### On a parabolic problem with nonlinear Newton boundary conditions

Miloslav Feistauer, Karel Najzar, Karel ˇSvadlenka

Abstract. The paper is concerned with the study of a parabolic initial-boundary value problem with nonlinear Newton boundary condition considered in a two-dimensional domain. The goal is to prove the existence and uniqueness of a weak solution to the problem in the case when the nonlinearity in the Newton boundary condition does not satisfy any monotonicity condition and to analyze the finite element approximation.

Keywords: parabolic convection-diffusion equation, nonlinear Newton boundary condi- tion, Galerkin method, compactness method, finite element approximation, error esti- mates

Classification: 35K60, 65N30, 65N15

Introduction

A number of problems of technology and science are described by partial dif- ferential equations equipped with nonlinear Newton boundary conditions. Let us mention, e.g. radiation and heat transfer problems ([2], [22], [27]), modeling of electrolysis of aluminium with turbulent flow at the boundary ([11], [29]) and some problems of elasticity ([18]). Our paper was inspired by some nonstandard applications in biology, where the nutrition of kernels of plants can be described by a parabolic partial differential equation equipped by mixed Dirichlet - nonlinear Newton boundary conditions (see, e.g., [1], [7]).

Up to now, elliptic problems equipped with Newton nonlinear boundary con- ditions have been analyzed analytically as well as numerically. In the analysis of these problems one meets a number of obstacles, particularly in the very topical case when the nonlinearity is unbounded and has a polynomial behaviour. The first results for a problem of this type were obtained in [11], where the existence and uniqueness of the solution of the continuous problem was proved with the aid of the monotone operator theory and the convergence of the approximate solutions to the exact one was established under the assumption that all integrals appearing in the discrete problem were evaluated exactly. In [12], the convergence of the finite element method was proved in the case that both volume and boundary integrals were calculated with the aid of quadrature formulae. In the analysis of the boundary terms it was not possible to apply the well-known Ciarlet–Raviart

theory ([5], [6]) of the finite element numerical integration because of the nonlin-
earity on the boundary. The convergence analysis was obtained with the aid of a
suitable modification of results from [33]. Furthermore, the work [13] is concerned
with the derivation of error estimates. They were obtained thanks to the uniform
monotonicity of the problem in [13]. However, in contrast to standard nonlinear
situations treated, e.g., in [4], [16], [17], [34], where strong monotonicity was used,
an optimalO(h) error estimate for linear finite elements was not achieved. The
order of convergence is reduced due to the fact that only uniform monotonicity
with growth of degree t^{2+α}, α > 0, holds now, and due to the nonlinearity in
the boundary integrals. Moreover, also the application of numerical integration
in the nonlinear boundary integral can lead to a further reduction of the rate of
convergence. The theoretically established decrease of the order of convergence
caused by the nonlinearity in the Newton boundary condition was confirmed with
the aid of numerical experiments in [15]. Finally, in [14], the effect of the approxi-
mation of a curved boundary is analyzed with the aid of Zl´amal’s concepts of ideal
triangulation and ideal interpolation ([35]). Let us also mention that another ap-
proach was used in [19] and [20], where the problem for the Laplace equation with
nonlinear Newton boundary condition was transformed to a nonlinear boundary
integral equation.

Practical applications often require the solution of nonstationary transient problems with Newton boundary conditions. In this paper we shall be concerned with the analysis of nonstationary convection-diffusion problem equipped with mixed Dirichlet - nonlinear Newton boundary conditions. In Section 1, the con- tinuous problem is formulated. The concept of a weak solution is introduced and some auxiliary results are established. We assume that the nonlinearity in the Newton boundary condition has a linear growth and is Lipschitz-continuous.

Section 2 is devoted to the proof of the existence and uniqueness of the weak solu- tion. In Section 3, under the assumption that the space domain is polygonal, the finite element solution is analyzed and error estimates for the semidiscretization in space are obtained.

1. Formulation of the problem

1.1 Function spaces and classical formulation.

Let Ω⊂R^{2} be a bounded domain with a Lipschitz-continuous boundary Γ =

∂Ω consisting of three parts Γ_{1}, Γ_{2}, Γ_{3}, see Figure 1. By Ω we denote the closure
of Ω. ForT > 0 let us denote by Q_{T} the space-time cylinder Ω×(0, T). Letn
denote the unit outer normal to Γ. ByNwe denote the set of all positive integers.

We introduce the following notation of function spaces:

C(Ω) — space of functions continuous in Ω;

C^{k}(Ω), k∈N— space of functions having continuous derivatives of order
kin Ω;

Figure 1: Computational domain

C_{0}^{∞}(0, T) — space of infinitely differentiable functions with compact support
in (0, T);

L^{p}(Ω), 1≤p <∞— space of measurable functions whosepth power is Lebesgue
integrable over Ω, equipped with the norm

(1.1) kukL^{p}(Ω)=

Z

Ω|u|^{p}dx
1/p

;

L^{2}_{α}(Ω), where α ∈ C(Ω), α1 ≥ α≥ α0 >0, α0, α1 = const, is the α-weighted
L^{2}-space which is a Hilbert space with the scalar product

(1.2) (u, v)α=

Z

Ω

α(x)u(x)v(x)dx;

W^{k,p}(Ω), 1≤p <∞— Sobolev space of functions fromL^{p}(Ω) whose distribution
derivatives of order≤kare elements ofL^{p}(Ω), equipped with the norm

(1.3) kukW^{k,p}(Ω)= X

|α|≤k

kD^{α}uk^{p}_{L}^{p}_{(Ω)}
1/p

,

where D^{α}u = ^{∂}^{α}^{1+}^{α}^{2}^{u}

∂x^{α}_{1}^{1}∂x^{α}_{2}^{2}, and α = (α_{1}, α_{2}), |α| = α_{1}+α_{2}. We set H^{k}(Ω) =
W^{k,2}(Ω). InH^{1}(Ω) we shall also work with the seminorm

(1.4) |u|H^{1}(Ω)=

Z

Ω|∇u|^{2}dx
1/2

;
H^{µ}(Ω), µ∈(^{1}_{2},1) — space of functionsu∈L^{2}(Ω), for which

(1.5)

I(u) =R

Ω

R

Ω

|u(x)−u(y)|^{2}

kx−yk^{2(1+µ)} dxdy1/2

<∞, with the norm
kukH^{µ}(Ω)=

kuk^{2}_{L}^{2}_{(Ω)}+I^{2}(u)1/2

.

Further, we shall introduce the Bochner spaces. Let X be a Banach space.

Then we define: C([0, T];X) — space of functions u: [0, T]→X, continuous, for which

(1.6) kukC([0,T];X)= sup

t∈[0,T]ku(t)kX <∞;

C^{1}([0, T];X) — space of functions u: [0, T] →X continuously differentiable in
[0, T];

L^{p}(0, T;X),1≤p <∞— space of functionsu: (0, T)→X, strongly measurable
such that

(1.7) kukL^{p}(0,T;X)=
Z _{T}

0 kuk^{p}_{X}dt
1/p

<∞.
Forp= 2,X=L^{2}(Ω) we haveL^{2}(0, T;L^{2}(Ω))≡L^{2}(Q_{T});

L^{∞}(0, T;L^{2}(Ω)) — space of functionsu: (0, T)→L^{2}(Ω) such that
(1.8) kukL^{∞}(0,T;L^{2}(Ω))= ess sup

t∈(0,T)ku(t)kL^{2}(Ω)<∞.

It is known thatL^{p}(Ω), 1< p <∞,H^{µ}(Ω),µ∈(^{1}_{2},1],L^{p}(0, T;X), 1< p <∞,
are reflexive Banach spaces.

In virtue of the Sobolev imbedding theorems,

H^{s}(Ω)֒→֒→H^{r}(Ω), r, s∈N, 0≤r < s,
(1.9)

(compact imbedding)
H^{1}(Ω) ֒→ L^{q}(Ω), q∈[1,∞).

(1.10)

(continuous imbedding)

By [26] (in the case of domains with infinitely smooth boundary) and [8] (for Lipschitz-continuous boundary),

(1.11) H^{µ}(Ω)֒→֒→H^{µ−ε}(Ω), if µ≥ε >0.

Hence, forµ∈(0,1) we have

(1.12) H^{1}(Ω)֒→֒→H^{µ}(Ω)֒→֒→H^{0}(Ω) =L^{2}(Ω).

Due to the theorem on traces (see [26] for domains with infinitely smooth bound-
aries and [23] for Lipschitz-continuous boundaries), for everyµ∈(^{1}_{2},1] the trace
mapping θ : H^{µ}(Ω) → L^{2}(∂Ω), is continuous. This means that there exists
C_{Tr}(µ)>0 such that

(1.13) kukL^{2}(∂Ω)≤C_{Tr}(µ)kukH^{µ}(Ω), u∈H^{µ}(Ω).

We setC_{Tr}=C_{Tr}(1).

Now we introduce the followinginitial-boundary value problem: Find a function
u=u(x, t) defined inQ_{T} such that

(1.14) α(x)∂u(x, t)

∂t = div (β(x)∇u(x, t) +v(x)u(x, t)) +q(x) in Q_{T},
β(x)∂u(x, t)

∂n +γ(x)u(x, t) =G(x, u(x, t)) on Γ1, (1.15)

u(x, t) = 0 on Γ_{2},
(1.16)

β(x)∂u(x, t)

∂n = 0 on Γ_{3},

(1.17)

u(x,0) =u^{0}(x), x∈Ω.

(1.18)

Let us assume that the functions from (1.14)–(1.18) have the following proper- ties:

α∈C(Ω), α_{1} ≥α≥α_{0}>0, α_{0}, α_{1} = const,
(1.19)

β ∈C^{1}(Ω), β_{1} ≥β ≥β_{0}>0, β_{0}, β_{1}= const,
(1.20)

v∈h

C^{1}(Ω)i2

, (1.21)

γ∈C(Γ_{1}), |γ| ≤γ_{1} = const,
(1.22)

q∈L^{2}(Ω),
(1.23)

u_{0}∈L^{2}(Ω).

(1.24)

Moreover, G: Γ_{1}×R→ R, G =G(x, u), is continuous, has a linear growth
and is Lipschitz-continuous with respect tou. This means that there existg≥0,
g∈L^{2}(Γ_{1}),K≥0,L_{G}, such that

|G(x, u)| ≤g(x) +K|u|, ∀x∈Γ1, ∀u∈R, (1.25)

|G(x, u)−G(x, u^{∗})| ≤L_{G}|u−u^{∗}|, ∀x∈Γ_{1}, ∀u, u^{∗}∈R.
(1.26)

(Let us note that (1.25) follows from (1.26).)

Classical solution of the above initial-boundary value problem is a function
u ∈ C^{2}(Q_{T}) satisfying equation (1.14), boundary conditions (1.15)–(1.17) and
initial condition (1.18) pointwise.

In the analysis of this problem it will be necessary to work with a number
of various constants. Constants with fixed meaning in the whole paper will be
denoted by symbolsK, L_{G},γ1,α0,α1,β0,C_{Tr}, C_{F}, C0, C1,. . . . On the other
hand, byC we shall denote a generic constant having, in general, different values
in different places.

1.2 Weak solution.

In order to define the concept of a weak solution, we introduce the space of test functions

(1.27) V ={ϕ∈H^{1}(Ω); ϕ|Γ2 = 0}.

Let us remind that the well-known Friedrichs inequality holds in this space: there
exists a constantC_{F} >0 such that

(1.28) kϕkH^{1}(Ω)≤C_{F}|ϕ|H^{1}(Ω) ∀ϕ∈V.

This implies that| · |H^{1}(Ω) is a norm onV equivalent with the normk · kH^{1}(Ω).
The norm| · |H^{1}(Ω) onV is induced by the scalar product ((·,·))_{V} defined by

(1.29) ((u, v))_{V} =

Z

Ω∇u· ∇v dx.

The weak formulation is derived in a standard way: We assume that uis a classical solution, multiply equation (1.14) by an arbitrary test function ϕ∈V, integrate over Ω and use Green’s theorem and conditions (1.15)–(1.17). We obtain the relation

Z

Ωα∂u

∂tϕ dx= Z

Ωdiv [β∇u+vu]ϕ dx+ Z

Ωqϕ dx

= Z

∂Ω[β∇u+vu]·nϕ dS− Z

Ω[β∇u+vu]· ∇ϕ dx+ Z

Ωqϕ dx

= Z

Γ1

[G(x, u)−γu+v·nu]ϕ dS+ Z

Γ3

v·nuϕ dS

− Z

Ω

β∇u· ∇ϕ dx− Z

∂Ω

v·nuϕ dS+ Z

Ω

div(vu)ϕ dx+ Z

Ω

qϕ dx.

Hence,

(1.30)

Z

Ω

α∂u

∂tϕ dx+ Z

Ω

[β∇u· ∇ϕ−div(vu)ϕ]dx

= Z

Γ1

[G(x, u)−γu]ϕ dS+ Z

Ω

qϕ dx.

Let us introduce the following notation:

(u, ϕ)_{α}=
Z

Ω

αuϕ dx, (1.31)

a_{0}(u, ϕ) =
Z

Ω

β∇u· ∇ϕ dx, (1.32)

a_{1}(u, ϕ) =−
Z

Ω

div(vu)ϕ dx, (1.33)

a(u, ϕ) =a_{0}(u, ϕ) +a_{1}(u, ϕ),
(1.34)

d_{0}(u, ϕ) =
Z

Γ1

G(x, u)ϕ dS, (1.35)

d_{1}(u, ϕ) =−
Z

Γ1

γuϕ dS, (1.36)

d(u, ϕ) =d_{0}(u, ϕ) +d_{1}(u, ϕ).

(1.37)

Then (1.30) can be written in the form

(1.38) d

dt(u(t), ϕ)_{α}+a(u(t), ϕ) =d(u(t), ϕ) + (q, ϕ) ∀ϕ∈V.

In what follows we prove several important properties of these forms.

Lemma 1. The formsa,dare defined foru, ϕ∈V. For these functions we have

|a_{0}(u, ϕ)| ≤C_{1}|u|H^{1}(Ω)|ϕ|H^{1}(Ω),
(1.39)

|a_{1}(u, ϕ)| ≤C_{2}|u|H^{1}(Ω)|ϕ|H^{1}(Ω),
(1.40)

|d_{0}(u, ϕ)| ≤C_{3}(1 +|u|H^{1}(Ω))|ϕ|H^{1}(Ω),
(1.41)

|d1(u, ϕ)| ≤C4|u|H^{1}(Ω)|ϕ|H^{1}(Ω),
(1.42)

a(u, u)≥β_{0}

2 |u|^{2}_{H}^{1}_{(Ω)}−C_{0}kuk^{2}_{L}^{2}_{(Ω)} for u∈V,
(1.43)

with constants C_{0}, . . . , C_{4} >0 independent of u, ϕand β_{0} >0 from (1.20). If,
moreover,

v·n≤0 on Γ1∪Γ3, (1.44)

divv≤0 in Ω, (1.45)

then

(1.46) a(u, u)≥β_{0}|u|^{2}_{H}^{1}_{(Ω)}, u∈V.

Proof: Using the Cauchy inequality, the theorem on traces, the Friedrichs in- equality and assumptions (1.20)–(1.25), we find that

|a_{0}(u, ϕ)| ≤
Z

Ω|β∇u· ∇ϕ|dx≤β_{1}
Z

Ω|∇u· ∇ϕ|dx

≤C1|u|H^{1}(Ω)|ϕ|H^{1}(Ω),

|a_{1}(u, ϕ)| ≤
Z

Ω|div(vu)ϕ|dx≤ Z

Ω|divvuϕ|dx+ Z

Ω|v· ∇u ϕ|dx

≤C^{v}(kukL^{2}(Ω)kϕkL^{2}(Ω)+|u|H^{1}(Ω)kϕkL^{2}(Ω))

≤C_{F}C^{v}(C_{F}+ 1)|u|H^{1}(Ω)|ϕ|H^{1}(Ω)=C_{2}|u|H^{1}(Ω)|ϕ|H^{1}(Ω),

|d_{0}(u, ϕ)| ≤
Z

Γ1

|G(u, x)ϕ|dS ≤ Z

Γ1

g|ϕ|dS+K Z

Γ1

|uϕ|dS

≤ kgkL^{2}(Γ1)kϕkL^{2}(∂Ω)+KkukL^{2}(∂Ω)kϕkL^{2}(∂Ω)

≤C_{Tr}C_{F}kgkL^{2}(Γ1)|ϕ|H^{1}(Ω)+KC_{Tr}^{2} C_{F}^{2}|ϕ|H^{1}(Ω)|u|H^{1}(Ω)

=C_{3}(1 +|u|H^{1}(Ω))|ϕ|H^{1}(Ω),

|d_{1}(u, ϕ)| ≤
Z

Γ1

|γuϕ|dS≤γ_{1}
Z

Γ1

|uϕ|dS≤γ_{1}kukL^{2}(∂Ω)kϕkL^{2}(∂Ω)

≤C_{4}|u|H^{1}(Ω)|ϕ|H^{1}(Ω),

where C^{v} = kvk_{C}^{1}_{(Ω)}, C_{Tr} is the constant from the theorem on traces (1.13)
andC_{F} is the constant from the Friedrichs inequality (1.28) andC_{1} =β_{1}, C_{2} =
C_{F}Cv(C_{F}+ 1),C_{3}=C_{Tr}C_{F}max{kgkL^{2}(Γ1), KC_{Tr}C_{F}},C_{4}=γ_{1}C_{F}^{2}C_{Tr}^{2} . Further,
we have

(1.47) a(u, u)≥β_{0}|u|^{2}_{H}^{1}_{(Ω)}−
Z

Ω

div(vu)u dx≥β_{0}

2 |u|^{2}_{H}^{1}_{(Ω)}−C_{0}kuk^{2}_{L}^{2}_{(Ω)},
withC0 =C^{v}(1 + 1/(2β0)), since

Z

Ω

div(vu)u dx≤ | Z

Ω

u^{2}divvdx|+|
Z

Ω

(v· ∇u)u dx|

≤Cv(kuk^{2}_{L}^{2}_{(Ω)}+
Z

Ω|∇u| |u|dx)

≤C^{v}(kuk^{2}_{L}^{2}_{(Ω)}+|u|H^{1}(Ω)kukL^{2}(Ω))

≤ β_{0}

2 |u|^{2}_{H}^{1}_{(Ω)}+C_{0}kuk^{2}_{L}^{2}_{(Ω)},

where we have used Young’s inequality:

(1.48) ab≤1

2εa^{2}+ 1

2εb^{2}, a, b≥0, ε >0
withε=β_{0}.

Under assumptions (1.44), (1.45), we obtain
(1.49) a(u, u)≥β_{0}|u|^{2}_{H}^{1}_{(Ω)}−

Z

Ω

div(vu)u dx≥β_{0}|u|^{2}_{H}^{1}_{(Ω)},
since

Z

Ω

div(vu)u dx= Z

∂Ω

v·nu^{2}dS−
Z

Ω

v· ∇u u dx

= Z

Γ1∪Γ3

v·nu^{2}dS−1
2

Z

∂Ω

v·nu^{2}dS−
Z

Ω

u^{2}divvdx

=1 2

Z

Γ1∪Γ3

v·nu^{2}dS+1
2

Z

Ωu^{2}divvdx≤0.

Definition of a weak solution. We say that uis a weak solution of problem (1.14)–(1.18), if

(a)u∈L^{2}(0, T;V)∩L^{∞} 0, T;L^{2}(Ω)
,

(b)usatisfies identity (1.38) for allϕ∈V in the sense of distributions on (0, T), i.e.,

(1.50)

−
Z _{T}

0

(u(t), ϕ)_{α}ϑ^{′}(t)dt+
Z _{T}

0

a(u(t), ϕ)ϑ(t)dt=
Z _{T}

0

d(u(t), ϕ)ϑ(t)dt +

Z T

0 (q, ϕ)ϑ(t)dt ∀ϑ∈C_{0}^{∞}(0, T), ∀ϕ∈V,
and,

(c)usatisfies the initial condition

(1.51) u(0) =u^{0}.

2. Existence of a weak solution

The goal of the next two paragraphs will be the proof of the existence and uniqueness of a weak solution of the problem (1.14)–(1.18):

Theorem 1. There exists a unique solution of problem(1.50),(1.51).

2.1 Proof of existence.

The existence of a weak solution will be proved with the aid of the Galerkin
method. The spaceV is separable and, hence, there exists its basis{w_{k}}^{∞}k=1such
that

(2.1) V =

[∞ k=1

X_{k}

H^{1}(Ω)

, where X_{k}= span{w_{1}, . . . , w_{k}}.

Let us define the Galerkin approximationu_{k} ∈C^{1}([0, T], X_{k}) which satisfies
the conditions

d

dt(u_{k}(t), w_{i})_{α}+a(u_{k}(t), w_{i}) =d(u_{k}(t), w_{i}) + (q, w_{i}),
(2.2)

i= 1, . . . , k, t∈(0, T),
u_{k}(0) =u^{0}_{k}=P_{k}u^{0}.

(2.3)

Here, the mappingP_{k} is theL^{2}-projection onX_{k}, i.e., for u∈L^{2}(Ω), we define
P_{k}u∈X_{k} so that

(2.4) (P_{k}u, ϕ) = (u, ϕ) ∀ϕ∈X_{k}.

In the sequel, we shall employ the following inequality from [3]:

(2.5) kwk^{2}_{L}^{2}_{(∂Ω)}≤C5kwkL^{2}(Ω)|w|H^{1}(Ω) ∀w∈V,
called the multiplicative trace inequality.

Now we derive a priori estimates of approximate solutionsu_{k}.

Lemma 2. There exists a constantC >0such that each solutionu_{k}of problem
(2.2),(2.3)satisfies the estimates

ku_{k}kL^{∞}(0,T;L^{2}(Ω))≤C,
(2.6)

ku_{k}kL^{2}(0,T;V)≤C ∀k= 1,2, . . . .
(2.7)

Proof: Conditions (2.2) can be written as

(2.8) (u^{′}_{k}(t), ϕ)_{α}+a(u_{k}(t), ϕ) =d(u_{k}(t), ϕ) + (q, ϕ) ∀ϕ∈X_{k}, t∈(0, T),
where we simply write u^{′}_{k} instead of ^{∂u}_{∂t}^{k}. Substituting u_{k}(t) forϕ in (2.8), we
obtain

(2.9) Z

Ω

αu^{′}_{k}(t)u_{k}(t)dx+a(u_{k}(t), u_{k}(t)) =d(u_{k}(t), u_{k}(t)) +
Z

Ω

qu_{k}(t)dx.

In virtue of Lemma 1, (2.5), the linear growth of the functionG, the Cauchy inequality and Young’s inequality, we find that

1 2

Z

Ω

α(u^{2}_{k})^{′}(t)dx+β0

2 |u_{k}(t)|^{2}_{H}^{1}_{(Ω)}−C_{0}ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}

≤ | Z

Γ1

G(x, u_{k}(t))u_{k}(t)dS−
Z

Γ1

γu^{2}_{k}(t)dS+
Z

Ω

qu_{k}(t)dx|

≤ | Z

Γ1

gu_{k}(t)dS|+K
Z

Γ1

u^{2}_{k}(t)dS+kqkL^{2}(Ω)ku_{k}(t)kL^{2}(Ω)+
Z

Γ1

|γ|u^{2}_{k}(t)dS

≤1

2kgk^{2}_{L}^{2}_{(Γ}_{1}_{)}+
1

2 +K

ku_{k}(t)k^{2}_{L}^{2}_{(Γ}_{1}_{)}+γ_{1}ku_{k}(t)k^{2}_{L}^{2}_{(Γ}_{1}_{)}
+kqkL^{2}(Ω)ku_{k}(t)kL^{2}(Ω)

≤ 1

2 +K+γ_{1}

ku_{k}(t)k^{2}_{L}^{2}_{(Γ}_{1}_{)}+1

2ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}+1

2kqk^{2}_{L}^{2}_{(Ω)}+1

2kgk^{2}_{L}^{2}_{(Γ}_{1}_{)}

≤C_{6}ku_{k}(t)kL^{2}(Ω)|u_{k}(t)|H^{1}(Ω)+1

2ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}+C_{7}

≤ 1

2 +C_{6}^{2}
β_{0}

ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}+β_{0}

4 |u_{k}(t)|^{2}_{H}^{1}_{(Ω)}+C_{7},

whereC_{6}=C_{5}(^{1}_{2} +K+γ_{1}),C_{7}= ^{1}_{2}(kqk^{2}_{L}^{2}_{(Ω)}+kgk^{2}_{L}^{2}_{(Γ}_{1}_{)}). Hence,
(2.10) d

dtk√

αu_{k}(t)k^{2}_{L}^{2}_{(Ω)}+β_{0}

2 |u_{k}(t)|^{2}_{H}^{1}_{(Ω)}≤2(1

2+C_{0}+C_{6}^{2}

β_{0})ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}+2C_{7}.
The integration with respect to time yields

α_{0}ku_{k}(t)k^{2}L^{2}(Ω)+β_{0}
2

Z _{t}

0 |u_{k}(ξ)|^{2}H^{1}(Ω)dξ≤2(1

2+C_{0}+C_{6}^{2}
β_{0})

Z _{t}

0 ku_{k}(ξ)k^{2}L^{2}(Ω)dξ
+ 2C_{7}T+α_{1}ku_{k}(0)k^{2}_{L}^{2}_{(Ω)},

and, thus,

(2.11) ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}+C10

Z t

0 |u_{k}(ξ)|^{2}_{H}^{1}_{(Ω)}dξ ≤C8

Z t

0 ku_{k}(ξ)k^{2}_{L}^{2}_{(Ω)}dξ+C9,
where C_{8} = 2(^{1}_{2}+C_{0}+^{C}_{β}^{6}_{0}^{2})/α_{0},C_{9} = (2C_{7}T+α_{1}ku_{k}(0)k^{2}_{L}^{2}_{(Ω)})/α_{0} and C_{10} =
β0/(2α0). Now we use Gronwall’s lemma in the form from [9]:

Lety, w, z, r∈C([0, T]),w, r≥0 andy(t) +w(t)≤z(t) +R_{t}

0r(s)y(s)ds. Then (2.12) y(t) +w(t)≤z(t) +

Z _{t}

0

r(ϑ)z(ϑ) exp (
Z _{t}

ϑ

r(s)ds)dϑ.

Here we setz(t) =C_{9},r(s) =C_{8}, y(t) =ku_{k}(t)k^{2}_{L}^{2}_{(Ω)},
w(t) =C_{10}Rt

0|u_{k}(ξ)|^{2}_{H}1(Ω)dξ. Then (2.11) implies that

(2.13)

ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}+C10

Z t

0 |u_{k}(ξ)|^{2}_{H}^{1}_{(Ω)}dξ

≤C_{9}+C_{8}C_{9}
Z _{t}

0

exp (
Z _{t}

ϑ

C_{8}ds)dϑ=C_{9}exp (C_{8}t)≤C_{9}exp (C_{8}T).

From this we get (2.14) max

t∈[0,T]ku_{k}(t)k^{2}_{L}^{2}_{(Ω)}+C_{10}
Z _{T}

0 |u_{k}(ξ)|^{2}_{H}^{1}_{(Ω)}dξ≤2C_{9}exp (C_{8}T) = const,

which immediately yields (2.6) and (2.7).

Let us continue in the proof of the existence of the weak solution. Since
{w_{1}, . . . , w_{k}}is a basis inX_{k}, there exist functionsζ_{1}(t), . . . , ζ_{k}(t) such that

(2.15) u_{k}(t) =

Xk

i=1

ζ_{i}(t)w_{i}.

Conditions (2.8) represent a system of ordinary differential equations for unknown
functions ζ_{i}(t), i= 1, . . . , k. Its right-hand side satisfies the Carath´eodory con-
ditions and is Lipschitz-continuous with respect toζ_{i},i= 1, . . . , k, which implies
the existence of a unique generalized (i.e. absolutely continuous) solution in some
time interval [0, T^{∗}]. From the uniform boundedness (2.6) and (2.7), it follows
that there exists a unique approximate solution u_{k} in the whole time interval
[0, T] (see e.g. [24]).

With the aid of a modification of Theorem 4.11 in [28, p. 290], there exists such
a basis{w_{i}}^{∞}_{i=1} inV that

((w_{i}, ϕ))_{V} = λ_{i}(w_{i}, ϕ)_{α} ∀ϕ∈V,
(2.16)

(w_{i}, w_{j})_{α} = δ_{ij} ∀i, j∈N,
(2.17)

(( w_{i}

√λ_{i}, w_{j}

pλ_{j}))_{V} = δ_{ij} ∀i, j∈N,
(2.18)

0< C≤λ1≤λ2 ≤ . . . and λr→ ∞ as r→ ∞, (2.19)

{wi/p

λi}^{∞}i=1 forms an orthonormal basis in V.

(2.20)

(The scalar product ((·,·))_{V} is defined in (1.29).)

Moreover, forX_{k}= span{w_{1}, . . . , w_{k}} and

(2.21) P_{k}^{α}v=

Xk

i=1

(v, w_{i})_{α}w_{i}:V →X_{k}⊂V,
we obtain

(2.22) |P_{k}^{α}v|H^{1}(Ω)≤ |v|H^{1}(Ω) ∀v∈V.

Actually, in virtue of (2.16), (2.18)–(2.21), forv∈V we have

(2.23)

|P_{k}^{α}v|^{2}_{H}^{1}_{(Ω)}=
Xk

i=1

(v, w_{i})^{2}_{α}((w_{i}, w_{i}))_{V}

= Xk

i=1

1

λ_{i}((v, w_{i}))^{2}_{V} =
Xk

i=1

((v, w_{i}

√λ_{i}))^{2}_{V} ≤ |v|^{2}_{H}^{1}_{(Ω)}.
Further, for everyϕ∈X_{k} we have

(2.24) (P_{k}^{α}v, ϕ)α=
Xk

i=1

(v, w_{i})α(w_{i}, ϕ)α= (v, ϕ)α.

Now let us return to the definition (2.2) of the approximate solutionu_{k}, rewrit-
ten in the form (2.8). Since X_{k} ⊂V ⊂ L^{2}_{α}(Ω) ≡L^{2}_{α}(Ω)^{∗} ⊂V^{∗}, the derivative
u^{′}_{k} =∂u_{k}/∂t can be considered as an element of V^{∗}. If we denote by h·,·i the
duality betweenV^{∗} andV in such a way that

(2.25) hϑ, ϕi= (ϑ, ϕ)α ∀ϕ∈V, ∀ϑ∈L^{2}_{α}(Ω),
then we have

(2.26) hu^{′}_{k}, ϕi= (u^{′}_{k}, ϕ)_{α} ∀ϕ∈V.

Letv ∈ V. Then, according to (2.24), (2.26) and (2.8), sinceu^{′}_{k} ∈X_{k}, we find
that

hu^{′}_{k}(t), vi= (u^{′}_{k}(t), v)_{α}= (u^{′}_{k}(t), P_{k}^{α}v)_{α}

=−a(u_{k}(t), P_{k}^{α}v) +d(u_{k}(t), P_{k}^{α}v) + (q, P_{k}^{α}v)_{L}2(Ω).
This, Lemma 1 and (2.22) imply that

(2.27) |hu^{′}_{k}(t), vi| ≤C(|u_{k}(t)|H^{1}(Ω)+ 1)|v|H^{1}(Ω) ∀v∈V, t∈(0, T),
where the constantC depends onC_{1}, . . . , C_{4}. Hence,

(2.28) ku^{′}_{k}(t)k^{2}_{L}^{2}_{(0,T}_{;V}^{∗}_{)}=
Z _{T}

0 ku^{′}_{k}(t)k^{2}V^{∗}dt≤2C^{2}
Z _{T}

0

|u_{k}(t)|^{2}_{H}^{1}_{(Ω)}+ 1
dt

= 2C^{2}T+ 2C^{2}ku_{k}k^{2}_{L}^{2}_{(0,T;V}_{)}, k= 1,2, . . . ,
which is bounded by a constant independent ofk, as follows from (2.7).

The obtained results can be summarized in the following way:

Theorem 2. The sequence {u_{k}}^{∞}_{k=1} is bounded in L^{∞}(0, T;L^{2}(Ω)) and in
L^{2}(0, T;V). The sequence{u^{′}_{k}}^{∞}_{k=1} is bounded inL^{2}(0, T;V^{∗}).

In what follows, we shall apply the well-known Aubin–Lions lemma (see, e.g., [25] or [10]):

Theorem 3. LetX_{0},X,X_{1} be Banach spaces with the following properties:

(a) X_{0}֒→X ֒→X_{1} (continuous imbedding),
(b) X0,X1 are reflexive,

(c) X0֒→֒→X (compact imbedding).

Let us put

(2.29) W =

v∈L^{2}(0, T;X_{0});∂v

∂t ∈L^{2}(0, T;X_{1})

. Then

(2.30) W ֒→֒→L^{2}(0, T;X).

Now we prove the following results:

Theorem 4. The sequence of approximate solutions {u_{k}}^{∞}k=1 is compact in
L^{2}(0, T;H^{µ}(Ω)) for µ ∈ (^{1}_{2},1). The sequence of traces {u_{k}|∂Ω×(0,T)}^{∞}_{k=1} is
compact inL^{2}(0, T;L^{2}(∂Ω)).

Proof: It is necessary to show that there exists a functionuand a subsequence
{u_{k}}^{∞}k=1 (for simplicity we use the same notation) such that

u_{k}→uin L^{2}(0, T;H^{µ}(Ω)) fork→ ∞,
(2.31)

u_{k}|∂Ω×(0,T)→u|∂Ω×(0,T)in L^{2}(0, T;L^{2}(∂Ω)) for k→ ∞.
(2.32)

Let us setX_{0} =V,X_{1} =V^{∗}, X =H^{µ}(Ω) for µ∈(^{1}_{2},1). Then, in view of the
results from Paragraph 1.1, the conditions (a), (b), (c) from the previous theorem
are satisfied, because V ⊂ H^{1}(Ω) ֒→֒→ H^{µ}(Ω) ֒→ L^{2}(Ω) ≡ L^{2}(Ω)^{∗} ֒→ V^{∗}.
Moreover, the trace operatorθ is defined onH^{µ}(Ω) andθ:H^{µ}(Ω)→L^{2}(∂Ω) is
a continuous mapping. By the Aubin–Lions lemma, there exists a subsequence of
approximate solutions{u_{k}}^{∞}_{k=1} such that

(2.33) u_{k}→u in L^{2}(0, T;H^{µ}(Ω)) as k→ ∞.
This means that

(2.34) ku_{k}−uk^{2}_{L}^{2}_{(0,T;H}^{µ}_{(Ω))}=
Z _{T}

0 ku_{k}(t)−u(t)k^{2}H^{µ}(Ω)dt→0 as k→ ∞.

Further, from the property (1.13) of the trace operator θ: H^{µ}(Ω) →L^{2}(∂Ω) it
follows that

(2.35) ku_{k}(t)|∂Ω−u(t)|∂ΩkL^{2}(∂Ω)

=kθu_{k}(t)−θu(t)kL^{2}(∂Ω)≤C_{Tr}(µ)ku_{k}(t)−u(t)kH^{µ}(Ω).
Hence,

ku_{k}−uk^{2}_{L}^{2}_{(0,T}_{;L}^{2}_{(∂Ω))}=
Z _{T}

0

Z

∂Ω|u_{k}−u|^{2}dS

dt

=
Z _{T}

0 ku_{k}(t)|∂Ω−u(t)|∂Ωk^{2}_{L}^{2}_{(∂Ω)}dt

≤C_{Tr}^{2} (µ)
Z T

0 ku_{k}(t)−u(t)k^{2}_{H}^{µ}_{(Ω)}dt→0 as k→ ∞,

what we wanted to prove.

SinceL^{2}(0, T;L^{2}(∂Ω)) =L^{2}(∂Ω×(0, T)), we obtain:

Corollary. It is possible to choose a subsequence{u_{k}}^{∞}k=1of approximate solu-
tions satisfying(2.31),(2.32) and

(2.36) u_{k}→u a.e. in ∂Ω×(0, T).

Remark. The convergence of the traces ofu_{k}(t) fork→ ∞can also be proved
by putting X_{0} = V, X = L^{2}(Ω), X_{1} = V^{∗} in the Aubin–Lions lemma. This
yields the strong convergence of a subsequence (sinceW ֒→֒→L^{2}(0, T;L^{2}(Ω))):

(2.37) u_{k}→u in L^{2}(0, T;L^{2}(Ω)).

Further we use the multiplicative trace inequality (2.5)

kwk^{2}_{L}^{2}_{(∂Ω)}≤C5kwkL^{2}(Ω)|w|H^{1}(Ω) ∀w∈V.

From this we have
kwk^{2}L^{2}(0,T;L^{2}(∂Ω))=

Z _{T}

0 kw(t)k^{2}L^{2}(∂Ω)dt≤C_{5}
Z _{T}

0 kw(t)kL^{2}(Ω)|w(t)|H^{1}(Ω)dt

≤C_{5}
Z T

0 kw(t)k^{2}_{L}^{2}_{(Ω)}dt

!1/2 Z T

0 |w(t)|^{2}_{H}^{1}_{(Ω)}dt

!1/2

(2.38)

=C5kwkL^{2}(0,T;L^{2}(Ω))kwkL^{2}(0,T;V).

This, the boundedness (2.7) of the sequence {u_{k}}^{∞}_{k=1} in L^{2}(0, T;V) and (2.37)
imply that

ku_{k}−uk^{2}_{L}^{2}_{(0,T;L}^{2}_{(∂Ω))}≤C_{5}ku_{k}−ukL^{2}(0,T;L^{2}(Ω))ku_{k}−ukL^{2}(0,T;V)

≤C_{5}(C+kukL^{2}(0,T;V))ku_{k}−ukL^{2}(0,T;L^{2}(Ω))→0,
as k→ ∞,
what we wanted to prove.

Now it is possible to pass to the limit in equation (2.8), rewritten in the form

(2.39)

− Z T

0 (u_{k}(t), w_{i})αϑ^{′}(t)dt+
Z T

0 a(u_{k}(t), w_{i})ϑ(t)dt

=
Z _{T}

0

d(u_{k}(t), w_{i})ϑ(t)dt+
Z _{T}

0

(q, w_{i})ϑ(t)dt ∀ϑ∈C_{0}^{∞}(0, T),
i= 1, . . . , k.

The sequenceu_{k} satisfies

u_{k}⇀ u *-weakly inL^{∞}(0, T;L^{2}(Ω)),
(2.40)

u_{k}⇀ u weakly inL^{2}(0, T;V),
(2.41)

u_{k}→u strongly inL^{2}(0, T;H^{µ}(Ω)), µ∈(1
2,1),
(2.42)

u_{k}→u strongly inL^{2}(Q_{T}),
(2.43)

u_{k}|∂Ω×(0,T)→u|∂Ω×(0,T) strongly in L^{2}(0, T;L^{2}(∂Ω)),
(2.44)

u_{k}→u a.e. in ∂Ω×(0, T).

(2.45)

Letϑ∈C_{0}^{∞}(0, T). It is obvious that the mappings
φ∈L^{2}(0, T;V)7→

Z _{T}

0

(φ(t), w_{i})αϑ^{′}(t)dt ∈R,
(2.46)

φ∈L^{2}(0, T;V)7→

Z T

0

a(φ(t), w_{i})ϑ(t)dt ∈R,
(2.47)

are continuous linear functionals onL^{2}(0, T;V). On the basis of the definition of
the weak convergence inL^{2}(0, T;V) we can immediately pass to the limit in the
first two terms in (2.39). Further, we split the third term in two parts:

(2.48)
Z _{T}

0

d(u_{k}(t), w_{i})ϑ(t)dt=
Z _{T}

0 {d_{0}(u_{k}(t), w_{i}) +d_{1}(u_{k}(t), w_{i})}ϑ(t)dt.

The part withd_{1} is again linear in u_{k}(t) and we proceed as above. Concerning
the part withd_{0} we have to prove that

(2.49) Z T

0 {d_{0}(u_{k}(t), w_{i})−d_{0}(u(t), w_{i})}ϑ(t)dt→0 as k→ ∞.
This is a consequence of the Lipschitz continuity of the function G, the Cauchy
inequality and (2.44):

(2.50)

Z _{T}

0 {d_{0}(u_{k}(t), w_{i})−d_{0}(u(t), w_{i})}ϑ(t)dt

= Z T

0

Z

Γ1

(G(x, u_{k}(t))−G(x, u(t)))w_{i}ϑ(t)dS

dt

≤ Z T

0

Z

Γ1

L_{G}|u_{k}(t)−u(t)| |w_{i}|dS

|ϑ(t)|dt

≤C Z

Γ1

|w_{i}|^{2}dS

1/2 Z _{T}

0

Z

Γ1

|u_{k}(t)−u(t)|^{2}dS dt
1/2

→0,
whereC=L_{G}kϑkL^{2}(0,T).

Summarizing the above results, we see that the limit functionusatisfies con-
ditionsu∈L^{2}(0, T;V)∩L^{∞}(0, T;L^{2}(Ω)) and (1.50) withϕ:=w_{i},i= 1,2, . . . .
This and (2.1) imply that (1.50) holds for allϕ∈V. It remains to verify condi-
tion (1.51). For eachv∈L^{2}(Ω),

(2.51) (P_{k}v, ϕ)→(v, ϕ), ∀ϕ∈V as k→ ∞.

Actually, for ϕ ∈ V there exists {ϕ_{k}}^{∞}_{k=1}, ϕ_{k} ∈ X_{k} such that ϕ_{k} → ϕ in V
(cf. (2.1)) and

(2.52)

|(P_{k}v−v, ϕ)|=|(P_{k}v−v, ϕ−ϕ_{k})|

≤C_{F}(kP_{k}vkL^{2}(Ω)+kvkL^{2}(Ω))kϕ−ϕ_{k}kV

≤2C_{F}kvkL^{2}(Ω)kϕ−ϕ_{k}kV →0 as k→ ∞.

Further, in view of (2.28), we can assume that the sequenceu_{k} is chosen in such
a way that

(2.53) u^{′}_{k}= ∂u_{k}

∂t ⇀eu in L^{2}(0, T;V^{∗}).

Then for allϕ∈V and allϑ∈C_{0}^{∞}(0, T)
Z T

0

(ϕ, u(t))ϑ^{′}(t)dt= lim

k→∞

Z T

0

(ϕ, u_{k}(t))ϑ^{′}(t)dt

=− lim

k→∞

Z _{T}

0

(u^{′}_{k}(t), ϕ)ϑ(t)dt=−
Z _{T}

0

(eu(t), ϕ)ϑ(t)dt, (2.54)

which means thatu^{′} =∂u/∂t=u, and hence,e

(2.55) u^{′}_{k}⇀ u^{′} in L^{2}(0, T;V^{∗}).

Obviously for all ϕ ∈ V and ϑ ∈ C_{0}^{∞}[0, T) with ϑ(T) = 0, ϑ(0) 6= 0, we have
ϕϑ(t), ϕϑ^{′}(t)∈L^{2}(0, T;V), and, hence,

(2.56)

Z T

0

u^{′}_{k}(t)−u^{′}(t), ϕ
ϑ(t)dt

=−(u_{k}(0)−u(0), ϕ)ϑ(0)−
Z _{T}

0

(u_{k}(t)−u(t), ϕ)ϑ^{′}(t)dt.

In virtue of (2.55) and (2.41), the integrals in the first expression as well as the
first integral in the last expression have zero limit as k → ∞. Hence, using
u_{k}(0) = P_{k}u^{0} and (2.51), we find that (u^{0}−u(0), ϕ) = 0 for all ϕ∈ V, which
means that

(2.57) u(0) =u^{0}.

Thus, we have proven thatuis a weak solution of problem (1.14)–(1.18).

2.2 Proof of uniqueness.

Let us assume that there exist two weak solutionsu1, u2 of problem (1.14)–

(1.18). This means that the following equations are satisfied:

(2.58) d

dt(u_{i}(t), ϕ)α+a(u_{i}(t), ϕ) =d(u_{i}(t), ϕ) + (q, ϕ) ∀ϕ∈V, i= 1,2,
in the sense of distribution on (0, T).

On the basis of results from [32, Chapter III, Lemma 1.2], or [21], (2.59) d

dt(u_{i}(t), ϕ)_{α}=h∂u_{i}

∂t (t), ϕi, ϕ∈V, i= 1,2, for a.e. t∈(0, T).

(See (2.25).) From (2.58) and (2.59), writingw=u1−u2, we obtain (2.60) h∂w

∂t(t), ϕi+a(w(t), ϕ) =d(u1(t), ϕ)−d(u2(t), ϕ),

∀ϕ∈V, for a.e. t∈(0, T).

Now, we substituteϕ:=w(t) and find from (2.60) that (2.61) h∂w

∂t(t), w(t)i+a(w(t), w(t)) =d(u_{1}(t), w(t))−d(u_{2}(t), w(t)),

for a.e. t∈(0, T).

From the above references it follows thatw∈C([0, T];L^{2}_{α}(Ω)) and

(2.62) d

dt Z

Ω

α|w(t)|^{2}dx= d

dt(w(t), w(t))_{α}

= 2h∂w

∂t(t), w(t)i for a.e. t∈(0, T).

This and (2.61) imply that (2.63) 1

2 d dt

Z

Ω

α|w(t)|^{2}dx+a(w(t), w(t))dx

= Z

Γ1

[G(x, u_{2}(t))−G(x, u_{1}(t))]w(t)dS−
Z

Γ1

γ|w(t)|^{2}dS for a.e. t∈(0, T).

The individual terms will be estimated with the aid of Young’s inequality (1.48), inequality (2.5), Lemma 1 and assumptions (1.19)–(1.23) and (1.26). Thus, for a.e. t∈(0, T),

d dt

Z

Ω

α|w(t)|^{2}dx+β_{0}|w(t)|^{2}_{H}^{1}_{(Ω)}

≤2(γ_{1}+L_{G})C_{5}kw(t)kL^{2}(Ω)|w(t)|H^{1}(Ω)+ 2C_{0}kw(t)k^{2}_{L}^{2}_{(Ω)}
(2.64)

≤ β0

2 |w(t)|^{2}_{H}^{1}_{(Ω)}+C11kw(t)k^{2}_{L}^{2}_{(Ω)},
whereC_{11}= 2C_{0}+ 2(L_{G}+γ_{1})^{2}C_{5}^{2}/β_{0}. Thus

d dt

Z

Ω

α|w(t)|^{2}dx+β_{0}

2 |w(t)|^{2}_{H}^{1}_{(Ω)}≤C_{11}
Z

Ω|w(t)|^{2}dx.

The integration with respect to time is possible and yields

(2.65) α0

Z

Ω|w(t)|^{2}dx+β0

2 Z t

0 |w(ξ)|^{2}_{H}^{1}_{(Ω)}dξ≤C11

Z t

0

Z

Ω|w(ξ)|^{2}dx dξ,
Z

Ω|w(t)|^{2}dx≤ C_{11}
α0

Z _{t}

0

Z

Ω|w(ξ)|^{2}dx dξ,
because|w(0)|^{2} = 0.

From the last inequality, using Gronwall’s lemma (2.12), we find that (2.66)

Z

Ω|w(t)|^{2}dx≤0, t∈(0, T).

This already implies thatw≡0 and, hence,u_{1}=u_{2}, which proves the uniqueness
of the weak solution.

3. Finite element approximation

Let us assume that the domain Ω is polygonal. By{Th}h∈(0,h0), h_{0} >0, we
denote a system of triangulations of Ω with standard properties from the finite
element theory (see, e.g., [5]): This formed by a finite number of closed triangles
K and

(a) Ω = [

K∈Th

K, (3.1)

(b) ifK_{1}, K_{2}∈ Th, K_{1}6=K_{2}, then eitherK_{1}∩K_{2}=∅

or K_{1}∩K_{2} is a common vertex orK_{1}∩K_{2} is a common side
of K_{1} andK_{2}.

Let the end points of Γ_{1}, Γ_{2}, Γ_{3} be vertices of the triangulationsTh.

Byh_{K} andϑ_{K} we denote the length of the maximal side and the magnitude
of the minimal angle ofK∈ Th, respectively, and set

(3.2) h= max

K∈Th

h_{K}, ϑ_{h}= min

K∈Th

ϑ_{K}.

Let us assume that the system{Th}h∈(0,h0)is regular. This means that there
exists a constantϑ_{0}>0 such that

(3.3) ϑ_{h}≥ϑ_{0} ∀h∈(0, h_{0}).

We define the following finite dimensional spaces:

(3.4) X_{h}={v_{h}∈C(Ω);v_{h}|K∈P1(K) ∀K∈ Th},
V_{h}=X_{h}∩V ={v_{h}∈X_{h};v_{h}|Γ2 = 0},
whereP_{1}(K) is the space of all linear polynomials onK.

Theapproximate solution is defined as a functionu_{h} with the following prop-
erties:

(a) u_{h}∈C^{1}([0, T];V_{h}),
(3.5)

(b) d

dt(u_{h}(t), ϕ_{h})_{α}+a(u_{h}(t), ϕ_{h}) =d(u_{h}(t), ϕ_{h}) + (q, ϕ_{h}) ∀ϕ_{h}∈V_{h},
(c) u_{h}(0) =u^{0}_{h}=π_{h}u^{0},

whereπ_{h} is a suitable interpolation operator fromV into V_{h}.

Similarly as in the case of the Galerkin approximation we can prove the exis- tence of a unique solution of thediscrete problem (3.5).

If we denote by{v_{1}, v_{2}, . . . , v_{N}} a basis of the spaceV_{h}, then there exist func-
tionsξ_{j}(t), j= 1, . . . , N, such that

(3.6) u_{h}(t) =

XN

j=1

ξ_{j}(t)v_{j}

and condition (3.5), (b) can be rewritten in the form

(3.7) d dt(

XN

j=1

ξ_{j}(t)v_{j}, v_{i})α+a(

XN

j=1

ξ_{j}(t)v_{j}, v_{i}) =d(

XN

j=1

ξ_{j}(t)v_{j}, v_{i}) + (q, v_{i}),
i= 1, . . . , N,
or

(3.8) XN

j=1

(v_{j}, v_{i})αdξ_{j}(t)
dt =−

XN

j=1

a(v_{j}, v_{i})ξ_{j}(t) +d(

XN

j=1

ξ_{j}(t)v_{j}, v_{i}) + (q, v_{i}),
i= 1, . . . , N.

This is a system of nonlinear ordinary differential equations which can be solved
by a suitable discrete method for the solution of ODE’s. Let us mention several
simple numerical schemes. To this end, we construct a partition{t_{k}}^{M}_{k=0} of the
time interval [0, T], wheret_{k}=kτ andτ =T /M.

We have several possibilities of the time discretization:

(1) We use the approximationξ_{j}^{k}≈ξ_{j}(t_{k}) and

(3.9) dξ_{j}(t_{k})

dt ≈ ξ_{j}^{k+1}−ξ_{j}^{k}
τ

and all other terms with ξ_{j} are considered on the time level t_{k}. In this way we
obtain a simple explicit forward Euler scheme whose stability is conditioned by a
rather restrictive limitation of the time stepτ.

(2) The use of the backward time difference

(3.10) dξ_{j}(t_{k+1})

dt ≈ ξ_{j}^{k+1}−ξ_{j}^{k}
τ

on the time levelt_{k+1} leads to fully implicit unconditionally stable scheme. This
requires to solve a nonlinear algebraic system on each time levelt_{k+1}for unknowns
ξ_{1}^{k+1}, . . . , ξ^{k+1}_{N} .