ψ − e ψ h ) k L ( τ ) + C k e ψ − e ψ h k W , ( τ ) . By using a priori energy estimate and H regularity, we have k∇ ( e ψ − e ψ h ) k L ( e Ω ) k∇ ( e ψ − e ψ h ) k L ( e Ω ) k∇ ( e ψ − I h e ψ ) k L ( e Ω ) Ch k e ψ k H ( e Ω ) Ch k e ϕ k L ( e Ω ) Ch. (5.45) Let ˜ I h be the Scott-Zhang interpolant. Then by the triangle and inverse inequalities,we have X τ ∈ e T k e ψ − e ψ h k W , ( τ ) C X τ ∈ e T (cid:18) k e ψ − ˜ I h e ψ k W , ( τ ) + k ˜ I h e ψ − e ψ h k W , ( τ ) (cid:19) C (cid:18) X τ ∈ e T k e ψ − ˜ I h e ψ k W , ( τ ) + h − k ˜ I h e ψ − e ψ h k W , ( e Ω ) (cid:19) C (cid:18) X τ ∈ e T k e ψ − ˜ I h e ψ k W , ( τ ) + h − k e ψ − ˜ I h e ψ k W , ( e Ω ) + h − k e ψ − e ψ h k W , ( e Ω ) (cid:19) . Similarly as (5.45), we can prove the following estimate: h − k e ψ − ˜ I h e ψ k W , ( e Ω ) + h − k e ψ − e ψ h k W , ( e Ω ) C, and by using the properties of ˜ I h (cf. [5, Theorem 4.8.3.8]), X τ ∈ e T k e ψ − ˜ I h e ψ k W , ( τ ) C X τ ∈ e T k e ψ k W , ( τ ) C k e ψ k H ( e Ω ) C k e ϕ k L ( e Ω ) C. Now we substitute these estimates into (5.44). This yields k e u − e u h k L ( e Ω ) C k u k L ∞ ( Ω ) . (5.46)Then, by substituting (5.40) and (5.46) into (5.39), we obtain E C k u k L ∞ ( Ω ) . (5.47)To estimate E , we use the fact that e u h − u h is discrete harmonic in Ω , i.e. Z Ω ∇ ( e u h − u h ) · ∇ χ h d x = Z Ω ∇ ( e u − u ) · ∇ χ h d x = 0 ∀ χ h ∈ ˚ S h ( Ω ) . Thus, by the weak discrete maximum principle proved in Theorem 1.1 and usingthe fact that u h = 0 and e u = 0 on ∂Ω , we have E = k e u h − u h k L ∞ ( Ω ) C k e u h − u h k L ∞ ( ∂Ω ) = C k e u h k L ∞ ( ∂Ω ) (use u h = 0 on ∂Ω )= C k e u h − e u k L ∞ ( ∂Ω ) (use e u = 0 on ∂Ω ) C k e u h − e u k L ∞ ( Ω ) = E , (5.48)which has already been estimated. Hence, substituting (5.47) and (5.48) into (5.36),we obtain k u − u h k L ∞ ( Ω ) C k u k L ∞ ( Ω ) . (5.49)This completes the proof of Theorem 5.1 in the case h h ∗ for some positiveconstant h ∗ . If h > h ∗ then we pick up a point x ∈ ¯ τ (in some tetrahedron τ ) satisfying | u h ( x ) | = k u h k L ∞ ( Ω ) . For such x we define a regularized Green’s function G asthe solution of(5.50) − ∆G ( x ) = ˜ δ ( x ) , x ∈ Ω,G ( x ) = 0 , x ∈ ∂Ω, where ˜ δ ∈ C ( Ω ) is the regularized Delta function concentrated at x , satisfyingsupp(˜ δ ) ⊂ ¯ τ and Z Ω χ h e δ d x = χ h ( x ) , ∀ χ h ∈ ˚ S h , k e δ k W l,p ≤ Kh − l − − /p ) for 1 ≤ p ≤ ∞ , l = 0 , , , . The construction of the function e δ can be found in [30, Lemma 2.2]. In particular,the construction of e δ can be done in any tetrahedron for the arbitrary mesh size h .We define G h = R h G ∈ ˚ S h , i.e.,(5.51) ( ∇ G h , ∇ χ h ) = (˜ δ, χ h ) ∀ χ h ∈ ˚ S h . The finite element function G h defined by the equation above satisfies the followingstandard energy estimate: k G h k H ( Ω ) C k ˜ δ k L ( Ω ) . Then using the Galerkin orthogonality, integration by parts, we obtain(5.52) u h ( x ) = ( ∇ u h , ∇ G h ) = ( ∇ u, ∇ G h ) = X τ ∈ T [( u, ∂ n G h ) ∂τ + ( u, − ∆G h ) τ ] k u k L ∞ ( Ω ) X τ ∈ T (cid:0) k ∂ n G h k L ( ∂τ ) + k ∆G h k L ( τ ) (cid:1) . Now, for h ≥ h ∗ , using the trace and inverse inequality we have X τ ∈ T (cid:2) k ∂ n G h k L ( ∂τ ) + k ∆G h k L ( τ ) (cid:3) Ch − X τ ∈ T k∇ G h k L ( τ ) Ch − k G h k W , ( Ω ) Ch − k G h k H ( Ω ) Ch − k ˜ δ k L ( Ω ) Ch − / ∗ , since k ˜ δ k L ( Ω ) Ch − / and h > h ∗ .Combining the two cases h h ∗ and h > h ∗ , we obtain the desired result ofTheorem 5.1. (cid:3)
6. Conclusion
In this article, we have proved the weak maximum principle of finite elementmethod (Theorem 1.1). The main difference between the current proof and theproof in [26] for two-dimensional polygons is that we have used L p estimates inplace of some L estimates in section 4, including (4.16), (4.18), (4.23), (4.24),(4.26), (4.28) and (4.29). As an application of the weak maximum principle of finiteelement methods, we have presented an L ∞ -stability of Ritz projection (Theorem5.1) by utilizing the argument in [26, Theorem 5.1]. Acknowledgement
We thank the anonymous referees for the valuable comments and suggestions.
References [1] T. Apel, A. R¨osch, and D. Sirch, L ∞ -error estimates on graded meshes with applica-tion to optimal control , SIAM J. Control Optim. 48 (2009), pp. 1771–1796.[2] T. Apel, M. Winkler, and J. Pfefferer, Error estimates for the postprocessing approachapplied to Neumann boundary control problems in polyhedral domains , IMA J. Numer.Anal. 38 (2018), pp. 1984–2025.[3] N. Behringer, D. Leykekhman, B. Vexler,
Global and local pointwise error esti-mates for finite element approximations to the Stokes problem on convex polyhedra ,arXiv:1907.06871.[4] J. Brandts, S. Korotov, Sergey and M. Kˇr´ıˇzek,
On nonobtuse simplicial partitions ,SIAM Rev. 51 (2009), pp. 317–335.[5] S. C. Brenner, L. R. Scott,
The mathematical theory of finite element methods.
Thirdedition. Texts in Applied Mathematics, 15. Springer, New York, 2008.[6] C. Christof, L ∞ -error estimates for the obstacle problem revisited , Calcolo 54 (2017),pp. 1243–1264.[7] P. G. Ciarlet, Discrete maximum principle for finite-difference operators , AequationesMath. 4 (1970), pp. 338–352.[8] P. G. Ciarlet and P. A. Raviart,
Maximum principle and uniform convergence for thefinite element method , Comput. Methods Appl. Mech. Engrg. 2 (1973), pp. 17–31.[9] M. Dauge,
Neumann and mixed problems on curvilinear polyhedra , Integr. Equat.Oper. Th. 15 (1992), pp. 227–261.[10] A. Draganescu, T. F. Dupont, and L. R. Scott,
Failure of the discrete maximumprinciple for an elliptic finite element problem , Math. Comp., 74 (2004), pp. 1–23.[11] A. Demlow,
Localized pointwise a posteriori error estimates for gradients of piece-wise linear finite element approximations to second-order quadilinear elliptic prob-lems , SIAM J. Numer. Anal. 44 (2006), pp. 494–514.[12] J. Frehse and R. Rannacher,
Asymptotic L ∞ -error estimates for linear finite elementapproximations of quasilinear boundary value problems , SIAM J. Numer. Anal. 15(1978), pp. 418–431.[13] P. Grisvard, Elliptic problems in nonsmooth domains.
Monographs and Studies inMathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.[14] J. Guzm´an, D. Leykekhman, J. Rossmann, and A. H. Schatz,
H¨older estimates forGreen’s functions on convex polyhedral domains and their applications to finite ele-ment methods , Numer. Math. 112 (2009), pp. 221–243.[15] W. H¨ohn, H. D. Mittelmann,
Some remarks on the discrete maximum-principle forfinite elements of higher order , Computing 27 (1981), pp. 145–154.[16] D. Jerison and C. E. Kenig,
The inhomogeneous Dirichlet problems in Lipschitz do-mains , J. Func. Anal. 130 (1995), pp. 161–219.[17] T. Kashiwabara and T. Kemmochi,
Maximum norm error estimates for the finiteelement approximation of parabolic problems on smooth domains , Preprint, 2018,arXiv:1805.01336[18] S. Korotov, Sergey and M. Kˇr´ıˇzek,
Acute type refinements of tetrahedral partitions ofpolyhedral domains , SIAM J. Numer. Anal. 39 (2001), pp. 724–733.[19] S. Korotov, Sergey, M. Kˇr´ıˇzek, and P. Neittaanm¨aki,
Weakened acute type conditionfor tetrahedral triangulations and the discrete maximum principle , Math. Comp. 70(2001), pp. 107–119.[20] D. Leykekhman and B. Vexler,
Finite element pointwise results on convex polyhedraldomains , SIAM J. Numer. Anal. 54 (2016), pp. 561–587. [21] D. Leykekhman and B. Vexler, Pointwise best approximation results for Galerkinfinite element solutions of parabolic problems , SIAM J. Numer. Anal. 54 (2016), pp.1365–1384.[22] B. Li,
Analyticity, maximal regularity and maximum-norm stability of semi-discretefinite element solutions of parabolic equations in nonconvex polyhedra , Math. Comp.88 (2019), pp. 1–44.[23] D. Meinder and B. Vexler,
Optimal error estimates for fully discrete Galerkin approx-imations of semilinear parabolic equations , ESAIM Math. Model. Numer. Anal. 52(2018), pp. 2307–2325.[24] J. A. Nitsche and A. H. Schatz,
Interior estimates for Ritz-Galerkin methods , Math.Comp. 28 (1974), pp. 937–958.[25] V. Ruas Santos,
On the strong maximum principle for some piecewise linear finiteelement approximate problems of nonpositive type , J. Fac. Sci. Univ. Tokyo Sect. IAMath. 29 (1982), pp. 473–491.[26] A. H. Schatz,
A weak discrete maximum principle and stability of the finite elementmethod in L ∞ on plane polygonal domains. I , Math. Comp. 34 (1980), pp. 77–91.[27] A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite elementmethods , Math. Comp. 31 (1977), pp. 414–442.[28] A. H. Schatz and L. B. Wahlbin,
Maximum norm estimates in the finite elementmethod on plane polygonal domains. I , Math. Comp. 32 (1978), pp. 73–109.[29] A. H. Schatz and L. B. Wahlbin,
On the quasi-optimality in L ∞ of the ˙ H -projectioninto finite element spaces , Math. Comp. 38 (1982), pp. 1–22.[30] V. Thom´ee and L. B. Wahlbin. Stability and analyticity in maximum-norm for simpli-cial Lagrange finite element semidiscretizations of parabolic equations with Dirichletboundary conditions. Numer. Math. , 87:373–389, 2000.[31] R. Vanselow,
About Delaunay triangulations and discrete maximum principles for thelinear conforming FEM applied to the Poisson equation , Appl. Math. 46 (2001), pp.13–28.[32] J. Wang, and R. Zhang,
Maximum principles for P -conforming finite element ap-proximations of quasi-linear second order elliptic equations , SIAM J. Numer. Anal.50 (2012), pp. 626–642.[33] J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusionequations , Math. Comp. 68 (1999), pp. 1429–1446.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA.
E-mail address : [email protected] Department of Applied Mathematics, The Hong Kong Polytechnic University, HungHom, Hong Kong.
E-mail address : buyang.li @@