Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory

Vol. 39, N 4, 2005, pp. 755–780 DOI: 10.1051/m2an:2005033

FINITE ELEMENT APPROXIMATION FOR DEGENERATE PARABOLIC EQUATIONS. AN APPLICATION OF NONLINEAR SEMIGROUP THEORY

Akira Mizutani

, Norikazu Saito

and Takashi Suzuki

Abstract. Finite element approximation for degenerate parabolic equations is considered. We pro- pose a semidiscrete scheme provided with order-preserving andL¹ contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L¹ and L^∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also deriveL¹ convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.

Mathematics Subject Classification. 35K65, 47H20, 65M60.

Received: June 1, 2004. Revised: March 2, 2005.

1. Introduction

This paper is concerned with the finite element method applied to the initial-boundary value problem for degenerate parabolic equation, 









u_t−∆f(u) = 0 in Ω×(0, T), f(u) = 0 on∂Ω×(0, T), u|t=0=u₀(x) on Ω,

(1.1)

where Ω⊂Rⁿ,n= 1,2,3, denotes a bounded domain with the Lipschitz boundary∂Ω,T an arbitrary positive constant, andf a non-decreasing continuous function defined onRsatisfyingf(0) = 0.

Problem (1.1) describes, for instance, the flow of homogeneous fluid in porous media if

f(u) =u|u|^γ−1 (1.2)

withγ >1, the fast diffusion if (1.2) with 0< γ <1, and the two phase Stefan problem in enthalpy formulation if f(u) =







α(u+ 1) (u≤ −1) 0 (−1< u <1) β(u−1) (u≥1)

(1.3)

withα, β >0. See, for more detail [14, 15, 32].

Keywords and phrases. Finite element method, degenerate parabolic equation, nonlinear semigroup.

1 Department of Mathematics, Faculty of Science, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan.

2 Faculty of Education, Toyama University, 3190 Gofuku, Toyama 930-8555, Japan. [email protected]

3 Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-1 Machikaneyama, Toyonaka, 560-0043, Japan.

c EDP Sciences, SMAI 2005

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2005033

L¹ theory to (1.1) was developed in early 1970’s in use of nonlinear semigroups. To summarise it, we set X = L¹(Ω) and introduce operators L and A in X byLv =−∆v forv ∈D(L) ={v ∈ W₀^1,1(Ω) | Lv ∈X} and Av =Lf(v) for v∈D(A) = {v ∈X |f(v)∈D(L)}, respectively. Then, problem (1.1) is reduced to the nonlinear evolution equation

du

dt +Au= 0 with u(0) =u₀ (1.4)

in X. Brezis-Strauss [6] proved that the operator−Aism-dissipativein X. This means that

v−ˆv_L1(Ω)≤ v−vˆ+λAv−λAˆv_L1(Ω) (1.5) holds for v,vˆ∈D(A) andλ >0, and alsoR(I+λA) =L¹(Ω) =D(A). Then, theory of Crandall-Liggett [12]

guarantees the generation of semigroup{S(t)}_t≥0 onX by

S(t) =s-lim

m→∞

I+ t

mA _−m

, (1.6)

andu(t) =S(t)u₀is regarded as the solution to (1.4). Another important property ofAis the order-preserving, that is,

g≥ˆg ⇒ (I+λA)⁻¹g≥(I+λA)⁻¹g.ˆ (1.7)

Relations (1.5) and (1.7) are summarised as [v−ˆv]₊

L¹(Ω)≤[v−vˆ+λAv−λAˆv]₊

L¹(Ω) (1.8)

forv,vˆ∈D(A), where [v]₊= max{0, v}. This implies [S(t)u₀−S(t)ˆu₀]₊

L¹(Ω)≤[u₀−uˆ₀]₊

L¹(Ω) (1.9)

by (1.6), whereu₀,uˆ₀∈X andt∈[0, T]. Inequality (1.9) means that{S(t)}_t≥0 is an order-preserving andL¹ contraction semigroup onX.

L^∞ stability of the resolvent,

(I+λA)⁻¹g

L^∞(Ω)≤ g_L∞(Ω), (1.10)

is also proven in [6], whereg∈X∩L^∞(Ω) andλ >0. This impliesL^∞stability of the semigroup

S(t)u_0L∞(Ω)≤ u_0L∞(Ω), (1.11) whereu₀∈X∩L^∞(Ω) andt∈[0, T].

So far, several schemes of time discretization have been examined. In fact, those structures of the problem, particularly (1.6), justify the backward difference approximation to (1.1), which was studied by [28]. Another scheme was obtained by the use of the nonlinear Chernoff formula of [5], where solution at each discrete time level is approximated by a linear elliptic equation. This approach was taken first by [3]. WhereasL¹framework was employed in [3,28],L²error estimates were obtained by [20,26,27] for modified schemes of [3]. Those works were done in the literature of porous media or that of Stefan problems. For fast diffusion problems, we refer to [23, 24].

On the other hand, for porous media and Stefan problems, fully discrete schemes where the space variable was discretized by finite element methods were also studied by many authors; [10, 13, 21, 30, 31, 34, 36]. Some of them gave error analysis in the H⁻¹ framework. We will mention a few remarks on such schemes in the next section, after having presented our scheme.

The present paper deals with a spatial discretization for (1.1), that is, du_h

dt +A_hu_h= 0 with u_h|_t=0=u_0h,

whereA_h,u_h, andu_0hstand for the finite element approximations ofA,u, and u₀, respectively.

Our purposes are twofold. Firstly, we introduce the spatial discretization A_h of A which preserves above mentioned properties. It can be done by making use of piecewise linear trial functions and the lumping mass technique, if a family of the triangulation{T_h}of Ω,h >0 being the discretization parameter, is of acute type (the definition will be recalled in Sect. 3). Actually, in Sections 2 and 3, we introduce A_h and prove that A_h satisfies the discrete analogue of (1.8) in a suitable Banach space X_h, respectively. From this, we immediately obtain the nonlinear semigroup{S_h(t)}_t≥0onX_hwhich is generated by−A_hand satisfies the discrete analogue of (1.9). Moreover, as will be mentioned in Section 4,A_h andS_h(t) areL^∞stable as well asAandS(t) are so.

The second purpose of this paper is to make error analysis. The goal of this end is to derive limh↓0 sup

t∈[0,T]u_h(t)−u(t)_L1(Ω)= 0. (1.12) Our main theorem (Th. 7.1) shows that (1.12) is valid, for example, if Ω⊂R²is convex,u₀is continuous on Ω with the value zero on∂Ω, f is strictly increasing, and{T_h}is provided with acuteness and quasi-uniformity.

Further an extension of Theorem 7.1 to the case where f is nondecreasing is also discussed (Prop. 7.1 and Lem. 7.2). However we have no error estimates and they will be studied in subsequent works. Proof of (1.12) follows the principle that the convergence of semigroup is a consequence of that of resolvents. Thus, Sections 5–7 are devoted to the proof of the convergence of the resolvent, the Yosida approximation, and the semigroup, respectively.

Finally, in Section 8, we present results of numerical experiments for the porous media nonlinearity. The time discretization makes use of the forward difference formula. We observe thatL¹ convergence of numerical solutions really takes place.

At this stage, we clarify our motivation of this work. As was mentioned above, several physical phenomena are modelled by (1.1), and therefore order-preserving andL¹ contraction properties are essential requirements from not only mathematical but also physical points of view. Consequently we are interested in discrete schemes which preserve such properties of the original problem. However it seems that little effort has been made in this direction. The first contribution of this paper is to give an discrete scheme enjoying a discrete version of order-preserving and L¹ contraction properties for a general nondecreasingf. Moreover our presented scheme is well suited for an actual computation. The second contribution is a convergence result of the form (1.12).

Our result can be applied to porous media and fast diffusion nonlinearities (1.2). Especially we do not know any convergence results for a spatial discretization to the fast diffusion problem at present. On the other hand, for the Stefan nonlinearity (1.3), our convergence result may be restrictive, since f and u₀ are assumed to be strictly increasing and continuous, respectively. The main interest here is to reveal a general nature of convergence rather than to go into details under specific assumptions on f. The convergence result itself is to be expected from semigroup theory. But, as is well-known, fundamental theorems in nonlinear semigroup theory were established by quite technical and somewhat tricky arguments. Therefore, it is not obvious that the similar argument works for discrete problems. For example, the effect of perturbation on f causes a new issue which have not appeared in the continuous problem (see Rem. 7.1). Thus, in the present paper, we will develop a discrete nonlinear semigroup theory. Also we note that, concerning the regularity of solutions, we only haveu(t)∈X and f(u(t))∈W₀^1,1(Ω), even ifu₀is continuous. Our argument does not require any redundant assumptions on the regularity of solutions.

Recently some of the authors and their colleague published the monograph [17], where finite element ap- proximation to (1.1) on flat torus with uniform triangulation is studied. Some lemmas and theorems described below are proven similarly, but we shall give them for completeness. Furthermore, the method of [17] for the convergence of resolvent is restrictive, and we shall provide new arguments here.

We follow the standard notation of [1]. We put · p= · _L^p_(Ω) forp∈[1,∞]. The spaceW₀^m,p(Ω) stands for the closure in W^m,p(Ω) ofC₀^∞(Ω), the set of C^∞ functions with compact supports in Ω. We write H^m(Ω) and H₀^m(Ω) instead ofW^m,2(Ω) andW₀^m,2(Ω), respectively, form≥0. The standard inner product ofL²(Ω) is denoted by (·,·). Furthermore we set

W ={v∈C(Ω)|v|∂Ω= 0}. (1.13)

Generic positive constants depending on Ω are denoted by C, C₁, and so forth. If it is necessary to specify the dependence on other parameters, say γ₁, γ₂, · · ·, then we write C(Ω, γ₁, γ₂,· · ·). We shall use the same symbolI to indicate the identity operator on any space.

2. Finite element approximation

For the sake of simplicity, in what follows, we suppose that Ω is an n-dimensional polyhedron. We take a family of triangulations {Th} = {Th}h↓0 defined on Ω, where each element σ ∈ Th is a closed simplex. The maximum side length of all elements inT_his denoted byh. We take the piecewise linear approximation, putting

X_h={χ∈W | χ is linear onσfor eachσ∈ Th}.

LetI_h be the set of vertices ofσ∈ T_h belonging to Ω. Fora∈I_h, the functionw_a∈X_his defined by w_a =

1 (ata)

0 (atb∈I_h\ {a}).

Then,{w_a|a∈I_h}forms a basis of X_h and the interpolation operatorπ_h:W →X_h is defined by π_hv=

a∈Ih

v(a)w_a.

Eacha∈I_h takes barycentric domainD_a. See [17], p. 203 for its precise definition. Let w_a(x) =

1 (x∈D_a) 0 (x∈Ω\D_a),

and denote by X_h the vector space spanned by {w_a|a∈I_h}. The linear transformation M_h : X_h → X_h, referred to as the lumping operator, is defined through w_a → w_a. Sometimes, we shall write χ_h for M_hχ_h, whereχ_h∈X_h.

The semidiscrete scheme studied in this paper is to solveu_h∈C¹([0, T];X_h) satisfying d

dt(u_h, v_h) + (∇π_hf(u_h),∇v_h) = 0 with (u_h(0), v_h) = (u_0h, v_h) (2.1) for any v_h ∈ X_h, whereu_0h ∈ X_h is an appropriate approximation of u₀ ∈ X. In order to convert (2.1) to the operator theoretic form, we introduce the following operators. Let L_h : X_h → X_h be the finite element approximationLdefined by

(L_hχ_h, v_h) = (∇χ_h,∇v_h)

forχ_h, v_h∈X_h. LetM_h^∗:X_h→X_h be the adjoint operator ofM_h associated with theL² inner product, and set

K_h=M_h^∗M_h:X_h→X_h. Then (2.1) is expressed as

K_hdu_h

dt +L_hπ_hf(u_h) = 0 with u_h(0) =u_0h. (2.2)

The operatorM_h is invertible inX_hand henceK_h⁻¹=M_h⁻¹(M_h^∗)⁻¹ is well-defined. Therefore, scheme (2.2) is equivalent to

du_h

dt +A_hu_h= 0 with u_h(0) =u_0h (2.3)

in X_h, where

A_hv=K_h⁻¹L_hπ_hf(v) is defined forv ∈W.

We here describe some examples ofu_0h∈X_h:

u_0h =P_hu₀ (ifu₀∈L²(Ω));

u_0h =R_hu₀ (ifu₀∈H₀¹(Ω));

u_0h =π_hu₀ (ifu₀∈W),

whereR_h:H₀¹(Ω)→X_h andP_h:L²(Ω)→X_h denote the Ritz and the orthogonal projection operators. They are defined by

(∇(v−R_hv),∇χ_h) = 0 (χ_h∈X_h) (2.4)

and

(v−P_hv, χ_h) = 0 (χ_h∈X_h), (2.5)

respectively. Further, if u₀ ∈ W₀^1,1(Ω), we can apply Scott and Zhang’s interpolation operator Π_h : W₀^1,1(Ω) → X_h and take u_0h = Π_hu₀. (For the precise definition of Π_h, see [35]. A version of such interpolation is described in [4].) In convergence analysis presented below, we assume u₀ ∈ W and take u_0h=π_hu₀.

Before concluding this section, we state a remark on another finite element scheme to (1.4). From the L² theoretical point of view, it may be natural to take

d

dt(u_h, v_h) + (∇f(u_h),∇vh) = 0 with (u_h(0), v_h) = (u_0h, v_h) (2.6) forv_h∈X_h. In this case, the operator theoretic representation reads

du_h

dt +L_hR_hf(u_h) = 0 with u_h(0) =u_0h.

Iff is locally Lipschitz continuous, scheme (2.6) is well-defined, because thenv_h∈X_h impliesf(v_h)∈H₀¹(Ω).

Namely, in this case (2.6) is conforming and was studied by [21, 30, 31, 34] including its time discretizations.

Based on the energy method, they discussed the stability, convergence, and error estimate in the L² norm for the porous media and the Stefan nonlinearities.

However, the linear partL_hR_h does not have such properties as (3.8), (3.9) and (3.10) given below. Thus, in general, v_h ∈ X_h → −L_hR_hf(v_h) ∈ X_h is not m-dissipative. This means that, even if an approximate solution converges to the original one, it is not certain that the approximate solution has order-preserving andL¹ contraction properties. On the contrary,−A_h ism-dissipative as will be shown in the next section.

3. Wellposedness

We pose on{Th}that

(H1) Acuteness. Given σ ∈ Th, a vertex P₀ ⊂ σ, and the opposite face F ⊂ σ to P₀, let S be a plane includingF. Then the foot of the perpendicular fromP₀ toS is always included inF.

Remark 3.1. Ifn= 1, (H1) always holds. If n= 2, it is equivalent to saying that eachσ∈ T_h is a right or an acute triangle. Generally, it corresponds to the non-negative type of Ciarlet and Raviart [9] or acuteness of Fujii [16].

This section is devoted to the proof of the following theorem, which is a discrete analogue of (1.8).

Theorem 3.1. Assume that (H1)holds. Then we have

M_hπ_h[v_h−vˆ_h]₊₁≤M_hπ_h[v_h−vˆ_h+λA_hv_h−λA_hˆv_h]₊

1, (3.1)

for v_h,ˆv_h∈X_h andλ >0. Furthermore, it holds that R(I+λA_h) =X_h.

This assures the unique solvability of (2.3). In fact,X_hforms a Banach space equipped with the norm χ_h1,h=

ΩM_hπ_h|χ_h| (3.2)

forχ_h∈X_h. Theorem 3.1 means that −A_h ism-dissipative inX_h with respect to this norm. Therefore, from the generation theorem of [12], scheme (2.3) is uniquely solvable globally in time and the solution is given as u_h(t) =S_h(t)u_0h, where

S_h(t) = lim

m→∞

I+ t

mA_{h −m}

. (3.3)

Combining (3.1) with (3.3), we deduce

[Sh(t)u_0h−S_h(t)ˆu_0h]_+1,h≤ [u0h−uˆ_0h]_+1,h (3.4) foru_0h,uˆ_0h∈X_h andt∈[0, T]. Therefore, it holds that

u_0h≥uˆ_0h ⇒ S_h(t)u_0h≥S_h(t)ˆu_0h In particular,S_h(t)u_0h≥0 follows fromu_0h≥0 and it holds that

S_h(t)u_0h1,h≤ u_0h1,h. At this stage, we assume that

(H2)Regularity. There is a positive constant ν₁ independent ofhsuch that ρ(σ)≥ν₁d(σ)

for any σ ∈ Th, where ρ(σ) and d(σ) indicate diameters of the inscribing and the circumscribing balls of σ, respectively.

Under such a reasonable assumption, we have a constantC >0 independent ofhsatisfying

C⁻¹χh₁≤ χh_1,h≤Cχh₁ (3.5)

forχ_h∈X_h. Hence, by (3.4), we obtain

[S_h(t)v_h−S_h(t)ˆv_h]₊

1≤C[v_h−vˆ_h]₊

1. Remark 3.2. Inequalities (3.5) follows from

C⁻¹χh_L1(σ)≤ Mhπ_h|χh|_L1(σ)≤Cχh_L1(σ) (χ_h∈X_h). (3.6) for any σ ∈ Th. Because {Th} is regular, inequality (3.6) is reduced to the caseσ = ˆσ, where ˆσ denotes the canonical reference element. The linear functions on ˆσ form a finite dimensional vector space Y. The desired estimate holds because Y is isometric to an Euclidean space, and any two norms onY are equivalent to each other. We also have

K_hχ_hp+K_h⁻¹χ_hp≤Cχ_hp (χ_h∈X_h, 1≤p≤ ∞). (3.7) under (H2). See [17], p.174.

Before stating the proof of Theorem 3.1, we collect some inequalities concerning linear part K_h⁻¹L_h, which hold under (H1). They are shown in [17], Sect. 5.1 and the proof is omitted here. First, discrete maximum principle

maxΩ (I+λK_h⁻¹L_h)⁻¹v_h≤max

Ω π_h[v_h]₊ (3.8)

holds, wherev_h∈X_h andλ >0. Here, well-definedness of (I+λK_h⁻¹L_h)⁻¹:X_h→X_h is included. It follows from (3.8) that

0≤v_h∈X_h, λ >0 ⇒ (I+λK_h⁻¹L_h)⁻¹v_h≥0. (3.9) Next, discreteL¹ contraction property is expressed as

0≤v_h∈X_h, λ >0 ⇒

ΩM_h(1 +λK_h⁻¹L_h)⁻¹v_h≤

ΩM_hv_h. (3.10)

The proof of (3.8) and (3.10) is explicitly mentioned for the case n = 2 in [17]. However, the other cases n= 1,3 can be done similarly under the assumption (H1). In (3.9) and (3.10), contribution of mass lumping is essential for λ > 0. If the consistent mass is employed, then (3.9) is restricted to the range 0< h²/λ1, while property (3.10) is not certain to hold. See Ciarlet-Raviart [9] and Fujii [16] for the former fact.

Thanks to (3.9) and (3.10), we can prove the following inequality, which is comparable to Kato’s one of [22].

Lemma 3.1. Assume that (H1)holds. Then we have

ΩM_hπ_h

(K_h⁻¹L_hπ_hv) sgn⁺v

≥0 (3.11)

for v∈W, where

sgn⁺v=

1 (v≥0) 0 (v <0).

Proof. First, we show that

ΩM_hπ_h

I+λK_h⁻¹L_h−1 v_h

+≤

ΩM_hπ_h[v_h]₊ (3.12)

holds forv_h∈X_hand λ >0. In fact, taking

v_h^±≡π_h[v_h]_±=±

a∈I_h^±

v_h(a)w_a, (3.13)

we have 0≤v_h^± ∈X_h and v_h =v⁺_h −v_h⁻, where I_h^± ={a∈I_h| ±v_h(a)≥0} and [ ·]_± = max{0,± · }. This implies

I+λK_h⁻¹L_h−1

v^±_h ≥0 by (3.9), and hence I+λK_h⁻¹L_h−1

v_h

+≤

I+λK_h⁻¹L_h−1 v_h⁺.

Becauseπ_h andM_h are order-preserving, we have M_hπ_h

I+λK_h⁻¹L_h−1 v_h

+≤M_h

I+λK_h⁻¹L_h−1 v_h⁺,

which implies

ΩM_hπ_h

I+λK_h⁻¹L_h−1 v_h

+≤

ΩM_hv_h⁺ by (3.10). This means (3.12).

Givenv∈W, we takev_h=π_hv andu_h=

I+εK_h⁻¹L_h−1

v_hforε >0. Because of v_h−u_h=ε

I+εK_h⁻¹L_h−1

K_h⁻¹L_hv_h, we have

ε

ΩM_hπ_h

I+εK_h⁻¹L_h−1

K_h⁻¹L_hv_h

·sgn⁺v_h

=

ΩM_hπ_h

(v_h−u_h)·sgn⁺v_h

=

ΩM_hπ_h[v_h]₊−

ΩM_hπ_h

u_h·sgn⁺v_h

≥

ΩM_hπ_h[v_h]₊−

ΩM_hπ_h[u_h]₊. The right-hand side is non-negative by (3.12), and hence

Ω

M_hπ_h

1 +εK_h⁻¹L_h−1

K_h⁻¹L_hv_h

·sgn⁺v_h ≥0.

Makingε↓0, we have

ΩM_hπ_h

K_h⁻¹L_hv_h

·sgn⁺v_h

≥0.

Hence noting

π_h

η·sgn⁺v

=

a∈Ih∩{πhv≥0}

η(a)w_a =π_h

η·sgn⁺π_hv

for anyη∈W, we obtain (3.11). The proof is complete.

Now we give the following.

Proof of Theorem 3.1. To prove (3.1), we show more generally that M_hπ_h[v−ˆv]₊

1≤M_hπ_h[v−vˆ+λA_hv−λA_hv]ˆ₊

1, (3.14)

where v,vˆ∈W andλ > 0. To this end, we suppose thatf is strictly increasing. Otherwise, we replace f by f_ε(u) =f(u) +εuand makeε↓0. Puttingg=v+λK_h⁻¹L_hπ_hf(v) and ˆg= ˆv+λK_h⁻¹L_hπ_hf(ˆv), we get that

M_hπ_h[v−ˆv]₊

1 =

ΩM_hπ_h

(v−ˆv)·sgn⁺(v−v)ˆ

=

ΩM_hπ_h

(g−ˆg)·sgn⁺(v−v)ˆ

−λ

ΩM_hπ_h

K_h⁻¹L_hπ_h(f(v)−f(ˆv))

·sgn⁺(v−ˆv) .

Here, we have sgn⁺w= sgn⁺(v−ˆv) holds forw=f(v)−f(ˆv)∈W, becausef is strictly increasing. Therefore, (3.11) guarantees that

ΩM_hπ_h

K_h⁻¹L_hπ_h(f(v)−f(ˆv))

·sgn⁺(v−v)ˆ

=

ΩK_h⁻¹π_h

K_h⁻¹L_hπ_hw·sgn⁺w

≥0.

This leads to

M_hπ_h[v−v]ˆ₊₁ ≤

ΩM_hπ_h

(g−g)ˆ ·sgn⁺(v−ˆv)

≤

ΩM_hπ_h[g−g]ˆ₊=M_hπ_h[v−vˆ+λA_hv−λA_hˆv]₊

1, and hence (3.1) follows.

Now we prove the maximality (I+λA_h)X_h=X_hforλ >0. Namely, giveng_h∈X_h, we show the existence of v_h∈X_hsatisfyingv_h+λA_hv_h=g_h. In fact,T_λ=I+λA_his a continuous mapping onX_h, a finite dimensional vector space provided with the norm · _1,h. In use of (3.1) we can take an open ballO ⊂X_hsufficiently large such thatg_h∈T_λ(∂O) for anyλ >0. We may suppose thatg_h∈ O. Then the topological degree deg (T_λ, g_h,O) is well-defined and its homotopy invariance implies

deg (T_λ, g_h,O) = deg (I, g_h,O) = 1.

This means that g_h∈T_λ(O), and the proof is complete.

4. L

stability

This section is devoted to the L^∞ stability of approximate solutions. Precisely, we show the following.

Theorem 4.1. Under the assumption (H1), it holds that

S_h(t)u_0h∞≤ u_0h∞, (4.1)

whereu_0h∈X_h andt∈[0, T].

Note that, as will be verified at the end of this section, (4.1) gives (I+λA_h)⁻¹g_h

∞≤ gh_∞, (4.2)

forg_h∈X_h, λ >0.

To prove Theorem 4.1, we make use of the nonlinear Chernoff formula, taking a finite element analogue of the time-discretization scheme of [3]. For the moment, we suppose that f is locally Lipschitz continuous. Let µ > 0 be the Lipschitz constant off on [−M, M], where M =u0h∞ for u_0h ∈X_h. We take τ =T /N for N ∈Nand putt_m=mτ for 0≤m≤N. Then, we introduce the regularizing parameters_τ >0 satisfying

limτ↓0s_τ = 0 and µτ

s_τ ≤1, (4.3)

and take{w^τ_h(t_m)}^N_m=0⊂X_h by

w^τ_h(t_m+1)−w^τ_h(t_m)

τ +

1−e^{−sτKh−1Lh} s_τ

π_hf(w_h^τ(t_m)) = 0

withw^τ_h(0) =u_0h, where {e^−sKh−1Lh}_s≥0 denotes the linear semigroup inX_h generated byK_h⁻¹L_h. We extend w^τ_h(t_m) to allt∈[0, T] as

w_h^τ(t) =

w^τ_h(0) (t= 0)

w^τ_h(t_m) (t_m−1< t≤t_m, 1≤m≤N) . (4.4) The following lemma is proven similarly to [3].

Lemma 4.1. In addition to the basic assumption on f, suppose that f is locally Lipschitz continuous on R.

Then w^τ_h(t)∈X_h is well-defined for all t∈[0, T], and moreover limτ↓0 sup

t∈[0,T]w^τ_h(t)−S_h(t)u_0h1,h= 0 (4.5) for u_0h∈X_h.

Proof. We have the formula

w_h^τ(t_m) =F_h(τ)^mu_0h, where

F_h(τ)φ_h=φ_h+ τ s_τ

e^{−sτKh−1Lh}π_hf(φ_h)−π_hf(φ_h)

. (4.6)

Since, byµτ /s_τ≤1, the mappingr→r−(τ /s_τ)f(r) is non-increasing, we have

−M− τ

s_τf(−M)≤u_0h− τ

s_τπ_hf(u_0h)≤M− τ

s_τf(M). (4.7)

On the other hand, (3.8) implies 0≤

I+λK_h⁻¹L_h−1

v_h^± ≤max_Ωv_h^± for v_h ∈ X_h and λ >0 with v^±_h ∈X_h defined by (3.13). In particular,

maxΩ

1 +λK_h⁻¹L_h−1

π_h[v]_±≤max

Ω π_h[v]_± holds for anyv∈W andλ >0. Then, the linear semigroup theory guarantees that

maxΩ e^−sKh−1Lhπ_h[v]_±≤max

Ω π_h[v]_±

for anys >0 andv∈W. Therefore, noting thatf(−M)≤π_hf(u_0h)≤f(M), we can deduce

f(−M)≤e^{−sτK−1h Lh}π_hf(u_0h)≤f(M). (4.8) Inequalities (4.7) and (4.8) imply

−M ≤w^τ_h(t₁) =u_0h+ τ s_τ

e^{−sτKh−1Lh}π_hf(u_0h)−π_hf(u_0h) ≤M,

which means F_h(τ)u_0h∞≤M. Therefore, we get by an induction that

w_h^τ(t_m)∞≤ u0h_∞. (4.9)

This allows us to assume that f is Lipschitz continuous with Lipschitz constantµ in R by replacingf(u) by f(±M) for±u≥M. Then,r→f(r) andr→r−(τ /s_τ)f(r) are non-decreasing onR, and it follows that

τ

s_τ |f(r)−f(s)|+

(r−s)− τ

s_τ(f(r)−f(s))

=|r−s| (4.10)

forr, s∈R. On the other hand, from (3.1) and (3.3) applied tof(u) =u, we have e^−sKh−1Lhπ_h[v]₊

1,h≤ π_h[v]_+1,h forv∈W. This, together with (4.10), gives that

F_h(τ)φ_h−F_h(τ)ψ_h1,h≤ τ

s_τ f(φ_h)−f(ψ_h)_1,h+

(φ_h−ψ_h)− τ

s_τ (f(φ_h)−f(ψ_h)) 1,h

= φ_h−ψ_h1,h (4.11)

forφ_h, ψ_h∈X_h.

Now we shall show (4.5). It is a consequence of the Chernoff formula, Theorem 3.1 of [5]. Namely, it suffices to prove that

limτ↓0

I+λ

τ(I−F_h(τ))]

₋₁

φ_h= (I+λA_h)⁻¹φ_h (4.12)

forφ_h∈X_h andλ >0. For this purpose, we put ψ_h= (I+λA_h)⁻¹φ_h, ψ_h^τ=

I+λ

τ (I−F_h(τ)) ₋₁

φ_h, φ^τ_h=ψ_h+λ

τ(I−F_h(τ))ψ_h. Then, we have

φ_h=ψ^τ_h+λ

τ (I−F_h(τ))ψ^τ_h and

φ_h−φ^τ_h=

1 +λ τ

(ψ_h^τ−ψ_h) +λ

τ (F_h(τ)ψ_h−F_h(τ)ψ_h^τ). Therefore, inequality (4.11) gives that

1 + λ

τ

ψ^τ_h−ψ_h1,h≤ φ_h−φ^τ_h1,h+λ

τ ψ_h^τ−ψ_h1,h, and hence

ψ^τ_h−ψ_h1,h≤ φ_h−φ^τ_h1,h. (4.13)

Inequality (4.13) provides an a priori estimate and hence the existence ofψ_h^τ follows similarly to the proof of Theorem 3.1.

Finally, by (4.6), we have

φ^τ_h=ψ_h− λ s_τ

e^{−sτKh−1Lh}π_hf(ψ_h)−π_hf(ψ_h)

and hence

limτ↓0φ^τ_h=ψ_h+λK_h⁻¹L_hπ_hf(ψ_h) =ψ_h+λA_hψ_h=φ_h.

Thus, we get (4.12) by (4.13) and the proof is complete.

Now, we give the following.

Proof of Theorem 4.1. Iff is locally Lipschitz continuous, then we have (4.5) and (4.9), which implies (4.1) by dimX_h<+∞.

If this is not the case, we take the Yosida approximation, a family{f_λ} converging tof locally uniformly as λ↓0. Namely, in use of the maximal monotone graphβ=f⁻¹, we define the inverse function off_λas

f_λ⁻¹≡β_λ= 1 λ

1−(1 +λβ)⁻¹

(4.14) which is non-decreasing,f_λ(0) = 0, and locally Lipschitz continuous. LetA^λ_hv=L_hπ_hf_λ(v). Then it generates the semigroup{S_h^λ(t)}t≥0 inX_hsatisfying

S_h^λ(t)u_0h

∞≤ u_0h∞

foru_0h∈X_h andt∈[0, T]. Makingλ↓0, we obtain (4.1) by dimX_h<+∞.

We proceed to the proof of (4.2). For this end, we take the duality map F :X_h →X_h^∗, regardingX_h as a closed subspace ofL^∞(Ω). Namely, forv_h, χ_h∈X_hit holds that

χ_h∈F(v_h) ⇐⇒ vh, χ_h=vh²_∞=χh²_∗,

where·,·denotes the pairing betweenX_handX_h^∗, and · _∗ the operator norm. See Miyadera [29],e.g., for the existence of such an operator. Then, by making use of (4.1), it holds that

S_h(τ)−1 τ

v_h, χ_h

= 1

τ {S_h(τ)v_h, χ_h − v_h, χ_h}

= 1 τ

S_h(τ)v_h, χ_h − v_h²_∞

≤ 1

τ {S_h(τ)v_h∞− v_h∞} χ_h∗≤0

for v_h ∈ X_h, τ > 0, and χ_h ∈ F(v_h). Hence, by making τ ↓ 0, we obtainA_hv_h, χ_h ≤ 0 for v_h ∈ X_h and χ_h∈F(v_h). The general theory of the duality map, say Corollary 2.7 of [29], guarantees that

gh_∞≤ (1 +λA_h)g_h∞ for anyg_h∈X_hand λ >0. Thus we establish (4.2).

5. Convergence of resolvent

Convergence of semigroup follows from that of resolvent. We assume the following condition concerning the domain Ω⊂Rⁿ:

(D) Ifn= 3 the Dirichlet problem

−∆w=g in Ω, w= 0 on ∂Ω admits the elliptic estimate

w_W2,p(Ω)≤C_pg_p forp∈(1, µ), whereµ > n= 3.

As for the triangulation, we suppose

(H3)Inverse inequality. There is a positive constant ν₂ independent ofhsuch that d(σ)≥ν₂h

for anyσ∈ Th.

This section is devoted to the

Theorem 5.1. IfΩis convex and provided with the property (D) (ifn= 3),{T_h}satisfies(H1),(H2)and(H3), andf is strictly increasing, then it holds that

limh↓0(I+λA)⁻¹g−(I+λA_h)⁻¹π_hg

∞= 0, (5.1)

whereg∈W andλ >0.

Several remarks are in order.

Remark 5.1. The family of triangulation{Th} satisfying (H2) and (H3) is often calledquasi-uniform.