=Paper=
{{Paper
|id=Vol-1623/papermp3
|storemode=property
|title=Optimization in Nonlinear Models of Mass Transfer
|pdfUrl=https://ceur-ws.org/Vol-1623/papermp3.pdf
|volume=Vol-1623
|authors=Roman Brizitskii, Zhanna Saritskaya
|dblpUrl=https://dblp.org/rec/conf/door/BrizitskiiS16
}}
==Optimization in Nonlinear Models of Mass Transfer==
https://ceur-ws.org/Vol-1623/papermp3.pdf
Optimization in Nonlinear Models of Mass Transfer
Roman Brizitskii1,2 and Zhanna Saritskaya1,2
1
Far-Eastern Federal University, Natural Sciences School,
Sukhanova St. 8, 690000 Vladivostok, Russia
2
Institute of Applied Mathematics FEB RAS,
Laboratory of computational aero-hydrodynamics,
Radio St. 8, 690000 Vladivostok, Russia
mlnwizard@mail.ru, zhsar@icloud.ccom
http://www.dvfu.ru, http://www.iam.dvo.ru
Abstract. Optimal control problem for convection–diffusion–reaction equa-
tion, in which reaction coefficient depends nonlinearly on substance’s concentra-
tion, is considered. Numerical algorithms for solving nonlinear boundary value
and optimal control problems are proposed for the equation under study. Sepa-
rately, the results of the numerical experiments about nonlinear boundary value
problems’ solvability are presented. For this purpose the FreeFem++ solver is
used. These studies allow to understand better the process of pollution’s spread
in the atmosphere and fight against its consequences. Particularly, they give
an opportunity to reveal and eliminate the sources of pollution using the mea-
sured impurity’s concentration in some available domain. Also the correctness
of mathematical models of mass transfer and optimal control problems, which
are considered in the paper, is justified.
Keywords: FreeFem++, nonlinear convection-diffusion-reaction equation, op-
timal control problem, multiplicative control problems, optimality system, nu-
merical algorithm
1 Introduction. Boundary value problem
In a bounded domain Ω ⊂ R3 with boundary Γ the following boundary value problem
is considered
−λ∆ϕ + u · ∇ϕ + kϕ = f in Ω, ϕ = 0 on Γ. (1)
Here function ϕ means polluting substance’s concentration, u is a given vector of
velocity, f is a volume density of external sources of substance, λ – constant diffusion
coefficient, function k = k(ϕ) is a reaction coefficient. This problem (1) will be called
problem 1 below.
This study of optimal control problems for a model (1) is intended to develop
efficient mechanisms to control chemical reactions’ behavior. The decision to choose a
velocity vector u as a control can signify the regularization of combustion process at
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
Optimization in Nonlinear Models of Mass Transfer 153
the expense of fuel feed’s intensity changing (see [1]). The efficiency criterion for such
kind of control is the measured concentration of unburned fuel in a subdomain.
It should be mentioned that some inverse problems can be reduced to the optimal
control ones as from the mathematical point of view optimal control problems are the
problems of cost functionals’ minimization on weak boundary problem’s solutions. At
the same time one or several functions can be changed in some certain convex closed
sets. The mentioned functions are usually called controls, cause their changing is exactly
the thing that influences on the minimum of the cost functional.
From the other side, one can thought that such functions are searched on the as-
sumption of the minimum of corresponding cost functionals, which attaches these ex-
tremum problems the meaning of indentification problems of the functions and inverse
problems. See [2–10] about similar methods and approaches.
Particularly, the optimal control problem, which is considered in this paper, can
represent the indentification problem for a velocity and a direction of wind or a fuel,
depending on the case and the situation.
With the help of this approach it’s possible to reveal the hidden sources of pollu-
tion, which are located in the places, inaccessible for observation (under water, on the
territory of the adjacent country). Then the data about the concentration of polluting
substance in the domain, which is accessible for measurement, about the direction and
the velocity of the wind or of the flow in the basin are used.
The results of numerical experiments, executed in FreeFEM++, are given for the
solving of nonlinear boundary value problem. We should note a quick convergence of
the simple iteration method while the initial approximation of the boundary value
problem’s solution was chosen not very successful. The computations are conducted
for a number of reaction coefficients, which depend nonlinearly on the substance’s
concentration at different boundary conditions. The chosen geometry of the domain and
the given velocity field simplify the understanding of numerical experiments’ results.
The numerical algorithm for solving the optimal control problem is presented. This
algorithm is based on using of optimality system, obtained for the extremum problem.
Sufficient conditions of such algorithms’ convergence were obtained in [20]. But these
conditions have the meaning of either the smallness conditions for the initial data of
a boundary value problem or demands greater values of the regularizer in an optimal
control problem, which spoils the quality of the last one. That’s why it’s interesting to
analyse the convergence of such algorithm depending on the initial approximation of
the optimal control problem’s solution. As in the case of the boundary value problem.
The reasoning of the correctess of the considered mathematical model implies the
following. The global solvability of problem 1, when reaction coefficients belong to
rather wide class of functions, is proved in [10–12]. In this paper it is shown that
power coefficients from [13–15] are particular cases of the reaction coefficients consid-
ered in [10–12], with which nonlocal uniqueness of boundary value problem’s solution
takes place. The solvability of multiplicative control problem with common reaction
coefficients is proved further. For a quadratic reaction coefficient optimality system is
obtained, on the analysis of which sufficient conditions for local uniqueness of multi-
plicative control problems’ solutions for particular cost functionals are received.
154 R. Brizitskii, Zh. Saritskaya
While studing problem 1 and optimal control problems Sobolev spaces will be used:
H s (D), Hs (D) ≡ H s (D)3 , s ∈ R and Lr (D), 1 ≤ r ≤ ∞, where D is either a domain
Ω or its boudary Γ . Scalar products in L2 (Ω), H 1 (Ω) and H1 (Ω) are denoted by (·, ·)
and (·, ·)1 , scalar products in L2 (Γ ) – by (·, ·)Γ , norm in L2 (Ω) – by k · k, norm or
semi-norm in H 1 (Ω) – by k · k1 or | · |1 .
It will be assumed that the domain Ω and its boundary Γ satisfy the following:
(i) Ω is a bounded domain in the space R3 with boundary Γ ∈ C 0,1 .
Let D(Ω) be the space of infinitely differentiable functions with finite support in
Ω, Lp+ (Ω) = {k ∈ Lp (Ω) : k ≥ 0}, p ≥ 3/2. Also let Z = {v ∈ L4 (Ω) : div v = 0 in Ω},
V ≡ Z ∩ H1 (Ω).
From Poincare-Friedrichs inequality and from the continuity of the embedding op-
erator H 1 (Ω) ⊂ L4 (Ω) this lemma follows:
Lemma 1. If conditions (i) hold, then there are such positive constants C0 , δ,
C4 and γ, depending on Ω, that for any functions ϕ, S ∈ H 1 (Ω), k ∈ Lp+ (Ω), where
p ≥ 3/2, u ∈ Z these relations are correct:
|(∇ϕ, ∇S)| ≤ kϕk1 kSk1 , kϕkL4 (Ω) ≤ C4 kϕk1 ,
|(kϕ, S)| ≤ C0 kkkLp (Ω) kϕk1 kSk1 , (2)
|(u · ∇ϕ, S)| ≤ γkukL4 (Ω) kϕk1 kSk1 ≤ γC4 kuk1 kϕk1 kSk1 ,
(u · ∇ϕ, ϕ) = 0 ∀ϕ ∈ H01 (Ω), (3)
and for any function S ∈ H01 (Ω) the inequality takes place
(∇S, ∇S) ≥ δkSk21 . (4)
From lemma 1 follows that while conditions (i) are satisfied with the constant
3/2
λ∗ = δλ when k ∈ L+ (Ω), then the coercitive inequality is met
λ(∇S, ∇S) + (kS, S) ≥ λ∗ kSk21 ∀S ∈ H01 (Ω). (5)
Let in addition to (i) the conditions hold:
(ii) f ∈ L2 (Ω), u ∈ Z.
(iii) k ∈ Lp+ (Ω), p ≥ 3/2, wherein function k = k(ϕ) is Lipschitz continuous of ϕ,
i.e. if kϕ1 k1 ≤ c and kϕ2 k1 ≤ c, then
kk(ϕ1 ) − k(ϕ2 )kLp (Ω) ≤ Lkϕ1 − ϕ2 kL4 (Ω) ∀ϕ1 , ϕ2 ∈ H01 (Ω).
Let’s multiply the equation in (1) by S ∈ H01 (Ω) and integrate over Ω. The following
will be got
λ(∇ϕ, ∇S) + (k(ϕ)ϕ, S) + (u · ∇ϕ, S) = (f, S) ∀S ∈ H01 (Ω). (6)
As a result, the weak formulation of problem 1 is obtained. It consists in finding function
ϕ ∈ H01 (Ω) from (6).
Definition 1. A function ϕ ∈ H01 (Ω) which satisfies (6) will be called a weak
solution of problem 1.
Optimization in Nonlinear Models of Mass Transfer 155
The following theorem takes place [12].
Theorem 1. If conditions (i)–(iii) hold, then a weak solution ϕ ∈ H01 (Ω) of problem
1 exists and the estimate takes place:
kϕk1 ≤ Mϕ = (1/λ∗ )kf k. (7)
If, besides, this condition is met
C0 Lkf k ≤ λ2∗ , (8)
then the problem 1’s weak solution is unique.
From [12–14] it ensues that the power dependence is interesting as an example of
particular cases of function k = k(ϕ), k = ϕ2 and k(ϕ) = ϕ2 |ϕ|, for instance. As
the case of quadratic reaction coefficient was analysed in detail in [12]. In particular,
it was shown that a function k = ϕ2 satisfies conditions (iii) and for this function
there is nonlocal uniqueness of problem 1’s weak solution, so let’s consider the function
k = ϕ2 |ϕ|.
For k = ϕ2 |ϕ| the equality is true:
k(ϕ1 ) − k(ϕ2 ) = ϕ21 (|ϕ1 | − |ϕ2 |) + (ϕ1 − ϕ2 )(ϕ1 + ϕ2 )|ϕ2 | a.e. in Ω
and also an estimate takes place:
Z 2/3
(ϕ1 − ϕ2 )3/2 ϕ31 dΩ ≤ kϕ1 − ϕ2 kL3 (Ω) kϕ1 k2L6 (Ω) .
Ω
In such case function k = ϕ2 |ϕ| satisfies the condition (iii).
When k = ϕ2 |ϕ| nonlocal uniqueness of problem 1’s solution takes place. Actually,
let k = ϕ2 |ϕ| and ϕ1 , ϕ2 ∈ H 1 (Ω) be two solutions of problem 1. Then their difference
ϕ = ϕ1 − ϕ2 ∈ H01 (Ω) satisfies the ratio
λ(∇ϕ, ∇h) + (ϕ31 |ϕ1 | − ϕ32 |ϕ2 |, h) + (u · ∇ϕ, h) = 0 ∀h ∈ H01 (Ω). (9)
It’s clear that
(ϕ31 |ϕ1 | − ϕ32 |ϕ2 |)(ϕ1 − ϕ2 ) = ϕ41 |ϕ1 | − ϕ32 |ϕ2 |ϕ1 − ϕ31 |ϕ1 |ϕ2 + ϕ42 |ϕ2 | a.e. in Ω
and on the strength of Young’s inequality
ϕ42 ϕ1 ≤ (4/5)ϕ52 + (1/5)ϕ51 , ϕ41 ϕ2 ≤ (4/5)ϕ51 + (1/5)ϕ52 a.e. in Ω.
In such case (ϕ31 |ϕ1 | − ϕ32 |ϕ2 |, ϕ) ≥ 0 a.e. in Ω. Assuming h = ϕ in (9), on the
strength of lemma 1 it can be concluded that ϕ = 0 or ϕ1 = ϕ2 in Ω.
From aforesaid and [12] follows
Theorem 2. Let conditions (i), (ii) hold. Then when k = ϕ2 and k = ϕ2 |ϕ|, there
is a unique weak solution ϕ ∈ H01 (Ω) of problem 1 and the estimate (7) is met.
Let’s separately consider the reaction coefficient k(ϕ), which generalizes the forth
power, but is not a function of ϕ in a common sense. Let k(ϕ) be an operator acting
form T to Lp+ (Ω), where p ≥ 3/2 and satisfying the following conditions:
156 R. Brizitskii, Zh. Saritskaya
(1) for all w1 , w2 ∈ Br = {w ∈ T : kwk1,Ω ≤ r} the following estimate holds:
kk(w1 ) − k(w2 )kLp (Ω) ≤ Lkw1 − w2 kL6 (Ω) ,
where L is a constant, depending on r, but not depending on w1 , w2 ;
(2) k(ϕ)ϕ satisfies monotony condition
(k(ϕ1 )ϕ1 − k(ϕ2 )ϕ2 , ϕ1 − ϕ2 ) ≥ 0 ∀ϕ1 , ϕ2 ∈ H01 (Ω).
Let’s consider a simple example of the operator k(ϕ), satisfying the conditions
(1), (2) and generalizing numerical functions: k(ϕ) = ϕ4 in subdomain Q ⊂ Ω and
3/2
k(ϕ) = k0 in Ω \ Q, where k0 ∈ L+ (Ω).
This example takes into account the influence on the chemical reaction’s velocity
not only of substance’s concentration, but also of inhomogeneity of chemical reaction’s
behavior in the considered domain. Conditions (1), (2) are also met for the reaction
coefficient k(ϕ) = α(x)ϕ4 , where α(x) ∈ L∞ + (Ω).
It should be noted that the reaction coefficient k(ϕ) = ϕ4 gives the convection-
diffusion-equation the maximum possible nonlinearity of 5th power for the solution
ϕ ∈ H 1 (Ω). For such strong nonlinearity the theory of problem 1’s solvability proving,
which was stated above , is unacceptable in view of the fact that the boundary value
problem’s operator is not compact. The solvability of problem 1 at k(ϕ), satisfying the
conditions (1), (2) follows from the results of [16].
The following theorem holds
Theorem 3. Let conditions (i), (1), (2) hold. Then there is a unique weak solution
ϕ ∈ H01 (Ω) of problem 1 and the estimate (7) is met.
2 Statement of optimal control problem and its solvability
Let’s formulate an optimal control problem for problem 1. For this purpose the whole
set of initial data will be devided into two groups: the group of fixed functions, in
which function f is included, and the group of controlling functions, in which u will be
included, assuming that it can be changed in some subset K.
Let’s introduce an operator F : H01 (Ω) × K → H −1 (Ω) by formula
hF (ϕ, u), Si = λ(∇ϕ, ∇S) + (u · ∇ϕ, S) + (k(ϕ)ϕ, S) − (f, S).
Then (6) can be rewritten in the following form:
F (ϕ, u) = 0. (10)
Let’s suppose that these conditions hold
(j) K ⊂ V is a nonempty convex closed set;
(jj) µi ≥ 0, i = 1, 2 and K is a bounded set µl > 0, l = 0, 1 and the functional I is
bounded below.
Treating (10) as a conditional restriction on the state ϕ ∈ H01 (Ω) and on the control
u ∈ K, the problem of conditional minimization can be formulated as follows:
µ0 µ1
J(ϕ, k) ≡ I(ϕ) + kuk21 → inf, F (ϕ, u) = 0, (ϕ, u) ∈ H01 (Ω) × K. (11)
2 2
Optimization in Nonlinear Models of Mass Transfer 157
The following cost functionals can be used in the capacity of the possible ones:
Z
2
I1 (ϕ) = kϕ − ϕd kQ = |ϕ − ϕd |2 dx, I2 (ϕ) = kϕ − ϕd k21,Q .
Ω
Here ϕd ∈ L2 (Q) is a given function in some subdomain - Q ⊂ Ω. The set of
possible pairs for the problem (11) is denoted by Zad = {(ϕ, u) ∈ H01 (Ω) × K :
F (ϕ, u)=0, J(ϕ, u)<∞}.
Theorem 3. Let conditions (i)–(iii) and (j), (jj) hold. Then there is at least one
solution of the optimal control problem (11).
Proof. Let (ϕm , um ) be a minimizing sequence, for which the following is true:
lim J(ϕm , um ) = inf J(ϕm , um ) ≡ J ∗ .
m→∞ (ϕm ,um )∈Zad
That and the conditions of theorem for the functional J from (11) imply the estimate
kum k1 ≤ c1 . From theorem 1 follows directly that kϕm k1 ≤c2 , where constant c2 doesn’t
depend on m.
Then the weak limits ϕ∗ ∈ H01 (Ω) and u∗ ∈ V of some subsequences of sequences
{ϕm } and {um } exist. Corresponding sequences will be also denoted by {ϕm } and
{um }. With this in mind it can be considered that
ϕm → ϕ∗ ∈ H 1 (Ω) weakly in H 1 (Ω) ang strongly in L4 (Ω), (12)
um → u∗ ∈ H1 (Ω) weakly in H1 (Ω) and strongly in L4 (Ω). (13)
Let’s show that F (ϕ∗ , u∗ ) = 0, i.e.
λ(∇ϕ∗ , ∇S) + (k(ϕ∗ )ϕ∗ , S) + (u∗ · ∇ϕ∗ , S) = (f, S) ∀S ∈ H01 (Ω). (14)
And it should be taken into account that ϕm and um satisfy the relations
λ(∇ϕm , ∇S) + (k(ϕm )ϕm , S) + (um · ∇ϕm , S) = (f, S) ∀S ∈ H01 (Ω). (15)
Let’s pass to the limit in (15) at m → ∞. All linear summands in (15) turn into
corresponding ones in (14). For the nonlinear summand (k(ϕm )ϕm , S) the inequality
takes place
|(k(ϕm )ϕm , S) − (k(ϕ∗ )ϕ∗ , S)| ≤ |(k(ϕm )(ϕm − ϕ∗ ), S)| + |(k(ϕm ) − k(ϕ∗ ), ϕ∗ S)|.
On the strength of lemma 1 and condition (iii) for the function k = k(ϕ) it is obtained
that
|(k(ϕm ) − k(ϕ∗ ), ϕ∗ S)| ≤ Lkϕm − ϕ∗ kL4 (Ω) kϕ∗ kL4 (Ω) kSkL4 (Ω) → 0 at m → ∞.
To apply the property (12) for the summand |(k(ϕm )(ϕm −ϕ∗ ), S)|, embedding density
by norm k · k1 will be used. Let {Sn } ∈ D(Ω) be such a sequence of functions that
kSn − Sk1 → 0 at n → ∞.
This inequality holds:
|(k(ϕm )(ϕm − ϕ∗ ), Sn )| ≤
158 R. Brizitskii, Zh. Saritskaya
kk(ϕm )kL3/2 (Ω) kSn kL12 (Ω) kϕm − ϕ∗ kL4 (Ω) → 0 at m → ∞.
As far as
||(k(ϕm )(ϕm − ϕ∗ ), Sn )| − |(k(ϕm )(ϕm − ϕ∗ ), S)|| ≤ |(k(ϕm )(ϕm − ϕ∗ ), Sn − S)| ≤
≤ kk(ϕm )kL3/2 (Ω) kϕm − ϕ∗ kL6 (Ω) kSn − SkL6 (Ω) → 0 at n → ∞, m = 1, 2, ...
Then
lim (k(ϕm )ϕm , S) = (k ∗ (ϕ∗ )ϕ∗ , S).
m→∞
For the nonlinear summand (um · ∇ϕm , S) this relation is satisfied
(um · ∇ϕm , S) − (u∗ · ∇ϕ∗ , S) =
= (u∗ · ∇(ϕm − ϕ∗ ), S) + ((um − u∗ ) · ∇ϕm , S) ∀S ∈ H01 (Ω). (16)
∗ 2
On the strength of (12) a weak convergence takes place: ∇ϕm → ∇ϕ in L (Ω),
according to which
(u∗ · ∇(ϕm − ϕ∗ ), S) = (∇(ϕm − ϕ∗ ), u∗ S) → 0 at m → ∞ ∀S ∈ H01 (Ω),
and from (13) follows that
|((um −u∗ )·∇ϕm , S)| ≤ k∇ϕm kL2 (Ω) kum −ukL4 (Ω) kSkL4 (Ω) → 0 as m → ∞ ∀S ∈ H01 (Ω).
Then, taking (16) into account, it is obtained
lim (um · ∇ϕm , S) = (u∗ · ∇ϕ∗ , S).
m→∞
As the functional J is weakly semicontinuous below on H01 (Ω) × V, then from
aforesaid follows that
J ∗ = lim J(ϕm , um ) = limm→∞ J(ϕm , um ) ≥ J(ϕ∗ , u∗ ) ≥ J ∗ .
m→∞
3 Optimality systems
Further the case of k(ϕ) = ϕ2 |ϕ| will be considered and the principle of Lagrange
multipliers for the problem (11) will be justified.
Let’s introduce a Lagrange multiplier (λ0 , θ) ∈ R × H01 (Ω) and a Lagrangian L :
H0 (Ω) × V × R × H01 (Ω) → R by formula
1
L(ϕ, u, λ0 , θ) = λ0 J(ϕ, u) + hθ, F (ϕ, u)i ≡ λ0 J(ϕ, u) + hF (ϕ, u), θi. (17)
A common analysis shows that Frechet derivative of the operator F with respect to ϕ
in (10) for k = ϕ2 |ϕ| in the point (ϕ̂, û) ∈ H01 (Ω) × V is a linear continuous operator
Fϕ0 (ϕ̂, û) : H01 (Ω) → H −1 (Ω), which assigns to each element τ ∈ H01 (Ω) an element
ˆl ∈ H −1 (Ω), where
hˆl, Si = λ(∇τ, ∇S) + 4(ϕ̂2 |ϕ|τ, S) + (u · ∇τ, S).
Optimization in Nonlinear Models of Mass Transfer 159
From lemma 1 follows that the operator Fϕ0 (ϕ̂, û) is an isomorphism. Then according
to [18, 19] the theorem takes place:
Theorem 5. While conditions (i), (ii) and (j), (jj) hold, let (ϕ̂, û) ∈ H01 (Ω) × V be
an element, on which the local minimum is achieved in the problem (11) if k = ϕ2 |ϕ|.
Then there is a unique nonzero Lagrange multiplier (1, θ), where θ ∈ H01 (Ω), such as
Euler–Lagrange equation is satisfied
hJϕ0 (ϕ̂, û), τ i + hFϕ0 (ϕ̂, û)τ, θi = 0 ∀τ ∈ H01 (Ω), (18)
and is equivalent to
λ(∇τ, ∇θ) + 4(ϕ̂2 |ϕ|τ, θ) + (u · ∇τ, θ) = −µ0 (ϕ̂ − ϕd , τ )Q ∀τ ∈ H01 (Ω), (19)
and also the minimum principle is true:
hL0u (ϕ̂, û, 1, θ), u − ûi ≥ 0 ∀ u ∈ V,
which is equivalent to the inequality
µ1 (û, u − û)1 + ((u − û) · ∇ϕ̂, θ) ≥ 0 ∀u ∈ V. (20)
The relation (19) together with the variational inequality (20) and the operational
restriction (10), which is equivalent to the ratio (6), are the optimality system for the
problem (11) when k = ϕ2 |ϕ|.
4 Computations
Two different nonlinear boundary value problems were solved in cases of reaction coef-
fecients k(ϕ) = ϕ2 and k(ϕ) = |ϕ|. For both cases the exact solution ϕe = 0.2(x2 + y 2 )
was taken. Also, the same u = (0; 0), ϕ0 = 0.1(x + y), λ = 10. The function f was
simply calculated in each case by substituting the function ϕ in the equation by the
exact solution. The domain is [−1; 1] × [0; 1]. The relative error is obtained by formula
kϕ−ϕe k2
kϕe k2 .
FreeFem++ listing
border c1(t=-1,1){x = t; y = 0; label=1;};
border c2(t=0,1){x = 1; y = t;label=2; };
border c3(t=-1,1){x=t;y=1;label=3;};
border c4(t=0,1){x=-1;y=t;label=4;};
int n=3;
mesh Th = buildmesh(c1(20*n) + c2(10*n) + c3(-20*n) + c4(-10*n));
fespace Vh(Th,P2);
Vh u1, u2, phi, phi1, s, err0, phi0;
u1 = 0;
u2 = 0;
int lambda = 10;
160 R. Brizitskii, Zh. Saritskaya
Vh f=-lambda*0.8+(0.2*(x^2+y^2))^3;
Vh phiex = 0.2*(x^2+y^2);
phi0 = 0.1*(x+y);
plot(phi0, cmm="phi0", wait = true, value = 1, fill = 1);
problem equation1(phi1,s)=
int2d(Th)(lambda*(dx(phi1)*dx(s) + dy(phi1)*dy(s)))
+ int2d(Th)(phi0^2*phi1*s)
+ int2d(Th)((u1*dx(phi1) + u2*dy(phi1))*s)
- int2d(Th)(f*s)
+ on(1,phi1=0.2*x^2)
+ on(2,phi1=0.2+0.2*y^2)
+ on(3,phi1=0.2+0.2*x^2)
+ on(4,phi1=0.2+0.2*y^2);
real E0, E01, L2err0;
int i;
for (i=0;i<=20;i++){
equation1;
err0=phi1-phiex;
E0 = sqrt(int2d(Th)(err0^2));
E01 = sqrt(int2d(Th)(phiex^2));
L2err0 = E0/E01;
cout <<"L2err0 = "< 0 and the fact that the set K ⊂ V is convex and closed,
we can introduce the projection operator P : V → K. Then the variational inequality
(24) is equivalent to the equation uk = P (ϕk−1 θk−1 /µ1 ), k ≥ 0.
The result about this algorithm’s convergence was obtained similarly to the [20].
Acknowledgments. This work was supported by the Russian Foundation for Basic
Research (project no. 16-01-00365) and Ministry of Education and Science of Russian
Federation (contract no. 14.Y26.31.0003).
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