Semigroup formulation of Rothe’s method:

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Semigroup formulation of Rothe’s method:

application to parabolic problems

Mari´an Slodiˇcka

Abstract. A semilinear parabolic equation in a Banach space is considered. The purpose of this paper is to show the dependence of an error estimate for Rothe’s method on the regularity of initial data. The proofs are done using a semigroup theory and Taylor spectral representation.

Keywords: error estimates, parabolic equation, backward Euler method Classification: 65M15, 35K22, 65M20

1. Introduction.

The Rothe method (also called method of lines or backward Euler method) is well known as an efficient theoretical tool for solving a wide range of evolution problems.

Moreover it has a strong numerical aspect. The aim of this paper is to investigate the Rothe method from the point of view of the semigroup theory.

LetXbe a Banach space with the normk k. The operatorA is assumed to be sectorial in X (cf. [3, D. 1.3.1]) with the domain D(A), where Reσ(A)> δ₀ >0.

We denoteX_α =D(A^α) forα∈R. The norm inX_α is defined bykvk_α =kA^αvk.

The problem we are considering is the abstract semilinear evolution equation (t∈ h0, Ti, 0≤α <1, β ≥α)

(1.1) ∂tu(t) +Au(t) =f(t, u(t)) u(0) =v∈X_β. The right-hand sidef :R×X_α→Xsatisfies

(1.2)₁ kf(t, x)−f(s, y)k ≤C |t−s|(1 +kxkα+kykα) +kx−ykα . Further

(1.2)₂ kf(t, z)kα≤C 1 +kzk_2α for anyt, s∈ h0, Ti,x, y∈X_α andz∈X_2α.

It is easy to find thatA generates an analytic semigroup in Xand there exists a global solution of (1.1) which can be described as follows:

u(t) =T(t)v+ Z t

0

T(t−s)f(s, u(s))ds (1.3)

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where

T(t) = e^−At= (2πi)⁻¹ Z

Γ

e^λt(λ+A)⁻¹dλ

and Γ is a curve in ̺(−A) running from∞e⁻^iφ to ∞e^iφ for any φ ∈ (π/2, π).

Without loss of generality we can put

(1.4) λ∈Γ⇔λ=−δ−s cosϕ±is sinϕ fors∈ h0,∞),ϕ∈(0, π/2), δ=δ(δ₀)>0.

The backward Euler method is applied to the discretization in time (1.5) (u_i−u_i−1)τ⁻¹+Au_i=f(t_i, u_i−1)

u₀=v fori∈N,τ is a time step, t_i=iτ.

There exists a great number of papers devoted to the study of error estimates for the method of lines applied to (1.1) e.g. [4]–[6], [11], [13], [14], etc. The proof technique demonstrated there needs certain regularity assumptions of initial data (practicallyv∈X₁) in order to derive some error estimates. For a global Lipschitz continuous right-hand sidef andv∈X₁ one can prove

ku(t_i)−u_ik ≤C τ.

On the other hand, there exist many papers concerned with nonsmooth data error estimates e.g. [1], [2], [7]–[10], [12], [15], [16], [18], [19], etc. Ifβ =α= 0 it is known that

ku(t_i)−u_ik_γ≤C (i−γ)⁻¹+τ^1−γlnτ⁻¹

, 0≤γ <1.

We have to point out the fact that any of the articles mentioned above cannot say anything about the error estimate (independent of t) for the backward Euler method applied to (1.1) in the case whenβ > α >0 and β is sufficiently small, i.e.

if we consider the initial data of low regularity andf depends on spatial derivatives ofu.

The aim of this paper is to demonstrate the influence of the parametersα, β,γ on the rate of convergence. Section 2 is devoted to the study of linear homogeneous problem. However some of our results demonstrated there can be derived by a shorter argumentation; we have chosen this way in order to prepare the basic facts for nonhomogeneous case. The main result is formulated in Theorem 1. Section 3 deals with the nonlinear case when the right-hand sidef depends onu. There are given the most important results from the practical point of view (cf. Theorem 4).

In fact, considering a concrete problem and applying the Sobolev imbedding theorem one can arrive at the error estimate in suitable function space likeL_p,W_p^k,C (cf. Appendix).

Our proof-technique is very compact and enables us to deal with the initial data of various regularity (from nonsmooth to smooth) and establish the rate of convergence practicallyτ^{min(1,β−γ)} uniformly with respect tot.

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Sometimes in the literature the Lipschitz condition off is assumed only locally inXwhich generalizes the semilinear parabolic equations covered. But in that case if f = f(u), the solution u = u(x, t) must be uniformly bounded in Ω× h0, Ti.

Thusv∈L_∞(Ω), whereas we can deal with initial data e.g.v∈H^2β(Ω)∩H˚^β(Ω), v /∈L_∞(Ω) andβ is a suitable small positive number.

From practical reasons it is important to discuss spatial discretization, too. This will be the subject of a future investigation.

Remark 1. C denotes a generic positive constant independent ofτ.

2. Homogeneous problem.

Throughout this section we suppose f = 0. Thus according to (1.5) one can deduce

(2.1) u_i= (I+τ A)⁻ⁱv.

Using operational calculus (cf. [17,§5.6]) we can prolong our approximate solution u_i from time steps into the whole intervalh0, Tias follows:

(2.2) Tτ(t)v = (I+τ A)^−t/τv= (2πi)⁻¹ Z

Γ(1−τ λ)^−t/τ(λ+A)⁻¹v dλ where Γ is taken from (1.4).

Let us note that the integral in (2.2) is absolutely convergent for every posi- tivet, τ. In spite of this we suppose without loss of generalityτ < τ₀<1. T(t) is an analytic semigroup, and the following estimates hold:

kT(t)k ≤C, t≥0, (2.3)

kA^δT(t)k ≤C_δt^−δ, δ≥0, t >0, (2.4)

k(T(t)−I)xk ≤δ⁻¹C_1−δt^δ kA^δxk, x∈X_δ, 0< δ≤1, t≥0, (2.5)

kA^δxk ≤CkAxk^δ kxk^1−δ, x∈X₁, 0≤δ≤1.

(2.6)

T_τ(t)v, as an approximate solution of (1.1) forf = 0, was introduced in [15]. We know thatT_τ(t),t≥0, is a semigroup for which the smoothing effect takes place.

More exactly, we can write (cf. [16, L. 1]

Lemma 1. Letγ≥0and t, τ >0 such that t > γτ. ThenT_τ(t)x∈X_γ for every x∈X.

Using this fact, we know that both solutions (exact and approximate, f = 0) become smoother with increasing time. Hence we can try to establish an error estimate in the norm ofX_γ. It is easy to see that this must depend on the parameter β,γ and probably on the timet. First, we state or prove some lemmas which play an important part in our proofs.

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Lemma 2. Ifλ∈C,Reλ <0andt, τ >0then

|(1−τ λ)^−t/τ −e^λt| ≤ |λ|² |Reλ|⁻² |(1−τReλ)^−t/τ −e^Re^λt|.

Proof: See [15].

Lemma 3. Letb >0and I(x;a, b) =

Z ∞ x

e^−zz^−a−1 e^z(1 +b⁻¹z)^−b−1 dz.

Then:

(i) I(0;a, b)≤b^−a (1−a)⁻¹+a⁻¹

foro < a <1, (ii) I(0;a, b)≤b⁻¹ fora= 0,

(iii) I(ε;a, b)≤b⁻¹a⁻¹ε^1−aforε >0,a≥1, (iv) I(0;a, b)≤C(a)b^−a(b+a)⁻¹ for0> a >−b.

Proof: Let us denote I_ε,N =

Z N ε

e^−zz^−a−1 e^z(1 +b⁻¹z)^−b−1 dz.

forb >0,a∈R,N > ε >0. One can easily find (z >0)

∂z e^z(1 +b⁻¹z)^−b−1

=b⁻¹ze^z(1 +b⁻¹z)^−b−1,

∂z

Z z N

e^−ss^−a−1ds= e^−zz^−a−1. Thus using integration by parts we have

(2.7) I_ε,N =Z ε 0

b⁻¹ze^z(1 +b⁻¹z)^−b−1dz Z N

ε

e^−zz^−a−1dz + +

Z _N

ε

Z _N

z

e^−ss^−a−1ds

b⁻¹ze^z(1 +b⁻¹z)^−b−1dz.

We consider the case 0≤a <1, first. We estimate I_ε,N ≤

Z _N

0

Z _N

z

e^−ss^−a−1ds

b⁻¹ze^z(1 +b⁻¹z)^−b−1dz≤

≤ Z ∞

0

Z ∞ z

e^−ss^−a−1ds

b⁻¹ze^z(1 +b⁻¹z)^−b−1dz.

Applying

Z ∞ z

e^−ss^−a−1ds≤z^−a−1 Z ∞

z

e^−sds≤z^−a−1e^−z

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we obtain (2.8) I_ε,N ≤

Z ∞

0 b⁻¹z^−a(1 +b⁻¹z)^−b−1dz=b^−a Z ∞

0 w^−a(1 +w)^−b−1dw.

The last estimate yields fora= 0 I_ε,N ≤

Z ∞ 0

(1 +w)^−b−1dw=b⁻¹ from which, taking the limit asN → ∞andε→0, we prove (ii).

If 0< a <1, then after integration by parts in (2.8), we can write I_ε,N ≤b^−a(b+ 1)(1−a)⁻¹

Z _∞

0

w^1−a(1 +w)^−b−2dw≤b^−a (1−a)⁻¹+ (a+b)⁻¹ . Now, lettingN → ∞andε→0 in the last inequality, we conclude (i).

Let us considera≥1. Using (2.7) one can find I_ε,N ≤

Z ε 0

b⁻¹(1 +b⁻¹z)^−b−1dz εe^ε Z ∞

ε

e^−zz^−a−1dz+

+ Z _∞

ε

Z _∞

z

e^−ss^−a−1ds

b⁻¹ze^z(1 +b⁻¹z)^−b−1dz.

Applying

Z _∞

z

e^−ss^−a−1ds≤e^−z Z _∞

z

s^−a−1ds=a⁻¹z^−ae^−z we have

I_ε,N ≤ Z ε

0 b⁻¹(1 +b⁻¹z)^−b−1dz a⁻¹ε^1−a+ +

Z _∞

ε

a⁻¹z^1−ab⁻¹(1 +b⁻¹z)^−b−1dz≤b⁻¹a⁻¹ε^1−a, from which, taking the limit asN → ∞, we prove (iii).

So we have to prove (iv) to conclude the proof. Analogously as for 0 ≤a <1 one can find

(2.9) I_ε,N ≤ Z ∞

0

Z ∞ z

e^−ss^−a−1ds b⁻¹ze^z(1 +b⁻¹z)^−b−1dz.

Let us denote

(2.10) x= [x] +{x}, x∈R

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where [x]∈Z,{x} ∈ h0,1). Further we set

(x−1)_k=

k

Y

i=1

(x−i), k≥1 and (x−1)₀= 1.

Using integration by parts one can write Z ∞

z

e^−ss^−a−1ds=

= e^−zz⁻¹

[−a]−1

X

k=0

(−a−1)_kz^−a−k+ (−a−1)_[−a]

Z _∞

z

e^−ss^{−a}−1ds and taking into account

Z ∞ z

e^−ss^{−a}−1ds≤e^−zz^{−a}−1 we get

(2.11)

Z ∞ z

e^−ss^−a−1ds≤e^−zz⁻¹

[−a]

X

k=0

(−a−1)_kz^−a−k.

Here, (2.9) and (2.11) yield forb >−a >0

I_ε,N ≤

[−a]

X

k=0

(−a−1)_kb^−a−k Z ∞

0

w^−a−k(1 +w)^−b−1dw≤

≤

[−a]

X

k=0

(−a−1)_kb^−a−k Z _∞

0

(1 +w)^{−a−b−k−1}dw≤

[−a]

X

k=0

(−a−1)_kb^−a(a+b)⁻¹.

Hence we have proved (b >−a >0)

I_ε,N ≤C(a)b^−a(a+b)⁻¹

and lettingε→0,N → ∞we conclude (iv).

Now we are ready to derive the main result at this paragraph.

Theorem 1. LetAbe a sectorial operator in a Banach spaceXwhereReσ(A)>

δ0 >0. Then(ν >0)

(i) kT_τ(t)−T(t)k ≤C τ t⁻¹ fort >0, (ii) kTτ(t)−T(t)k−ν ≤C τ^min(1,ν) fort >0, (iii) kT_τ(t)−T(t)k_ν ≤C τ^1−ν(t−ντ)⁻¹ fort > ντ.

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Proof: (ii) Using the spectral representation ofT_τ(t),T(t) and (2.12) A(λ+A)⁻¹=I−λ(λ+A)⁻¹, λ∈Γ, one can write

A^−ν T_τ(t)−T(t)

=

= (−1)^[ν]+1(2πi)⁻¹ Z

Γ

(1−τ λ)^−t/τ −e^λt

λ^−[ν]−1A^1−{ν}(λ+A)⁻¹dλ where [ ],{ }are defined by (2.10). Thus taking into account

(2.13) kA^ξ(λ+A)⁻¹k ≤C |λ|^ξ−1 λ∈Γ, 0≤ξ≤1 and Lemma 2, we have

kT_τ(t)−T(t)k_−ν ≤C Z

Γ

(1−τ λ)^−t/τ −e^λt

|λ|^−ν−1|dλ| ≤

≤C t^ν Z _∞

δt

e^−zz^−ν−1 e^z(1 +τ t⁻¹z)^−t/τ −1 dz.

Puttinga=ν andb=tτ⁻¹, in virtue of Lemma 3, we conclude (ii).

The assertions (i), (iii) can be proved in the same way.

3. Nonhomogeneous problem.

Throughout this paragraph we suppose that the function f satisfies (1.2) i.e.

(1.2)₁ and (1.2)₂. Considering the discretization scheme (1.5) one can easily find

(3.1) u_i=Tτ(t_i)v+

i−1

X

k=0

Tτ(t_i−t_k)f(t_k+1, u_k)τ.

Applying the semigroup theory we know that if

(3.2) 0≤α <1, α≤β <1

then there exists a global unique solution u(t) of (1.1) defined by (1.3).

Let us prove some discrete analogs of Gronwall’s lemma, first.

Lemma 4. Let {A_i}, {a_i} be sequences of nonnegative real numbers satisfying (q≥0)

(3.3) a_i≤A_i+

i−1

X

j=1

a_jq, i∈N.

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Then

a_i≤A_i+ e^qi

i−1

X

j=1

A_jq, i∈N.

Proof: Let us fixi≥2. We define

S(n, i) =

i−1

X

j=1 j−1

X

k1=1 k1−1

X

k2=1

· · ·

kn−1−1

X

kn=1

A_k_n for 0< n≤i−2,

S(n, i) =

i−1

X

j=1

A_j for n= 0, S(n, i) = 0 for n > i−2.

Iterating (3.3) (i−1)-times one can find

(3.4) a_i≤A_i+q

i−1

X

k=0

q^kS(k, i).

Using mathematical induction with respect ton ≥0 and taking into account the fact that ^x_y

= 0 forx < yone can prove

(3.5) S(n, i) =

i−1

X

j=1

A_j

i−1−j n

, n≥0.

Hence, (3.4) and (3.5) yield

a_i≤A_i+q

i−1

X

j=1

A_jn 1 +

i−j−1

X

k=1

q^k

i−1−j k

o

=

=A_i+q

i−1

X

j=1

A_jn 1 +

i−j−1

X

k=1

q^k(i−j)^k k!

k

Y

l=1

1− l i−j

o≤

≤A_i+q

i−1

X

j=1

A_je^q(i−j)≤A_i+ e^qi

i−1

X

j=1

A_jq.

Lemma 5. Let{An},{wn}be sequences of nonnegative real numbers satisfying

(3.6) wn≤An+C

n−1

X

k=1

(tn−t_k)^a−1w_kτ

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for0< τ <1,0< a≤1,C >0,t_n=nτ≤T. Then w_n≤C

A_n+

n−1

X

k=1

A_kτ+

n−1

X

k=1

(t_n−t_k)^a−1A_kτ

, C=C(a, T).

Proof: Let us put m= min{n∈N; na≥1}. Iterating (3.6) once, we have wn≤C

An+

n−1

X

k=1

(tn−t_k)^a−1A_kτ+

n−1

X

k=1 k−1

X

l=1

(tn−t_k)^a−1(t_k−t_l)^a−1w_kτ²

≤

≤C An+

n−1

X

k=1

(tn−t_k)^a−1A_kτ+

n−1

X

k=1

(tn−t_k)^2a−1w_kτ . Thus, after (m−1) iterations we obtain

w_n≤C A_n+

n−1

X

k=1

(t_n−t_k)^a−1A_kτ+

n−1

X

k=1

w_kτ .

The rest of the proof follows from the last inequality and Lemma 4.

As is standard practice, we need a priori estimates for exact and approximate solutions.

δ0 >0. Suppose(1.2), (3.2). Then for0≤γ≤β we have (i) ku(t)kγ≤C for t >0,

(ii) ku_ikγ≤C for i∈N, (iii) ku(t)k_2α≤C t^(β−2α)⁻ for t >0,

where u(t), u_i are defined by (1.3), (3.1), respectively, and (x)− = min(0, x) for x∈R.

Proof: (i) Using (1.3) one can write

(3.7) ku(t)k_γ≤C

1 + Z t

0

(t−s)^−γku(s)k_αds .

Setting γ =α in (3.7) and using Gronwall’s lemma, we getku(t)kα ≤C. Hence (3.7) yields (i).

(ii) Let us note that

kT_τ(t_i)k_γ≤ kT(t_i)k_γ+kT_τ(t_i)−T(t_i)k_γ≤C t^−γ_i +τ^−γ(i−γ)⁻¹

≤C t^−γ_i . The rest can be proved analogously as (i).

(iii) In view of (1.3) we can write A^2αu(t) =A^2α−βT(t)A^βv+

Z t 0

A^αT(t−s)A^αf(s, u(s))ds.

Hence

ku(t)k_2α≤C t^(β−2α)⁻+C Z _t

0

(t−s)^−α 1 +ku(s)k_2α ds.

Now, applying Gronwall’s lemma we conclude the proof.

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δ₀ >0. Suppose(1.2), (3.2). Then for0≤γ < β we have

ku(t+h)−u(t)kγ≤C h^β−γ, 0≤t < t+h≤T.

Proof: We can write

(3.8) u(t+h)−u(t) = T(h)−I T(t)v+

Z _t+h

t

T(t+h−s)f(s, u(s))ds+

+ Z t

0

T(h)−I

T(t−s)f(s, u(s))ds=I₁+I₂+I₃. Now we estimate all the addends in (3.8).

kI₁k_γ≤ k T(h)−I

A^γ−βT(t)A^βvk ≤C h^β−γ. For the second item we have

kI₂k_γ≤ Z t+h

t

kA^γT(t+h−s)k kf(s, u(s))kds≤

≤C Z _t+h

t

(t+h−s)^−γds≤C h^β−γ. Estimating the last term we find

kI3kγ≤C h^β−γ Z t

0 kA^γT(t−s)f(s, u(s))kds≤C h^β−γ,

which concludes the proof.

We are ready now to give the estimate of (u(t_i)−u_i) in the norm ofX_γ. It is very important for practical reasons to have a positive rate of convergence uniformly for everyi∈N.

δ0 >0. Suppose(1.2), (3.2). Then fori∈Nand0≤γ < β we have ku(t_i)−u_ik_γ≤C τ^β−γ.

Proof: (i) Let us rewrite (u(t_i)−u_i) into the following form

(3.9) u(t_i)−u_i=

7

X

j=1

I_j

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where

I₁ = T(t_i)−Tτ(t_i) v I₂ =

i−1

X

k=1

Tτ(t_i−t_k) f(t_k+1, u(t_k))−f(t_k+1, u_k) τ,

I3 =

i−1

X

k=1

T(t_i−t_k)−Tτ(t_i−t_k)

f(t_k+1, u(t_k))τ,

I₄ =

i−1

X

k=1

Z _t_k+1

tk

T(t_i−s) f(s, u(s))−f(t_k+1, u(s)) ds,

I5 =

i−1

X

k=1

Z tk+1

tk

T(t_i−s) f(t_k+1, u(s))−f(t_k+1, u(t_k)) ds,

I₆ =

i−1

X

k=1

Z _t_k+1

tk

T(t_i−s)−T(t_i−t_k)

f(t_k+1, u(t_k))ds, I₇ =

Z τ 0

T(t_i−s)f(s, u(s))ds−T_τ(t_i)f(τ, v)τ.

Let us start estimating all the items in (3.9). Using Theorem 1, we have kI₁kγ=kA^γ−β T(t_i)−Tτ(t_i)

A^βvk ≤C τ^β−γ, (3.10)₁

kI₁kα=k T(t_i)−Tτ(t_i)

A^αvk ≤C τ t⁻¹_i . (3.10)₂

The second term can be estimated as follows:

kI₂kγ≤

i−1

X

k=1

kA^γTτ(t_i−t_k) f(t_k+1, u(t_k))−f(t_k+1, u_k) kτ.

Thus

(3.11) kI₂k_γ≤C

i−1

X

k=1

(t_i−t_k)^−γku(t_k)−u_kk_ατ.

Further

A^γI₃=

i−1

X

k=1

A^γ T(t_i−t_k)−Tτ(t_i−t_k)

f(t_k+1, u(t_k))τ.

From this, applying Theorem 1, Theorem 2 we have (3.12)₁ kI₃kγ≤C τ^1−γ

i−1

X

k=1

(t_i−t_k−γτ)⁻¹τ≤C τ^β−γ.

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Analogously

A^αI3 =

i−1

X

k=1

T(t_i−t_k)−Tτ(t_i−t_k)

A^αf(t_k+1, u(t_k))τ.

Hence

(3.12)₂ kI₃kα≤C τ

i−1

X

k=1

(t_i−t_k)⁻¹ 1 +ku(t_k)k_2α τ≤

≤C τlnτ⁻¹ 1 +t^(β−2α)_i ⁻ . For the forth item, one can write

A^γI4 =

i−1

X

k=1

Z tk+1

tk

A^γT(t_i−s) f(s, u(s))−f(t_k+1, u(s)) ds,

kI₄kγ≤C τ Z ti

0

(t_i−s)^−γds≤C τ.

(3.13)

We rewriteI₅ into the following form

A^γI₅ =

i−1

X

k=1

Z tk+1

tk

A^γT(t_i−s) f(t_k+1, u(s))−f(t_k+1, u(t_k)) ds.

So, applying [3, T. 3.5.2], we obtain

kI₅k_γ≤C

i−1

X

k=1

Z tk+1

tk

(t_i−s)^−γku(s)−u(t_k)k_αds≤

≤C

i−1

X

k=1

Z tk+1

tk

(t_i−s)^−γ Z s

tk

k∂_ξu(ξ)kαdξ ds≤C τ Z ti

τ

(t_i−s)^−γs⁻¹ds.

Hence

(3.14) kI5kγ≤C τlnτ⁻¹t^−γ_i . The sixth term is estimated as follows:

A^γI₆ =

i−1

X

k=1

Z tk+1

tk

I−T(s−t_k)

A^γT(t_i−s)f(t_k+1, u(t_k))ds,

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thus

(3.15)₁ kI₆k_γ≤C τ^β−γ

i−1

X

k=1

Z _t_k+1

tk

(t_i−s)^−βkf(t_k+1, u(t_k))kds≤C τ^β−γ.

On the other hand A^αI₆ =

i−1

X

k=1

Z _t_k+1

tk

I−T(s−t_k)

T(t_i−s)A^αf(t_k+1, u(t_k))ds,

hence

(3.15)₂ kI6kα≤C τ

i−1

X

k=1

Z tk+1

tk

(t_i−s)⁻¹ 1 +ku(t_k)k2α ds≤

≤C τlnτ⁻¹ 1 +t^(β−2α)_i ⁻ . For the last addendum in (3.9), one can easily find

(3.16) kI7kγ≤C τ t^−γ_i .

Collecting all the results from (3.9), (3.10)₂, (3.11), (3.12)₂, (3.13), (3.14), (3.15)₂ and (3.16) forγ=αwe obtain

ku(t_i)−u_ik_α≤C τ+τlnτ⁻¹+τ t⁻¹_i +τlnτ⁻¹t^−ν_i +

i−1

X

k=1

(t_i−t_k)^−αku(t_k)−u_kk_ατ

for any 0≤ν <1. Thus Lemma 5 implies

(3.17) ku(t_i)−u_ikα≤C τ+τlnτ⁻¹+τ t⁻¹_i +τlnτ⁻¹t^−ξ_i for any 0≤ξ <1.

Now using (3.9), (3.10)₁, (3.11), (3.12)₁, (3.13), (3.14), (3.15)₁ and (3.16) we get

(3.18) ku(t_i)−u_ikγ≤C τ^β−γ+

i−1

X

k=1

(t_i−t_k)^−γku(t_k)−u_kkατ .

At the end, inserting (3.17) into (3.18) we conclude the proof.

4. Appendix.

In this appendix we present some ideas connected with the previous sections. At the end we give some simple examples in order to demonstrate our results.

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Remark 2. The restrictionβ <1 in (3.2) is probably due to our proof-technique.

We have not been able to remove it in general but it can be a good pastime to do

so.

Remark 3. The error estimate at time steps described in Theorem 4 can be easily prolonged to the whole intervalh0, Ti. In fact, if we define

un(t) =u_i−1+ (t−t_i−1)τ⁻¹(u_i−u_i−1) t∈ ht_i−1, t_i), i∈N, then fort∈ ht_i−1, t_i)

u(t)−u_n(t) =u(t)−u(t_i−1) +u(t_i−1)−u_i−1+

+ (t−t_i−1)τ⁻¹ u_i−u(t_i) +u(t_i)−u(t_i−1) +u(t_i−1)−u_i−1 . Now, applying Theorems 3 and 4 we can estimateku(t)−un(t)kγ.

There arise the following questions in many applications: “How is the minimal regularity of initial data, if we want to derive the error estimate independent oftin some functional space?” “How is the rate of convergence?” The answer, in many concrete cases, can be found using our results and the Sobolev imbedding theorem.

In the following examples, let us suppose that Ω ⊂ R^N is a bounded domain with sufficiently smooth∂Ω. Let us denote

X_δ=W^2δ,p(Ω)∩W˚^δ,p(Ω)

forδ≥0,p >1,N ≥1,X=X₀ =L_p(Ω). The norm inX_δ is equivalent to the one inW^2δ,p(Ω) where

kwk^p_Wk+s,p(Ω)=kwk^p_Wk,p(Ω)+ X

|α|=k

Z

Ω

Z

Ω|D^αw(x)−D^αw(y)|^p|x−y|^−n−psdx dy fork∈N, 0< s <1. Further we denote

B=

N

X

i=1

a_i∂_x_i, a_i∈R. Example 1.

du(t)

dt −△u(t) =f(t, u(t)) in Ω u= 0 on ∂Ω,

u(0) =v∈X_β, 0< β <1,

wheref as a real function is global Lipschitz continuous in all variables. Then ku(t_i)−u_ik_L_p_(Ω)≤C τ^β.

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Example 2.

du

dt−△u= sin(Bu) in Ω u= 0 on ∂Ω, u(0) =v∈X_β. If 1/2≤β <1, 0≤γ < β then

ku(t_i)−u_ik_W^2γ,p_(Ω)∼=ku(t_i)−u_ikγ≤C τ^β−γ. IfN/(2p)< γ < β <1, 1/2≤β then

ku(t_i)−u_ik_L∞(Ω)≤C ku(t_i)−u_ikγ≤C τ^β−γ.

These results cannot be obtained by the classical proof technique in Rothe’s method because of the low regularity of the initial datav (cf. [4], [6]).

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[16] ,Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data, Comment. Math. Univ. Carolinae32(1991), 703–713.

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Institute of Applied Mathematics, Comenius University, Mlynsk´a dolina, 842 15 Bratislava, Czechoslovakia

(Received October 11, 1991)