    # Weak discrete maximum principle of finite element methods in convex polyhedra

We prove that the Galerkin finite element solution u_h of the Laplace equation in a convex polyhedron , with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree r> 1, satisfies the following weak maximum principle: u_h_L^∞()< Cu_h_L^∞(∂) , with a constant C independent of the mesh size h. By using this result, we show that Ritz projection operator R_h is stable in L^∞ norm uniformly in h for r≥ 2, i.e. R_hu_L^∞()< Cu_L^∞() . Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.

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## 1. Introduction

Let be a finite element space of piecewise polynomials of degree subject to a quasi-uniform tetrahedral partition of a convex polyhedron , where denotes the mesh size of the tetrahedral partition. Let be the subspace of consisting of functions with zero boundary values.

A function is called discrete harmonic if it satisfies the following equation:

 (1.1) (∇uh,∇χh)=0∀χh∈˚Sh.

In this article, we establish the following result, called weak maximum principle of finite element methods.

###### Theorem 1.1.

A discrete harmonic function

satisfies the following estimate:

 (1.2) ∥uh∥L∞(Ω)⩽C∥uh∥L∞(∂Ω),

where the constant is independent of the mesh size .

As an application of the weak maximum principle, we show that the Ritz projection defined by

 (∇(u−Rhu),∇vh)=0∀vh∈˚Sh

is stable in for finite elements of degree , i.e.

 ∥Rhu∥L∞(Ω)⩽C∥u∥L∞(Ω)∀u∈H10(Ω)∩L∞(Ω).

Although this result is well-known for smooth domains [26, 28], for convex polyhedral domains the result was available only with an additional logarithmic factor [19, Theorem 12].

In the finite element literature, the maximum principle has attracted a lot of attention; see [7, 8, 24, 29, 30], to mention a few. However, the sufficient conditions for discrete maximum principle put serious restrictions on the geometry of the mesh. For piecewise linear elements in two-dimensions, the angles of the triangles must be less than , or the sum of opposite angles of the triangles that share an edge must be less than (for example, see [30, §5]). For quadratic elements in two dimensions, discrete maximum principe holds only for equilateral triangles . The situation in three dimensions is more complicated [4, 17, 18, 31], essentially it is hard to guarantee the discrete maximum principe even for piecewise linear elements. In this respect, there stands out the work of Schatz , who proved that a weak maximum principle in the sense of (1.2) holds for a wide class of finite elements on general quasi-uniform triangulation of any two dimensional polygonal domain . By utilizing the weak maximum principle, Schatz also established the stability of the Ritz projection in and norms. Such stability results have a wide range of applications, for example to pointwise error estimates of finite element methods for parabolic problems [21, 16, 20], Stokes systems , nonlinear problems [11, 10, 22], obstacle problems , optimal control problems [1, 2], to name a few. As far as we know,  is the only paper that establishes weak maximum principle and stability estimate (without the logarithmic factor) for the Ritz projection on nonsmooth domains.

In three dimensions the situation is less satisfactory. The stability of the Ritz projection in and norms are available on smooth domains [26, 28] and convex polyhedral domains [13, 19]. However, on convex polyhedral domains in , the -stability constant depends logarithmically on the mesh size , and it is not obvious how the logarithmic factor can be removed there. There are no results on the weak maximum principles in three dimensions even on smooth domains or convex polyhedra. The objective of this paper is to close this gap for convex polyhedral domains. In order to obtain the result, we have to modify the argument in  by considering error analysis in norm for some . In the case of convex polyhedral domains, the -norm based argument used in  would yield an additional logarithmic factor. Unfortunately, the current analysis does not allow us to extend the results to nonconvex polyhedral domains or graded meshes. These would be the subject of future research.

The paper is organized as follows. In section 2 we state some preliminary results that we use later in our arguments. In section 3, we reduce the proof of the weak discrete maximum principle to a specific error estimate. Section 4 is devoted to the 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 to showing the stability of the Ritz projection in norm uniformly in for higher order elements.

In the rest of this article, we denote by a generic positive constant, which may be different at different occurrences but will be independent of the mesh size .

## 2. Preliminary results

In this section, we present several well-known results that are used in our analysis. First result concerns global regularity of the weak solution to the problem

 (2.3) (∇v,∇χ)=(f,χ)∀χ∈H10(Ω).

On the general convex domains we naturally have the regularity (cf. ). However, on convex polyhedral domains, we have the following sharper regularity result (cf. [9, Corollary 3.12]).

###### Lemma 2.1.

Let be a convex polyhedron. Then there exists a constant depending on such that for any and , the solution of (2.3) is in and

 ∥v∥W2,p(Ω)⩽C∥f∥Lp(Ω).

The next result addresses the problem (2.3) when the source function is supported in some part of . The following lemma traces the dependence of the stability constant on the diameter of the support.

###### Lemma 2.2.

For any bounded Lipschitz domain , there exist positive constants and (depending on ) such that for and such that with , the solution of (2.3) satisfies

 ∥v∥W1,p(Ω)⩽Cd∗∥f∥Lp(Ω).
###### Proof.

For any there holds

 |(∇v,∇χ)|=|(f,χ)| ⩽∥f∥Lp(Sd∗(x0))∥χ∥Lp′(Sd∗(x0)) ⩽Cd∗∥f∥Lp(Sd∗(x0))∥∇χ∥Lp′(Sd∗(x0)) ⩽Cd∗∥f∥Lp(Ω)∥∇χ∥Lp′(Ω).

If then we let be the weak solution of

 {Δχ=∇⋅→winΩχ=0on∂Ω

The solution defined above satisfies

 ∇⋅(→w−∇χ)=0,

and, according to [15, Theorem B], there exists a positive constant such that

 ∥∇χ∥Lp′(Ω)⩽C∥→w∥Lp′(Ω)for32−α⩽p⩽3+α.

By using these properties, we have

 |(∇v,→w)|=|(∇v,∇χ)| ⩽Cd∗∥f∥Lp(Ω)∥∇χ∥Lp′(Ω)⩽Cd∗∥f∥Lp(Ω)∥→w∥Lp′(Ω).

The duality pairing estimate above implies the desired result. ∎

The next lemma concerns basic properties of harmonic functions on convex domains. The result is essentially the same as in [27, Lemma 8.3].

###### Lemma 2.3.

Let and be two subdomains satisfying , with

 Dd={x∈Ω:dist(x,D)⩽d},

where is a positive constant. If and is harmonic on , i.e.

 (∇v,∇w)=0,∀w∈H10(Dd),

then the following estimates hold:

 (2.4a) |v|H2(D) ⩽Cd−1∥v∥H1(Dd), (2.4b) ∥v∥H1(D) ⩽Cd−1∥v∥L2(Dd).

Finally, we need the best approximation property of the Ritz projection in norm. In , the best approximation property of the Ritz projection in norm was established on convex polyhedral domains. Together with the standard best approximation property in norm we obtain

 (2.5) ∥v−Rhv∥W1,p(Ω)⩽Cminχ∈˚Sh∥v−χ∥W1,p(Ω)∀v∈H10(Ω)∩W1,p(Ω),

for any . Extension of the above result to follows by duality (cf. [5, §8.5]). These can be summarized as below.

###### Lemma 2.4.

On a convex polyhedron , the following estimate holds for any fixed :

 ∥v−Rhv∥W1,p(Ω)⩽Ch∥v∥W2,p(Ω)∀v∈H10(Ω)∩W2,p(Ω).

## 3. Basic estimates

In [26, Corollary 5.1], the following interior error estimate was established

 ∥u−uh∥L∞(Ω1)⩽Chl|lnh|¯r|u|Wl,∞(Ω2)+Cd−3/q−p∥u−uh∥W−p,q(Ω2),

for , where for , for and , with and . Choosing , and in the above estimate, we obtain that there exists a constant independent of such that

 (3.1) ∥uh∥L∞(Ω1)⩽Cd−32∥uh∥L2(Ω2).

Let be a point satisfying

 |uh(x0)|=∥uh∥L∞(Ω)withd=dist(x0,∂Ω).

If then we can choose and . In this case, the following interior estimate holds (cf. [26, Corollary 5.1] and [25, Lemma 2.1 (ii)]):

 |uh(x0)|⩽Cd−32∥uh∥L2(Sd(x0)).

Otherwise, we have . In this case, the inverse inequality of finite element functions implies

 |uh(x0)|⩽Ch−32∥uh∥L2(Sh(x0)).

Hence, either or , the following estimate holds:

 (3.2) |uh(x0)|⩽Cρ−32∥uh∥L2(Sρ(x0)),withρ=d+2kh.

To estimate the term on the right hand side of the inequality above, we use the following duality property:

 ∥uh∥L2(Sρ(x0))=supsupp(φ)⊂Sρ(x0)∥φ∥L2(Sρ(x0))⩽1|(uh,φ)|,

which implies the existence of a function with the following properties:

 (3.3) supp(φ)⊂Sρ(x0),∥φ∥L2(Sρ(x0))⩽1

and

 (3.4) ∥uh∥L2(Sρ(x0))⩽2|(uh,φ)|.

For this function , we define to be the solution of the PDE problem (in weak form):

 (3.5) (∇v,∇χ)=(φ,χ)∀χ∈H10(Ω),

and let be the finite element solution of

 (∇vh,∇χh)=(φ,χh)∀χh∈˚Sh.

Then is the Ritz projection of , satisfying

 (3.6) (∇(v−vh),∇χh)=0∀χh∈˚Sh.

Let be the solution of the PDE problem (in weak form)

 (3.7)

Then the continuous maximum principle of the PDE problem implies

 (3.8) ∥u∥L∞(Ω)⩽∥uh∥L∞(∂Ω).

Notice, that is the Ritz projection of , i.e.

Therefore, we have

 ∥uh∥L2(Sρ(x0)) ⩽2|(uh,φ)| (here we used (3.4)) =2|(uh−u,φ)+(u,φ)| =2|(∇(uh−u),∇v)+(u,φ)| (here we used (3.5)) =2|(∇uh,∇v)+(u,φ)| (here we used (3.7)) ⩽2|(∇uh,∇v)|+2∥u∥L∞(Ω)∥φ∥L1(Ω) (3.9) ⩽2|(∇uh,∇v)|+Cρ32∥uh∥L∞(∂Ω)∥φ∥L2(Sρ(x0)),

where we have used (3.8) and the Hölder inequality in deriving the last inequality.

To estimate , we note that

 (∇uh,∇v) =(∇uh,∇(v−vh)) (here we use (1.1) and vh∈˚Sh) =(∇(uh−χh),∇(v−vh)) ∀χh∈˚Sh.(here we use % (???)).

We simply choose to be equal to at interior nodes and on ; thus is zero when , and

 ∥uh−χh∥L∞(Ω)⩽∥uh∥L∞(∂Ω).

If we define

 Λh={x∈Ω:dist(x,∂Ω)⩽h},

then

 |(∇uh,∇v)| ⩽∥∇(uh−χh)∥L∞(Λh)∥∇(v−vh)∥L1(Λh) ⩽Ch−1∥uh−χh∥L∞(Λh)∥∇(v−vh)∥L1(Λh) (3.10) ⩽Ch−1∥uh∥L∞(∂Ω)∥∇(v−vh)∥L1(Λh).

Then, substituting (3) and (3) into (3.2), we obtain

 (3.11) ∥uh∥L∞(Ω)⩽C(ρ−32h−1∥∇(v−vh)∥L1(Λh)+1)∥uh∥L∞(∂Ω).

The proof of Theorem 1.1 will be completed if we can establish

 (3.12) ρ−32h−1∥∇(v−vh)∥L1(Λh)⩽C.

## 4. Estimate of ρ−32h−1∥∇(v−vh)∥L1(Λh)

Let and for We define a sequence of subdomains

 Aj={x∈Ω:dj+1⩽|x−x0|⩽dj},j=0,1,2,…

For each we denote to be a subdomain slightly bigger than , defined by

 Alj=Aj−l∪⋯∪Aj∪Aj+1∪⋯∪Aj+ll=1,2,…

Let , with denoting the greatest integer not exceeding . Then

 2ρ⩽dJ+1⩽4ρ

and

 (4.13) measure(Aj∩Λh)≤Chd2j.

By using these subdomains defined above, we have

 ρ−32h−1∥∇(v−vh)∥L1(Λh) ⩽ρ−32h−1(J∑j=0∥∇(v−vh)∥L1(Λh∩Aj)+∥∇(v−vh)∥L1(Λh∩S4ρ(x0))) ⩽Cρ−32h−1J∑j=0h12dj∥∇(v−vh)∥L2(Λh∩Aj) (4.14) +Cρ−12h−12∥∇(v−vh)∥L2(Λh∩S4ρ(x0)),

where the Hölder inequality and (4.13) were used in deriving the last inequality.

Using global error estimate in norm, Lemma 2.1 with and (3.3), we obtain

 ρ−12h−12∥∇(v−vh)∥L2(Λh∩S4ρ(x0)) ⩽Cρ−12h−12h∥v∥H2(Ω) ⩽Cρ−12h−12h∥φ∥L2(Ω) ⩽C,

where we have used and in deriving the last inequality. Substituting the last inequality into (4) yields

 (4.15) ρ−32h−1∥∇(v−vh)∥L1(Λh)⩽Cρ−32h−12J∑j=0dj∥∇(v−vh)∥L2(Aj)+C.

Now, we use the following interior energy error estimate (proved in [23, Theorem 5.1], also see [25, Lemma 2.1 (i)]):

 ∥∇(v−vh)∥L2(Aj) ⩽C∥∇(v−Ihv)∥L2(A1j)+Cd−1j∥v−Ihv∥L2(A1j)+Cd−1j∥v−vh∥L2(A1j) ⩽(Ch+Ch2d−1j)∥v∥H2(A2j)+Cd−1j∥v−vh∥L2(A1j) (4.16) ⩽Chd12−3pj∥v∥W1,p(A3j)+Cd−1j∥v−vh∥L2(A1j)for65

where we have used and the following inequality in deriving the last inequality:

 (4.17) ∥v∥H2(A2j)⩽Cd12−3pj∥v∥W1,p(A3j)for65

The inequality above follows from Lemma 2.3, the Hölder inequality and Sobolev embedding, i.e.

 ∥v∥H2(A2j) ⩽Cd−2j∥v∥L2(A3j) ⩽Cd−2+32−3qj∥v∥Lq(A3j) if q>2 ⩽Cd12−3pj∥v∥W1,p(A3j) for 3q=3p−1 and 652).

This proves that (4) holds for .

By applying Lemma 2.2 to (4) with , we obtain

 ∥∇(v−vh)∥L2(Aj) ⩽Chd−32jρ∥φ∥L32(Sρ(x0))+Cd−1j∥v−vh∥L2(A1j) (4.18) ⩽Chd−32jρ32+Cd−1j∥v−vh∥L2(A1j),

where the last inequality is due to the following Hölder inequality:

 ∥φ∥L32(Sρ(x0))⩽Cρ12∥φ∥L2(Sρ(x0))with∥φ∥L2(Sρ(x0))⩽1.

From (4) we see that

 (4.19) dj∥∇(v−vh)∥L2(Aj)⩽Cρ32h12(hdj)12+C∥v−vh∥L2(A1j).

Then, substituting (4.19) into (4.15), we have

 ρ−32h−1∥∇(v−vh)∥L1(Λh) ⩽CJ∑j=0(hdj)12+Cρ−32h−12J∑j=0∥v−vh∥L2(A1j) (4.20) ⩽C+Cρ−32h−12J∑j=0∥v−vh∥L2(A1j).

It remains to estimate . To this end, we let be a smooth cut-off function satisfying

 χ=1onA1jandχ=0% outsideA2j.

Then

 ∥v−vh∥L6(A1j) ⩽∥χ(v−vh)∥L6(Ω) ⩽∥χ(v−vh)∥H1(Ω)(Sobolev % embedding H1(Ω)↪L6(Ω)) (4.21) ⩽∥∇(v−vh)∥L2(A2j)+Cd−1j∥v−vh∥L2(A2j).

By using (4

) and the interpolation inequality (for

)

 (4.22) ∥v−vh∥L2(A1j)⩽∥v−vh∥1−θLp(A1j)∥v−vh∥θL6(A1j)with12=1−θp+θ6,

we obtain

 ∥v−vh∥L2(A1j) =(ε−θ1−θ∥v−vh∥Lp(A1j))1−θ(ε∥∇(v−vh)∥L2(A2j)+Cεd−1j∥v−vh∥L2(A2j))θ ⩽ε−θ1−θ∥v−vh∥Lp(A1j)+ε∥∇(v−vh)∥L2(A2j)+Cεd−1j∥v−vh∥L2(A2j),

where can be an arbitrary positive number. By choosing with , we obtain

 ∥v−vh∥L2(A1j) ⩽(ρdj)−θσ1−θd−θ1−θj∥v−vh∥Lp(A1j) +(ρdj)σ(dj∥∇(v−vh)∥L2(A2j)+C∥v−vh∥L2(A2j)).

Hence,

 ρ−32h−12J∑j=0∥v−vh∥L2(A1j)

where we have used (4.19) in deriving the last inequality. Note that

 J∑j=0(ρdj)σ∥v−vh∥L2(A2j) ⩽C(ρdj)σ∥v−vh∥L2(S8ρ(x0))+2J∑j=0(ρdj)σ∥v−vh∥L2(A1j).

Combining the last two estimates, we obtain

 ρ−32h−12J∑j=0∥v−vh∥L2(A1j) +Cρ−32h−12(ρdj)σ∥v−vh∥L2(S8ρ(x0))+Cρ−32h−12J∑j=0(ρdj)σ∥v−vh∥L2(A1j).

If for sufficiently large constant , then the last term can be absorbed by the left side. Hence, we have

 J∑j=0ρ−32h−12∥v−vh∥L2(A1j) (4.23) +Cρ−32h−12(ρdj)σ∥v−vh∥L2(S8ρ(x0)).

It remains to estimate and . To this end, we let be a function satisfying

 ∥v−vh∥Lp(A1j) ⩽2(v−vh,ψ)and∥ψ∥Lq(A1j)⩽1,with1p+1q=1.

Let be the solution of

 {−Δw=ψinΩ,w=0on∂Ω,

Then using Lemma 2.4 and Lemma 2.1, we obtain

 (v−vh,ψ) =(∇(v−vh),∇w) =(∇(v−vh),∇(w−Ihw)) ⩽∥∇(v−vh)∥Lp(Ω)∥∇(w−Ihw)∥Lq(Ω) ⩽Ch2∥v∥W2,p(Ω)∥w∥W2,q(Ω) ⩽Ch2∥φ∥Lp(Ω)∥ψ∥