65
Acta Math. Univ. Comenianae
Vol. LXV, 1(1996), pp. 65–85
ERROR ESTIMATES OF A FULLY DISCRETE LINEAR
APPROXIMATION SCHEME FOR STEFAN PROBLEM
A. HANDLOVIČOVÁ
1. Introduction
In this paper we analyze the accuracy of a fully discrete linear approximation
scheme, which can be used in solving nonlinear Stefan-like parabolic problems with
nonlinear boundary condition
∂t u(t, x) − ∆β(u(t, x))) = f (t, x, β(u)) on Q := [0, T ] × Ω
u(0, x) = up (x)
on Ω
−∂ν β(u(t, x)) = cg β(u(t, x)) + ac (t, x) for x ∈ Γ, t ∈ (0, T )
where u : (0, T ) × Ω → < is unknown function, Ω ⊂ <d is a polygonal convex
domain with the boundary Γ, 0 < T < ∞, ν is the outward normal to Γ, f (t, x, s)
and ac (t, x) are Lipschitz continuous functions and β : < → < is a nondecreasing
Lipschitz continuous function. Finally cg is a real number and cg ≥ 0.
There are several linear approximation schemes, deal with the Stefan like problems or with the problems concerning non linear diffusion. Among them linear
approximation scheme based on so-called nonlinear Chernoff’s formula with constant relaxation parameter µ have been studied especially in [1], [12], [14], [15],
where also some energy error estimates have been investigated.
Another linear approximation schemes have been proposed in [5], [6], [7] and
[3]. [8] investigates problems with elliptic operator and a nonlinearity also on the
boundary (function g(t, x, s)). Jäger-Kačur approximation scheme [5] is of the
type
µi (θi − β(ui−1 )) − τ ∆θi = τ f (ti , x, β(ui−1 ))
−∂ν θi = g(ti , x, θi−1 )
on Γ,
or
in Ω,
ui = 0
on Γ,
1
|β(ui−1 + µi (θi − β(ui−1 ))) − β(ui−1 )| ≤ α|θi − β(ui−1 )| + o √
,
n
ui := ui−1 + µi (θi − β(ui−1 )), i = 1, . . . , n, u0 = u0 (x)
Received February 18, 1996.
1980 Mathematics Subject Classification (1991 Revision). Primary 65N15.
66
A. HANDLOVIČOVÁ
where θi is the approximation of the function β(u) at time ti , µi ∈ L∞ (Ω), 0 <
δ ≤ µi ≤ K and K −1 , 1 − α, δ are sufficiently small positive constants. By o( n1 )
(Landau’s symbol), we denote sequence cn , n = 1, 2, . . . such that ncn → 0 for
n → ∞.
This scheme has the disadvantage of not being explicit with respect to θi and
µi and therefore some iterative method in order to determine them must be used.
In papers mentioned above the convergence and energy error estimates for this
semi-discretization scheme have been investigated for both strictly monotone [5]
and nondecreasing [7] function β. In [4] linear approximation scheme based on
that from [5] is used but spreads it of two aspects:
— a fully discretization scheme, that means not only discretization in time, but
also finite element method for space discretization is involved,
— linearization is used not only for nonlinear function β but also for nonlinearities in right hand side of the equation and the nonlinearity in the boundary
condition as well.
If Vh0 and Vh1 are function spaces defined by (2.1) and Ph0 and Ph1 are defined
by (2.6) and (2.8) then this full discrete scheme is as follows:
For 1 ≤ i ≤ n find {ui (x), θi (x)} such that ui ∈ Vh0 and θi ∈ Vh1 , and for the
functions µi , ρi ∈ Vh0 it holds
(1.1)
u0 := Ph0 up (x),
θ0 := Ph1 (β(up (x)))
(1.2)
µi Ph0 θi , ψ + τ h∇θi , ∇ψi + τ < cg (θi ) + aci , ψ >h,Γ
= hµi β(ui−1 ), ψi + τ hf (β(ui−1 )), ψi + τ ρi (Ph0 θi − β(ui−1 )), ψ
+ hpi , ψi for all ψ ∈ Vh1 ,
||β(ui−1 + µi (Ph0 θi − β(ui−1 ))) − β(ui−1 )||L2 ,µi
1
(1.3)
,
≤ α||Ph0 θi − β(ui−1 )||L2 ,µi + o √
n
(1.4)
ui := ui−1 + µi (Ph0 θi − β(ui−1 )),
for i = 1, . . . , n, where functionals pi ∈ H ∗ satisfy
(1.5)
| hpi , ψi | ≤ ||pi ||H ∗ (Ω) ||ψ||H for all ψ ∈ H 1 (Ω) and
1
||pi ||H ∗ = O
, where σ > 1
nσ
aci = ac (ti , x),
0 < δ ≤ µi ≤ K, |ρi | ≤ K.
In this scheme function θi is used for linearization of the nonlinearity β, functions
µi ρi represent aproximations of β 0 1(u) and f 0 (u) respectively. K −1 , 1 − 2α, α, δ are
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
67
sufficiently small positive constants and || · ||L2 ,µi denotes weight function space
L2 with the weight function µi (e.g. [10]).
The values of functions θi , µi , ρi can be determined by an iterative process
(see [4]). In paper [4] the convergence of this fully discrete method is proved. The
aim of this paper is to prove some error estimates for this method.
Let eθ (t, x) and eu (t, x) have the same meaning as in (2.20). The main result
of this paper are the following error estimates for the scheme (1.2)–(1.4):
Let the relation between time and space discretization be
(1.6)
4
τ ≤ C∗ h 3 .
Then (under assumptions and notations stated in Section 2)
Z t
||eu ||L∞ (0,T ;L2 (Ω) + ||eθ ||L2 (I,L2 (Ω)) + ess sup ∇
eθ dt
Z t
+ ess sup eθ dt
0≤t≤T
0
0≤t≤T
0
L22 (Ω)
1
≤ Ch 3 .
L2 (Γ)
In Section 2 we give some notations, basic assumptions and summarize previous
results. In Section 3 we prove the error estimates.
2. Basic Notations, Assumptions and Preliminary Results
Let us assume
(HΩ )
Ω ⊂ <d (d ≥ 1) is polygonal convex domain with the boundary Γ,
Q := I × Ω, where I = (0, T ), 0 < T < ∞ and T is fixed.
Further we shall use function spaces and their symbols as in [10]: L2 (Ω), L22 (Ω)
with norm ||·||, L2 (I, L2 (Ω)) = L2 (Q) with norm ||·||L2 (I,L2 (Ω)) , H 1 (Ω) := W 1,2 (Ω)
with norm || · ||H . Dual space to H 1 (Ω) we denote H ∗ (Ω). Finally we shall use
functional space L2 (Γ) and its norm we denote || · ||Γ .
We use notation h·, ·i both for inner product in L2 (Ω) and duality pairing between H 1 (Ω) and H ∗ (Ω),
(Hβ )
β : < → <, β(0) = 0 is nondecreasing Lipschitz continuous function
0
satisfying 0 ≤ lβ ≤ β (s) ≤ Lβ < ∞ for almost all s ∈ < and there exist
c1 , c2 > 0, that for all s ∈ < it holds |β(s)| ≥ c1 |s| − c2 ,
(Hf )
f : I ×Ω×< → < is uniformly Lipschitz continuous function with Lipschitz
constant Lf , and there exists constant C such that ||f (t, x, 0)|| ≤ C for
a.a. t ∈ I, x ∈ Ω.
(Hg )
cg ≥ 0 and ac ∈ L∞ (I, H(Ω)) and ac is Lipschitz function with Lipschitz
constant La .
68
A. HANDLOVIČOVÁ
Let us denote
1
ḡ =
τ
i
Z
Ii
(cg β(u(t, x)) + ac (t, x)) dt,
gi = cg θi (x) + ac (ti , x) = cg θi + aci
(Hup )
up ∈ H 1 (Ω), β(up ) ∈ H 1 (Ω).
Let {Sh }h be a family of triangulations
h
Sh = {Sk }K
k=1
h
of Ω consisting of d-simplices Ω̄ = ∪K
k=1 Sk and h stands for the mesh size h =
supk=1,...,Kh diamSk .
(HSh ) The family {Sh } is regular and also the assumption for inverse inequality
holds in the sense of Ciarlet ([2, Chapter 3]).
Further we denote
Vh0 := {ψ : ψ/Sk is constant for all k = 1, . . . , Kh },
(2.1)
Vh1 := {ψ ∈ C 0 (Ω̄); ψ/Sk ∈ P1 (Sk ) for all k = 1, . . . , Kh },
where P1 (S) = {q : S → R|q is polynomial of degree at most one}
On the boundary Γ is defined discrete inner product by
(2.2)
hψ, φih,Γ :=
Kh Z
X
k=1
Sk ∩Γ
Πh (ψφ) ds
for any piecewise continuous functions ψ, φ ∈ Ω̄ where Πh stands for the local
linear interpolation operator ([2]).
Notice that the integral in (2.2) can be evaluated easily by means of the vertex
quadrature rule which is exact for functions from Vh1 ([2, p. 182]).
It holds ([20], [15]) for any ψ, ϕ ∈ Vh1
(2.3)
| hψ, ϕih,Γ | ≤ C3 ||ψ||H ||ϕ||H
It is also well known the error bound takes into account the effect of numerical
integration ([15]) for all ψ, ϕ ∈ Vh1
(2.4)
| hψ, ϕiΓ − hψ, ϕih,Γ | ≤ C5 h||ψ||H ||ϕ||H
The estimate (2.4) holds also for functions ϕ ∈ Vh1 and ψ = b · v, where v ∈ Vh1
and b ∈ Vh0 ; ||b||L∞ ≤ K, K > 0.
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
69
If we denote by a(·, ·) the inner product in H 1 (Ω)
a(w, z) := h∇w, ∇zi + hw, zi
and by G : H ∗ (Ω) → H 1 (Ω) the Green’s operator defined by
a(Gψ, z) = hψ, zi for all z ∈ H 1 (Ω), ψ ∈ H ∗ (Ω),
then the associated discrete operator Gh : H ∗ (Ω) → Vh1 is defined by
a(Gh ψ, ξ) = a(Gψ, ξ) for all ξ ∈ Vh1 , ψ ∈ H ∗ (Ω).
Because of assumptions above the operator G is regular in the sense of Ciarlet ([2]),
namely: for any ψ ∈ L2 (Ω) we have Gψ ∈ H 2 (Ω) and ||Gψ||H 2 (Ω) ≤ C||ψ||L2 (Ω) .
If {Sh }h and G are regular, then it holds
(2.5)
||[G − Gh ]ψ||H s ≤ C6 h2−(r+s) ||ψ||B −r for 0 ≤ s, r ≤ 1,
where H s and B −s are the following function spaces 0 ≤ s ≤ 1: H s := [H 1 (Ω),
L2 (Ω)]1−s , B −s := [L2 (Ω), H ∗ (Ω)]s , (see [11]).
Let us now introduce the L2 -projection operator Ph0 onto Vh0 , which for any
z ∈ L2 (Ω) is defined by
(2.6)
0
Ph z, ψ = hz, ψi for any ψ ∈ Vh0
and satisfies
(2.7)
||z − Ph0 z||H −s (Ω) ≤ C8 hr+s ||z||H r (Ω) , 0 ≤ s, r ≤ 1.
Let us also introduce the discrete H 1 -projection operator Ph1 , which for any z ∈
H 1 (Ω), Ph1 z ∈ Vh1 is defined by
(2.8)
a(z − Ph1 z, ψ) = 0 for any ψ ∈ Vh1 .
Moreover from approximation property of the operator Gh it holds ([15])
(2.9)
||z − Ph1 z||H s (Ω) ≤ C9 h2−(r+s) ||z||H 2−r (Ω) , 0 ≤ s, r ≤ 1,
We shall use the following well-known inequalities
(2.10)
(2.11)
(2.12)
a2
εb2
+
for any a, b ∈ < and ε > 0,
2ε
2
2a(a − b) = a2 − b2 + (a − b)2 for any a, b, ∈ <.
1
2
2
2
||v||Γ ≤ C10 ε||∇v|| + ||v||
ε
ab ≤
70
A. HANDLOVIČOVÁ
for any v ∈ W 1,2 (Ω) and sufficiently small ε > 0.
The inequality (2.12) see e.g. [13, p. 15].
We conclude with some notations concerning the time discretization. Let τ = Tn
be the time step and ti = iτ , Ii = (ti−1 , ti ] for 1 ≤ i ≤ n.
R
We also set z i := z(·, ti ), z̄ i := τ1 Ii z(·, t) dt for any continuous (resp. integrable) function in time defined in Q and ∂z i := (z i − z i−1 )/τ , 1 ≤ i ≤ n for any
given family {z i}ni=0 . Similarly we denote δui = (ui − ui−1 )/τ for i = 1, . . . , n and
ui , ui−1 ∈ Vh0 .
For simplicity we shall denote f (x, t, s) by f (s) The constant C will be generic
constant independent on time and space discretization parameters τ and h.
Further we denote γ := (τ, h) a pair of discretization parameters, where τ and
h have the same meaning as before. Then Rothe’s function is
(2.13)
θ(γ) (t, x) := θi−1 (x) +
t − ti−1
(θi − θi−1 )
τ
for t ∈ hti−1 , ti i, θi ∈ Vh1 a i = 1, . . . , n.
Step function we define as
(2.14)
θ̄(γ) (t, x) = θi (x) for t ∈ (ti−1 , ti i , i = 1, . . . , n,
θ̄(γ) (0, x) = β(u0 (x)).
We define also the function
(2.15)
Ph0 θ̄(γ) (t, x) = Ph0 θi for t ∈ (ti−1 , ti i , i = 1, . . . , n,
Ph0 θ̄(γ) (0, x) = β(u0 (x)).
Analogously we can define also Rothe’s and step function u(γ) (t, x) and ū(γ) (t, x).
(γ)
Time derivative for Rothe’s function u(γ) we denote ut and for t ∈ (ti−1 , ti )
we have
ui − ui−1
(γ)
ut =
.
τ
The dual space to the space Vh1 we denote Vh1,∗ .
All functions defined on Q can be prolongated by zero value outside Q.
Variational formulation of our problem is:
Problem (P). Find {u(t, x), θ(t, x)} such that
(2.16)
u(t, x) ∈ L2 (I; L2 (Ω)), ∂t u(t, x) ∈ L2 (I; H ∗ (Ω))
and u(0, x) = up (x)
(2.17)
θ(t, x) ∈ L2 (I; H 1 (Ω))
(2.18)
θ(t, x) = β(u(t, x)) for a.e.(t, x) ∈ Q
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
71
and for all ϕ ∈ L2 (I, H 1 (Ω)) the following equation holds (arguments are ommitted)
Z (2.19)
I
Z
Z
Z
∂u
, ϕ dt + h∇θ, ∇ϕi dt + hg(θ), ϕiΓ dt = hf (θ), ϕi dt.
∂t
I
I
I
Finally we denote
eθ (t, x) = β(u(t, x)) − θ̄(γ) (t, x)
(2.20)
eu (t, x) = u(t, x) − ū(γ) (t, x) for x ∈ Ω t ∈ I
We conclude this section with the basic results of [4].
Theorem. Assume that (HΩ ), (Hβ ), (Hup ), (Hf ), (Hg ), (HSh ) hold. Let θi ,
β(ui ), µi fulfill (1)–(1.5). Then for τ ≤ τ0 and h ≤ h0 such that (1.6) is fulfilled
and τ0 , h0 are sufficiently small, there exists a constant C, independent of the
discretization parameters such that
(2.21)
(2.22)
max ||β(ui )|| +
1≤i≤n
n
X
||ui − ui−1 ||2 +
i=1
n
X
τ ||∇θi ||2 ≤ C,
i=1
max
i=1,...,n
||Ph0 θi ||
+ max ||ui || ≤ C,
i=1,...,n
m
X
(2.23)
τ ||θi ||2 ≤ C,
i=1
m
X
(2.24)
τ ||θi ||2H ≤ C.
i=1
3. Error Estimates
We choose ϕχ[ti−1 ,ti ] for arbitrary ϕ ∈ H 1 (Ω) and i = 1, . . . , n as a test function
in (2.19). In the sense of previous notations we obtain
(3.1)
i i
∂u , ϕ + ∇β̄ , ∇ϕ + ḡ i , ϕ Γ = f¯i , ϕ
Variational formulation of the fully discretized problem (1.1)–(1.4) is
(3.2)
hδui , ϕi + h∇θi , ∇ϕi + hgi , ϕih,Γ
= hf (β(ui−1 ), ϕi + ρi (Ph0 θi − β(ui−1 )), ϕ + hpi , ϕi ,
for arbitrary ϕ ∈ Vh1 .
Now we prove the following lemma
72
A. HANDLOVIČOVÁ
Lemma 1. Let the assumptions (HΩ ), (Hβ ), (Hg ), (Hf ),(Hup ), (HSh ) and the
relation between time and space discretization fulfil (1.6). Than it holds
2
2
Z t
Z t
||eθ ||2L2 (I,L2 (Ω)) + ess sup ∇
e
dt
+
ess
sup
e
dt
θ
θ
0≤t≤T
0≤t≤T
0
0
Γ
1
h2
≤C τ2 +h+
:= Cψ(h, τ ).
τ
Proof. We substract variational identity (3.2) from (3.1), multiply by τ and
sum over i = 1, . . . , k. We have
(3.3)
k
u − uk , ϕ + τ
*
+
* k
+
k
X
X
i
i
∇
(β̄ − θi ), ∇ϕ + τ
(ḡ − gi ), ϕ
i=1
i=1
Γ
+
* k
+
* k
X
X
i
(f¯ − f (ti , β(ui−1 ))), ϕ − τ
pi , ϕ
=τ
i=1
*
+
* k
+
k
X
X

−τ
gi , ϕ
−
gi , ϕ
i=1
−τ
* k
X
Γ
i=1
+
ρi (Ph0 θi − β(ui−1 )), ϕ
i=1


h,Γ
+ up − Ph0 up , ϕ .
i=1
Now we choose as a test function ϕ = τ (Ph1 β̄ k − θk )
k = 1, . . . , m, and multiply by τ . We obtain
(3.4)
m Z
X
k=1
Ik
heu , eθ i dt +
m
X
*
τ2
+
k=1
= −τ 2
−
m
X
* k
+
X
2
i
1 k
τ
(ḡ − gi ), Ph β̄ − θk
i=1
* k
X
pi , Ph1 β̄ k − θk
i=1
k=1
Z
m
X
k=1
− τ2
+
k
X
∇
(β̄ i − θi ), ∇(Ph1 β̄ k − θk )
i=1
k=1
m
X
∈ Vh1 , sum again over
Ik
m
X
k=1
+
Γ
m
X
−−
τ uk − uk , Ph1 β̄ k − β̄ k dt
k=1
k
u − u, eθ dt
*
* k
+
+
k
X
X
1
k
1
k

gi , Ph β̄ − θk
−
gi , Ph β̄ − θk
i=1
Γ
i=1
h,Γ


LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
−τ
2
* k
m
X
X
k=1
+
m
X
73
+
ρi (Ph0 θi
−
β(ui−1 )), Ph1 β̄ k
− θk
i=1
* k
+
X
i
1 k
¯
τ
(f − f (ti , β(ui−1 ))), Ph β̄ − θk
2
k=1
m
X
+τ
i=1
up − Ph0 up , Ph1 β̄ k − θk .
k=1
We will estimate each term of this identity, so we denote (3.4) formally as follows
I + II + III = IV + · · · + X
First we define a function
h(s) = s −
where
H = max
β(s)
,
H
1
; Lβ
δ
(Lβ is Lipschitz constant for β and for δ it holds 0 < δ ≤ µi ≤ K for all i = 1, . . . , n.
For function h(s) it holds
0 ≤ h0 (s) ≤ 1 for a.a. s ∈ <.
(3.5)
If we use (1.4) for t ∈ Ik we get
eθ = β(u) − θk = β(u) − β(uk−1 ) −
uk − uk−1
+ (Ph0 θk − θk )
µk
and also
eu =
eθ
+ h(u) − h(uk−1 ) +
H
1
1
− 1 (uk − uk−1 ) + (θk − Ph0 θk )
Hµk
H
for all k = 1, . . . , n.
We will estimate first term as follows
I=
m Z
1 X
||eθ ||2 dt
H
k=1 Ik
m
XZ 1
1
0
+
h(u) − h(uk−1 ) + (
− 1)(uk − uk−1 ) + (θk − Ph θk ), eθ dt
Hµk
H
Ik
k=1
1
= I1m + I21 ,
H
74
A. HANDLOVIČOVÁ
2
where Im
1 = keθ kL2 (0,tm ;L2 (Ω)) and
I21 =
m Z
X
k=1 Ik
m Z
X
hh(u) − h(uk−1 ), β(u) − β(uk−1 )i dt
−
+
−
k=1 Ik
m Z
X
k=1 Ik
m Z
X
k=1
h(u) − h(uk−1 ),
uk − uk−1
µk
dt
h(u) − h(uk−1 ), Ph0 θk − θk dt
1−
Ik
1
Hµk
(uk − uk−1 ), eθ
dt +
m Z
1 X
θk − Ph0 θk , eθ dt
H
Ik
k=1
Finally we have
I≥
Im
1
+ lβ kh(u) − h(uk−1 )k2L2 (0,tm ,L2 (Ω))
H
m Z X
uk − uk−1
−
h(u) − h(uk−1 ),
dt
µk
k=1 Ik
m Z
X
+
h(u) − h(uk−1 ), Ph0 θk − θk dt
−
k=1 Ik
m Z
X
k=1
+
Ik
1−
1
Hµk
(uk − uk−1 ), eθ
dt
m Z
1 X
θk − Ph0 θk , eθ dt
H
Ik
k=1
1
= I1m + I1 + I2 + I3 + I4 + I5
H
The terms I2 –I5 we give on the right hand side of identity (3.4) and then we
estimate them. First we realize that
h(u) − h(uk−1 ) = u − uk−1 −
eθ
uk − uk−1
P 0 θk − θk
−
+ h
.
H
Hµk
H
So we have
m Z X
m X
uk − uk−1
uk − uk−1
I2 = −
u,
dt + τ
uk−1 ,
µk
µk
k=1 Ik
k=1
m Z
1 X
uk − uk−1
+
eθ ,
dt
H
µk
Ik
k=1
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
m m X
uk − uk−1 uk − uk−1
τ X
uk − uk−1
0
+τ
,
−
Ph θk − θk ,
Hµk
µk
H
µk
k=1
k=1
If we now use (2.21), (2.24) and (2.7), we obtain
" m
# 12
m X
X
1
uk − uk−1 2
|I2 | ≤ kukL2(Q) τ
kuk − uk−1 k
+τ
uk−1 ,
δ
µk
k=1
k=1
m
C8 h τ X
1 m
+
I1 + Cτ +
||θk ||2H
16H
2H
k=1
m X
1
3
1 uk − uk−1
1 m
≤
I + Cτ 2 + Ch2 + C τ 4 uk−1 , τ 4
16H 1
µk
2
k=1
m
m
X
X
1
3
1
1 m
uk − uk−1 2
≤
I1 + Cτ 2 + Ch2 + Cτ 2
||uk−1 ||2 + Cτ 2
||
||
16H
µk
k=1
k=1
1 m
1
≤
I + Cτ 2 + Ch2
16H 1
Analogously we estimate the term I3
2
C8 h τ
|I3 | ≤ ||u||L2 (Q)
m
X
! 12
||θk ||2H
+ C8 hτ
k=1
+
m
X
||uk−1 ||||θk ||H
k=1
m
m
k=1
k=1
X
X
1 m
4
3
||θk ||2H +
||θk ||2H + Cτ
I1 + C8 h2 τ
C8 h2 τ
16H
H
2H
1 m
≤ Ch + Ch2 + Cτ +
I .
16H 1
For term IV we use similarly (2.21)–(2.24) and (2.10)
|I4 | ≤
m Z
X
Ik
k=1
≤ τ 4H
||uk − uk−1 ||||eθ || dt
m
X
||uk − uk−1 ||2 +
k=1
1 m
1 m
I1 ≤ Cτ +
I .
16H
16H 1
We estimate the last term
m
|I5 | ≤
X
1 m
I1 + 4HC8 h2 τ
||θk ||2H
16H
k=1
1 m
≤
I + 4HC8 Ch2 .
16H 1
75
76
A. HANDLOVIČOVÁ
For the term II of identity (3.4) we use the definition of operator Ph1 . We get
II = τ
2
* k
m
X
X
+
∇(Ph1 β̄ i
−
θi ), ∇(Ph1 β̄ k
− θk )
i=1
k=1
* k
+
m
X
X
1 i
i
1 k
+τ
(Ph β̄ − β̄ ), Ph β̄ − θk
2
k=1
i=1
= II1 + II2 .
For the first term we use well known identity
(3.6)
2
m
X
k=1
ak
k
X
!
ai
i=1
=
m
X
!2
ak
+
k=1
Further we use also well known regularity
have from (2.19) for ϕχ[0,t] ([14]).
m
X
a2k for ak ∈ <.
k=1
Rt
0
β(u) ∈ L∞ (0, T ; H 2(Ω)), that we
2
m
m
τ2 τ2 X
X
1 k
II1 =
∇(Ph β̄ − θk ) +
||∇(Ph1 β̄ k − θk )||2
2 2
k=1
k=1
Z tm
Z tm
2
2
1
1
1
2
m
2
≥ ∇
Ph eθ (t) dt ≥ ∇
eθ (t) dt
− Ch =: II − Ch .
2
2
0
0
Second term we estimate as follows
2
m X
k
m
X
τ X 1 k
1 i
i ||Ph β̄ − θk ||2
|II2 | ≤ τ 4H
(Ph β̄ − β̄ ) +
16H
k=1 i=1
k=1
k Z
2
m
X
X
≤ τ 4H
(Ph1 β(u(t)) − β(u(t)))
k=1 i=1 Ii
Z tm
1
+
||Ph1 β(u(t)) − β(u(t))||2 + ||eθ ||2 dt
8H 0
m X
k Z
X
≤ τ 4HT
||Ph1 β(u(t)) − β(u(t))||2
3
k=1 i=1
Ii
1 2 2
1 m
C h ||β(u)||2L2 (I,H 1 (Ω)) +
I
8H 9
8H 1
1 2 2
1 m
≤ 4HT 2 C92 h2 ||β||2L2 (I,H 1 (Ω)) +
C h ||β(u)||2L2 (I,H 1 (Ω)) +
I
8H 9
8H 1
1 m
≤ Ch2 +
I ,
8H 1
+
77
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
where we have used the properties of function β(u), relations (2.9), (2.10) and
well-known inequality
m
X
!2
ak
≤m
k=1
m
X
a2k for arbitrary real numbers ak , k = 1, . . . , m.
k=1
We estimate third term
* k
+
* k
+
m
m
X
X
X
X
III = τ 2
(ḡ i − gi ), Ph1 β̄ k − β̄ k
+ τ2
(ḡ i − gi ), β̄ k − θk
i=1
k=1
k=1
Γ
i=1
Γ
= III1 + III2 ,
where III1 we estimate using (Hg )
2
m
m X
k
4 X
1X 1 k
τ
|III1 | ≤
||Ph β̄ − β̄ k ||2Γ +
(ḡ i − gi ) = III11 + III12 ,
2
2
i=1
k=1
k=1
Γ
where
III11
1
≤
2τ
Z
tm
0
||Ph1 β(u(t))
If we now use regularity
Rt
0
−
β(u(t))||2Γ
III12 ≤
Z
tm
0
||β||2H 2 (Ω) .
β(u) ∈ L∞ (I, H 2 (Ω)), again, we have
III11 ≤ C
c2g τ 4
C10 C92 h2
dt ≤
2
τ
h2
τ
k
k
2
2
m X
m X
X
X
i
4
i
āc − ac (ti )
(β̄ − θi ) + τ
k=1
i=1
Γ
k=1
i=1
Γ
= IIIA + IIIB
IIIA ≤
c2g τ 2
k Z
2
m X
X
(β(u(t)) − θi ) dt
Ii
k=1
≤ c2g T τ 2
i=1
m X
k Z
X
k=1 i=1
Ii
Γ
||β(u(t)) − θi ||2Γ
≤ 2c2g T 2 τ ||β(u)||2L2 (I,L2 (Γ)) + ||θ̄(γ) ||2L2 (I,L2 (Γ)) ≤ Cτ,
78
A. HANDLOVIČOVÁ
where we have used again the properties of β(u) and θ̄(γ) .
IIIB ≤
2T L2a τ 2
2
m X
k Z
X
2
(t − ti ) dt ≤ Cτ 2 .
Ii
k=1
i=1
Γ
Now we estimate III2
III2 =
m
X
* k
+
X
i
k
τ
(ḡ − gi ), β̄ − θk
2
i=1
k=1
= cg τ 2
m
X
k=1
Γ
* k
+
* k
+
m
X
X
X
i
k
2
i
¯
¯
i
k
(β̄ − θi ), β̄ − θk
+τ
(a c − ac ), β − θk
i=1
Γ
k=1
i=1
Γ
= III21 + III22 .
We use the property Hg
III21
* k
+
m
X
X
i
k
= cg τ
(β̄ − θi ), β̄ − θk
2
k=1
i=1
Γ
Z tm
2
cg 2
cg k
2
= τ
||β̄ − θk ||Γ +
eθ dt
2
2
0
Γ
k=1
Z
2
tm
cg τ 2
cg =
eθ dt
III211 +
.
2
2
0
Γ
m
X
Next term we estimate as follows
2
m
m
k
τ 2 cg X ¯
4τ 2 X X i
2
i
|III22 | ≤
||βk − θk ||Γ +
āc − ac 8
cg
k=1
k=1 i=1
Γ
k
2
Z
m X
2
X
τ cg
La
III211 + Cτ 2
≤
|t − ti | dt
8
τ
I
i
i=1
k=1
Γ
τ 2 cg
III211 + Cτ.
≤
8
Whole third term of (3.4) is now estimated.
We can estimate the term IV
k
2
m
τ2 X X |IV | ≤
pi ∗
2
i=1
k=1
H (Ω)
+
m
τ2 X 1 k
||Ph β̄ − θk ||2H = IV1 + IV2 ,
2
k=1
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
79
Using the properties of pi we have
IV1 ≤ τ T
m X
k
X
||pi ||2H ∗ (Ω) ≤ T 2
k=1 i=1
m
X
||pk ||2H ∗ (Ω) ≤ Cτ
k=1
and using the properties of β and θ̄(γ) together with (2.9) it holds
IV2 ≤ τ 2
m
X
||Ph1 β̄ k ||2H + τ 2
k=1
m
X
||θk ||2H ≤ τ (2 + C9 )||β||2L2 (I,H 1 (Ω)) + τ C ≤ Cτ.
k=1
The fifth term of (3.4) we estimate following the similar estimate of [14]

|V | ≤ C ||u||L2 (0,tm ;L2 (Ω) +
m
X
! 12  " m
# 12
X
2
1
k
k
2

τ ||uk ||
τ ||Ph β̄ − β̄ ||
≤ Ch.
k=1
k=1
The term V I we can estimate as in [3]
m Z
X
k
|V I| = u − u, eθ dt
k=1 Ik
m Z Z
X
tk
∂s u ds, eθ (t) dt
=
Ik
t
k=1
≤ τ k∂t u kL2 (0,tm ,H ∗ (Ω)) keθ kL2 (0,tm ,H(Ω)) ≤ Cτ,
that holds due to the properties of ∂t u, β(u)and a result of (2.24).
For the effect of a numerical integration we use (2.4) and obtain
k m X
X
|V II| ≤ C5 τ h
gi ||Ph1 β̄ k − θk ||H
k=1 i=1
H
2
m X
k
m
X
C
C5 X
5
≤ τ 2 h2
gi + τ 2
||Ph1 β̄ k − θk ||2H
2
2
2
i=1
k=1
k=1
H
= V II1 + V II2 ,
where V II2 we estimate in the same way as in IV2 and in V II1 we use (Hg )
V II1 ≤ 2C5 τ 2 h2
m
X
k=1
≤ Ch2
m
X
k=1

2
2 
k
m
X
X
c2g θi + aci 
i=1
H
k=1
h
||θk ||2H + ||ack ||2H ≤ C
τ
2
H
80
A. HANDLOVIČOVÁ
For the estimation of eighth term we use (2.10), (1.4) and properties of ρi too.
We have
k
2
m
m
4Hτ 3 X τ X 1 k
X ρi
|V III| ≤
(ui − ui−1 ) +
||Ph β̄ − θk ||2 ,
2
µ
16H
i=1 i
k=1
k=1
where second term we have already estimated by a value
as in II2 . The first term we easily estimate using (2.21)
1 m
8H I1
+ Ch2 in the way
k
2
m
m
4Hτ 3 X 4HK 2 T 2 X
X ρi
(ui − ui−1 ) ≤
τ
||uk − uk−1 ||2 ≤ Cτ.
2
2
µ
δ
i
i=1
k=1
k=1
If we use (2.10) in the ninth term, it holds
k
2
m X
m
X
X
1
i
|IX| ≤ 4Hτ
||Ph1 β̄ k − θk ||2
τ
(f¯ − f (ti , β(ui−1 )) +
16H
i=1
3
k=1
k=1
= IX1 + IX2 ,
1 m
where the second term is again less then 8H
I1 + Ch2 . For the first term of this
inequality we derive
Z
1
|f¯i − f (ti , β(ui−1 ))| ≤
|f (t, β(u)) − f (ti , β(ui−1 ))| dt
τ Ii
Z
Lf
≤
|β(u) − β(ui−1 )) dt + Lf τ
τ Ii
Z
Z ui − ui−1 Lf
Lf
dt
≤
|eθ | dt + Lf τ +
τ Ii
τ Ii µi
+ Lf |θi − Ph0 θi |
Z
ui − ui−1 Lf
+ Lf |θi − Ph0 θi |,
=
|eθ | dt + Lf τ + Lf τ Ii
µi
where we have used Lipschitz continuity of the function f , (1.4) and the definition
of eθ . Finally we have
|f¯i − f (ti , β(ui−1 ))|2 ≤
4L2f
τ2
Z
Ii
+ 4L2f
4L2f
≤
τ
Z
Ii
2
|eθ | dt + 4L2f τ 2
|ui − ui−1 |2
+ 4L2f |θi − Ph0 θi |2
µ2i
e2θ dt + 4L2f τ 2 + 4L2f
|ui − ui−1 |2
µ2i
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
81
+ 4L2f |θi − Ph0 θi |2 .
For the first term of the inequality in estimate of the ninth term we have
2
m
k
X
X
i
IX1 = 4H
τ
τ (f¯ − f (ti , β(ui−1 )))
i=1
k=1
≤ 4HT
m
X
τ2
||f¯i − f (ti , β(ui−1 )||2
i=1
k=1
≤ 16HL2f T
k
X
m
X
τ
i=1
k=1
+ 16HL2f T
2
k Z
X
e2 dt
θ
m
X
k=1
≤ Cτ
m
X
τ2
Ii
k X
ui − ui−1 2
|| + ||θi − Ph0 θi ||2 + Cτ 2
||
µi
i=1
I1k + C(τ + h2 ),
k=1
where we have again used properties (2.21) and (2.7). Finally for this term we
have
m
X
1 m
I1 + Cτ
I1k + C(τ + h2 )
|IX| ≤
8H
k=1
The last term of (3.4), we estimate again using (2.10) and we have
|X| ≤ τ 4H
m
X
m
||up − Ph0 up ||2 +
k=1
X
1
||Ph1 β̄ k − θk ||2 ,
τ
16H
k=1
The estimate of the second term is well-known:
estimate easily using (2.7) and a property H(up )
|X| ≤
1 m
8H I1
+ Ch2 . First term we
1 m
I + Ch2 .
8H 1
For now on we have estimated all terms of our identity (3.4). This result we can
write formally
1 m
2
I +I1 + IIm
2 + III21 ≤ −I2 − I3 − I4 − I5 + Ch − II2 − III1
H 1
− III22 + IV + V + V I + V II + V III + IX + X
thus
1 m
2
I + IIm
2 + III21 ≤ |I2 | + |I3 | + |I4 | + |I5 | + Ch |II2 | + |III1 |
H 1
+ |III22 | + |IV | + |V | + |V I| + |V II| + |V III| + |IX| + |X|
82
A. HANDLOVIČOVÁ
Now substituting estimates for all terms we obtain inequality
Z tm
Z tm
2
2
3 m 1
cg I + ∇
eθ dt +
eθ dt
8H 1
2
2
0
0
Γ
m
2
X
h
≤ C(h + h2 +
+ τ 2 + τ ) + Cτ
I1k .
τ
k=1
Using Gronwall’s lemma assuming without lost of generality that h < 1 and τ < 1,
we get the final result
||eθ ||2L2 (I,L2 (Ω))
Z t
2
Z t
2
+ ess sup ∇
eθ dt + ess sup eθ dt
0≤t≤T
0≤t≤T
0
0
Γ
2
1
h
.
≤C τ2 +h+
τ
This proved the assertion of lemma.
Now the following lemma results
Lemma 2. Under the assumptions of Lemma 1 it holds
||eu ||2L∞ (0,T ;H ∗ (Ω)) ≤ Cψ(h, τ ).
Proof. We use not only result of Lemma 1 but also the properties of Green’s
operator G and a discretization operator Gh from (2.5). We easily have
1
||ϕ||H ∗ (Ω) = hϕ, Gϕi 2 for any ϕ ∈ H ∗ (Ω)
Chosen now in (3.3) as a test function Gh (uk −uk ) ∈ Vh1 , after some rearrangement
we have
(3.7)
||eku ||2H ∗ (Ω) = ||uk − uk ||2H ∗ (Ω) = uk − uk , G(uk − uk )
* k
+
X
k
k
i
k
= eu , [G − Gh ]eu − τ ∇
(β̄ − θi ), ∇Gh (u − uk )
i=1
* k
+
X
i
k
−τ
(ḡ − gi ), Gh (u − uk )
i=1
Γ
* k
+
X
i
k
+τ
(f¯ − f (ti , β(ui−1 ))), Gh (u − uk )
i=1
−τ
* k
X
i=1
+
k
pi , Gh (u − uk )
83
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
*
+
* k
+
k
X
X
+ −τ 
gi , Gh (uk − uk )
−
gi , Gh (uk − uk )
−τ
* k
X
i=1
i=1
Γ


h,Γ
+
ρi (Ph0 θi − β(ui−1 )), Gh (uk − uk )
i=1
+ up − Ph0 up , Gh (uk − uk ) ,
Formally we denote it as
||eku ||2H ∗ (Ω) = XI + XII + XIII + · · · + XV III
Now the idea is as in the previous lemma; we will estimate all members of this
equation. XI we easily estimate using (2.5)
XI ≤ C6 h2 ||eku ||2 ≤ Ch2 ,
because both functions uk and uk too are bounded in L2 (Ω).
≤
XII
Z
||∇Gh eku || ∇
tk
0
5C7
1
k
k
eθ ≤ 10C7 a(Gh eu , Gh eu ) + 2
1 k 2
≤
+ ψ(h, τ ).
||e || ∗
10 u H (Ω)
Z
∇
0
tk
2
eθ Here the estimate of Gh and Lemma 1 were used.
2
k
X
1
i
(ḡ − gi ) + ||Gh eku ||2H
10
i=1
Γ
k
2
Z tk
2
2 2 X
τ
5C
1
10
2 2
≤ 5C10
cg eθ dt +
āic − aic + ||eku ||2H ∗ (Ω)
2
10
0
Γ
i=1
5C10 τ 2
XIII ≤
2
Γ
≤ Cψ(h, τ ) +
2
5C10
L2a T 2 meas(Γ)2 τ 2
1
+ ||eku ||2H ∗ (Ω) .
10
Now we again use the estimation as in IX
k
2
X
1
i
(f¯ − f (β(ui−1 ))) + ||eku ||2H ∗ (Ω)
10
i=1
Z
2
k
2 X
15Lf k 15L2f τ X
e2θ dt + Cτ 3 +
≤
||ui − ui−1 ||2
2 i=1 Ii
2δ 2 i=1
5τ 2
|XIV | ≤
2
+
k
15L2f C82 h2 X
1
τ ||θi ||2H + ||eku ||2H ∗ (Ω)
2
10
i=1
84
A. HANDLOVIČOVÁ
≤ C(||eθ ||2L2 (Q) + τ 3 + τ + h2 ) +
≤ C(ψ(h, τ ) + τ 3 + τ + h2 ) +
1 k 2
||e || ∗
10 u H (Ω)
1 k 2
||e || ∗ .
10 u H (Ω)
For XV we use (2.10) and properties of pi
XV ≤ Cτ 2 +
1 k 2
||e || ∗ .
10 u H (Ω)
Numerical integration error bounds effect, (2.11), (2.10) and (Hg ) we use for XVI
k
X
XVI ≤ τ hC5 gi ||Gh eku ||H
i=1
H
!
k
k
2
2
X
X
5C5 τ h
1
2
2
i 2
≤
cg ||θi ||H +
||ac ||H + ||eku ||2H ∗ (Ω)
2
10
i=1
i=1
≤ Ch2 +
1 k 2
||e || ∗ .
10 u H (Ω)
Analogously we can estimate in XVII, using namely (1.4), (2.10), (1.5), (2.21)
XVII ≤
k
5K 2 τ X
1
||ui − ui−1 ||2 + ||eku ||2H ∗ (Ω)
2δ 2 i=1
10
≤ Cτ +
1 k 2
||e || ∗ .
10 u H (Ω)
Finally from the properties Ph0 (2.7) we conclude
5C82 h2
1
1
||up ||2H + ||eku ||2H ∗ (Ω) ≤ Ch2 + ||eku ||2H ∗ (Ω) .
2
10
10
As a result we obtain
XVIII ≤
||eku ||2H ∗ (Ω) ≤ C(ψ(h, τ ) + h2 + τ + τ 2 + τ 3 ).
If we assume that, h < 1 a τ < 1 we immediately have the result of lemma.
Theorem 1. Let the assumptions (HΩ ), (Hβ ), (Hup ), (Hg ), (Hf ), (HSh ) hold
and for space and time step the following relation holds
4
τ ≤ C∗ h 3 ,
where constant C∗ is independent on τ, h. Then
||eu ||L∞ (0,T ;L2 (Ω) + ||eθ ||L2 (I,L2 (Ω))
Z t
+ ess sup ∇
eθ dt
0≤t≤T
0
L22 (Ω)
Z t
1
+ ess sup eθ dt
≤ Ch 3
0≤t≤T
0
Proof. The proof results immediately from Lemmas 1 and 2.
Γ
LINEAR APPROXIMATION SCHEME FOR STEFAN PROBLEM
85
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A. Handlovičová, Slovak Technical University, Dept. of Mathematics, Radlinského 11, 813 68
Bratislava, Slovakia