Abstract: - This work is devoted to the reﬁnement of the approximate solution, obtained by the pro-

jection method. The proposed approach uses expanding the design space by adding new coordinate

functions. As a result, it is possible to clarify previously obtained solution using small computer re-

sources. Applying this approach to the ﬁnite element method allows produce a local reﬁnement of the

mentioned solution. Suggested Approach illustrated in the ﬁnite element method for a boundary value

problem second order in one-dimensional and two-dimensional cases.

Key-Words: - a posteriori improvement, reﬁnement calculations, variational-grid methods

1 Introduction

The research in the ﬁeld of artiﬁcial intelligence

covers many areas of the development of science

and practice. These studies give opportunities to

optimize eﬀorts aimed at the eﬀective solution of

diﬃcult problems. Here we give examples of some

studies in the mentioned area.

Paper, [1], predicts the future of excitation

energy transfer with artiﬁcial intelligence-based

quantum dynamics.

In the paper, [2], the authors propose algo-

rithm with fast convergence rate for federated

edge learning.

In paper, [3], the authors propose a new algo-

rithm of machine learning techniques. Based on

recent advancements in music structure analysis,

the authors of paper, [4], automate the evaluation

process by introducing a collection of metrics.

The development of a disagreement-based on-

line learning algorithm is given in [5].

In [6], the authors develop a deep learning-

based approach to model.

Let us mention some works in which itis possi-

ble to eﬀectively use learning systems and means

of artiﬁcial intelligence.

The paper, [7], is devoted to the solution of the

nonstationary integro-diﬀerential equation with a

degenerate elliptic diﬀerential operator.

It seems that adaptive methods developed

with the help of artiﬁcial intelligence greatly sim-

plify the solution of problems of mathematical

physics.

The article, [8], investigate the approxi-

mate solution to a nonlinear Volterra integro-

diﬀerential equation.

In the article, [9], the authors have proposed a

highly eﬃcient and accurate collocation method.

It can also be assumed that the mentioned

means would be useful in solving the problems

considered in the works, [10], [11].

This paper is devoted to the posteriori im-

provement of the approximation in the projec-

tion method for solving a linear equation with a

self-adjoint positive deﬁnite operator. Improve-

ment was obtained by expanding the projection

space. The mentioned extension is a linear span,

an original projection space and an added element

energy space. The consideration of a given priori

parameterized class of elements that allow you to

build an adaptive method with the mentioned im-

provement. The proposed approach is applied to

the methods ﬁnite elements for one-dimensional

boundary second order tasks.

The consistent application of this approach al-

lows us to optimize the process of reﬁning the nu-

merical solution of the boundary value problem

without signiﬁcantly increasing the requirements

for computing system resources. The proposed

approach leads to the construction of adaptive

sequences of ambient spaces. These spaces form

a telescopic system of nested spaces, on the basis

of which an adaptive wavelet packet is built for

the economical transmission of information.

2 Background

Consider a Hilbert space Hwith a scalar prod-

uct (u, v). Let Abe a positively self-adjoint de-

ﬁned operator with domain D(A). Consider the

energy space HAof the operator A. We denote

the norm and the scalar product in the space YA

IuIand [u, v] respectively, so that IuI=p[u, u].

A Posteriori Improvement in Projection Method

YURI K. DEM'YANOVICH, IRINA G.BUROVA

Saint Petersburg State University Saint Petersburg, RUSSIA

Received: December 13, 2022. Revised: May 29, 2023. Accepted: June 21, 2023. Published: July 24, 2023.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

544

Volume 22, 2023

Here u, v ∈HA. In the discussed conditions the

solution of the problem

Au =f, f ∈H(1)

is equivalent to the solution of the next problem

min

u∈HA

F(u),(2)

F(u) = IuI2−2(u, f).(3)

Denote u∗the solution to problem (2) (and

hence also the solution to problem (1)). Let S

be a subspace of HA. Consider the solution eu∗to

the problem

min

u∈SF(eu).(4)

Thanks to the well-known representation

F(u) = Iu∗−uI2−Iu∗I2(5)

problem (4) is equivalent to problem

min

u∈S

Iu∗−euI,(6)

Iu∗−eu∗I≤Iu∗−euI∀eu∈S. (7)

It is clear from (4) that eu∗satisﬁes the identity

[eu∗,ev] = (f, ev)∀ev∈S. (8)

Consider the element ϕof the energy space

HA, assuming that ϕ /∈S. Let S1be the linear

span of the space Sand element ϕ,

def

=L{S, ϕ},(9)

where L{. . .}means the linear span of the ob-

jects, which are in curly brackets.

Consider the function

F(t)def

=F(eu∗+tϕ).(10)

Let us pose the question of ﬁnding the point

t∗of the minimum of this function,

min

t∈R1F(t).(11)

Since F(t) is a quadratic function and

limt→±∞ F(t)=+∞, a desired point exists and

it is unique. We have

F(eu∗+tϕ) = Ieu∗+tϕI2−2(eu∗+tϕ, f) =

=Ieu∗I2+ 2t[eu∗, ϕ] + t2IϕI2−2(eu∗, f)−2t(ϕ, f).

(12)

Lemma 1. Representation

F(t) = At2+ 2Bt +C, (13)

is right. Here

A=IϕI2, B = [eu∗, ϕ]−(eu∗, f ),

C=Ieu∗I2−2(eu∗, f).(14)

Proof. Formulas (13) – (14) follow from rela-

tions (3), (9), and (12).

Lemma 2. Relations

t∗=(f, ϕ)−[eu∗, ϕ]

IϕI2,(15)

F(t∗) = −{(f, ϕ)−[eu∗, ϕ]}2

IϕI2+F(eu∗).(16)

are fulﬁlled.

Proof. Obviously, the minimum point t∗of

quadratic form (13) has the form t∗=−B/A, and

the value of this quadratic form at the mentioned

point is equal to Φ(t∗) = −B2/A+C. Taking into

account formulas (14), we obtain the equalities

(15) and (16).

Theorem 1. A formula

F(eu∗+t∗ϕ) = F(eu∗)−{(f, ϕ)−[eu∗, ϕ]}2

IϕI2(17)

is correct.

Proof. Formula (17) is obtained from relations

(9) and (16).

Formula (17) means that for ϕ /∈Sthe energy

initial approximation eu∗can be reduced by the

transition to a new approximation eu∗+t∗ϕ. The

inequality

Ieu∗+t∗ϕ−u∗I2=Ieu∗−u∗I2−E, (18)

follows from (17). Here

Edef

={[eu∗, ϕ]−(f, ϕ)}2

IϕI2.(19)

The element ϕis called qualifying element,

and the number Eis local energetic clariﬁcation.

Remark 1. By selecting the element ϕ∈HA,

the energy approximation will be greatly im-

proved. Thus, it is necessary to ﬁnd

Mdef

= max

ϕ∈HA

{(f, ϕ)−[eu∗, ϕ]}2

IϕI2.

Due to the relation (f, ϕ)−[u∗, ϕ] we have

M= max

ϕ∈HA, ϕ6=0

{[u∗−eu∗, ϕ]}2

IϕI2.

It is clear that

M= max

ψ∈HA,IψI=1{[u∗−eu∗, ψ]}2.(20)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

545

Volume 22, 2023

It can now be seen that the maximum occurs at

ψ=ψ∗

def

=u∗−eu∗

Iu∗−eu∗I,

so M=Iu∗−eu∗I2. In this way, the energy ap-

proximation is maximally improved if, in formula

(17) takes ϕ=ϕ∗

def

=u∗−eu∗. According to for-

mula (15) for ϕ=ϕ∗we have

t∗=(f, ϕ∗)−[eu∗, ϕ∗]

IϕI2=[u∗−eu∗, ϕ∗]

Iϕ∗I2= 1,

for ϕ=ϕ∗.

From formulas (5) and (17) we obtain the min-

imum value −Iu∗I2of the functional F(u) in

space HA,F(eu∗+t∗ϕ∗) = −Iu∗I2and eu∗+t∗ϕ∗=

u∗. The result obtained indicates the logical in-

tegrity. The approach is under consideration.

3 Reﬁnement space

Consider in the energy space HAalso one ﬁnite-

dimensional subspace Φ with the properties Φ ∩

S={0}. Let ϕ1, . . . , ϕNbe the basis of this

space. Denote S1is the linear span of the spaces

Sand Φ

S1=L{S, Φ}.(21)

In what follows, tmeans a N-dimensional vec-

tor, t∈RN,t= (t1, t2, . . . , tN). The symbol ·

is used to denote the linear combination of ele-

ments ϕ1, . . . , ϕN,t·ϕ=PN

i=1 tiϕi. Consider

the function

F(t) = F(˜u∗+t·ϕ),(22)

where F(u) is the energy functional (3). Since

F(t) is a quadratic function with a positive deﬁ-

nite principal part, then there exists and uniquely

problem solution t∗

min

t∈RNF(t).(23)

We have

F(˜u∗+t·ϕ) =

=I˜u∗+t·ϕI2−2(˜u∗+t·ϕ, f) =

=I˜u∗I2−2(˜u∗, f)+2

i=1

{[˜u∗, ϕi]−(˜

f, ϕi)}ti+

i,j=1

[ϕi, ϕj]titj.

(24)

Consider matrix A0, vector B0and constant

C0deﬁned by equalities

A0= ([ϕi, ϕj])N

i,j=1, B0= ([˜u∗, ϕi]−(f, ϕi))N

i=1,

(25)

C0=I˜u∗I2−2(˜u∗, f).(26)

Lemma 3.

F(t)=(A0t, t) + 2(B0, t) + C0,(27)

where parentheses mean scalar preproduction

to RN.

Proof. Formula (27) follows from relations

(22) and (24) – (26).

Lemma 4.

t∗=A−1

0B0,F(t∗) = C0−(B0, A−1

0B0).(28)

Proof. From the positive deﬁniteness of the

quadratic form (27) follows the existence and

uniqueness points of minimum t∗. Equating to

zero the partial derivatives with respect to ti,

i= 1,2, . . . , N, we get t∗in the ﬁrst relation (28).

Substituting t∗into (27) gives the second of the

relation (28).

Theorem 2.

F(˜u∗+t∗·ϕ) = F(˜u∗)−(B0, A−1

0B0).(29)

Proof. Formula (29) is obtained from relations

(21) and (25) – (28).

Formula (29) means that under the conditions

considered, the energy initial approximation ˜u∗

can be reduced by transition to a new approxi-

mation ˜u∗+t∗·ϕ. From (29) follows the relation

I˜u∗+t∗·ϕ−u∗I2=I˜u∗−u∗I2−E, (30)

where

E= (B0, A−1

0B0),(31)

while the matrix A0and the vector B0are deﬁned

by relations (25).

The element t∗·ϕis called qualifying element,

and the number Eis local energetic clariﬁcation.

4 Triangulation and its reﬁnement

with stable boundary

We consider regular rectilinear triangulations of

polygonal domains on the plane R2(otherwise

called triangular sets). Triangulation is consid-

ered regular if each edge is a side of some trian-

gle, and the end of the edge cannot be an interior

point of another rib. Proper triangulation allows

for reﬁnement, which are again regular triangula-

tions.

Let Mbe a closed polygonal domain of the

plane R2. We denote its boundary by ∂M. Con-

sider some triangulation Tof M. Denote the

s-skeleton of the triangulation Tby Ts,s∈

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

546

Volume 22, 2023

{0,1,2}(i.e. T0is the set of vertices, T1is the

set of edges, and T2is the set of triangles). We

will assume that T0={vi|i= 1,2, . . . , K},T1=

{ls|s= 1,2, . . . , L},T2={Tj|j= 1,2, . . . , N}.

Triangulation Tis also called a two-

dimensional (triangular) complex. A set-

theoretic closure union of points of triangles in the

triangular complex Tis called the body of the tri-

angular complex. The triangular body complex

is denoted by |T |. It matches with area M. The

triangular body border complex Tis the set ∂M.

Sets ∂T0def

={vi|vi∈ T 0, vi⊂∂M},∂T1def

{ls|ls∈ T 1, ls⊂∂M} with induced incidence

relations generate a one-dimensional complex ∂T

subdividing the boundary ∂Mof the region un-

der consideration M. Complex ∂Tis called the

induced boundary complex. Its body coincides

with the boundary ∂Mof the domain M.

Reﬁning Ttriangulation Tis a new simplicial

complex. If at the same time the induced bound-

ary complex is preserved,

∂T=∂T,(32)

then the triangulation of Tis called reﬁnement

with a stable boundary for the triangulation T.

In what follows, we associate the vertex viwith

the vector ri, emanating from the origin and end-

ing at the mentioned vertex. In the case under

consideration, a topological triangulation struc-

ture is described by an incidence matrix MTof

the size 3 ×N, whose i-th line contains the ver-

tices of the i-th triangle.

Remark 3. Changing the order of matrix

rows MTmatches the renumbered triangles, and

changing the order of the elements in any row

of the matrix does not change the incidence cor-

respondences between the triangle and vertices.

Thus, a matrix derived from the original one by

the mentioned transformations describes the old

topological structure of the triangulation. We

also note, that the duplication of the rows in the

incidence matrix does not violate the topologi-

cal structure of triangulation (because the ”mul-

tiplicity” triangles are not considered).

The barycentric star of vertex r∈ T 0is de-

noted by Z(r). The closure for the union of the

points of its triangles is called the body of the

barycentric star. The body of a barycentric star

is denoted by |Z(r)|.

Two triangles are considered adjacent if they

have a common edge. We will assume that each

triangle Tof the triangulation under consider-

ation has sneighboring triangles, where s∈

{1,2,3}. If s= 3 then triangle Tis called the

inner triangle of the triangulation T, and for

s∈ {1,2}the triangle Tis called boundary tri-

angle.

The set of neighboring triangles for Twith the

addition of the given triangle T, we call the a

star triangle T. The star of the triangle Twill

be denoted by Z(T). The set-theoretic union tri-

angles from Z(T) is called the body of the star

Z(T), and is denoted by |Z(T)|. Note that both

the barycentric vertex star and the star triangles

are regular triangulations (triangular complexes).

Therefore, it is possible to reﬁne these triangula-

tions.

Consider an interior triangle Twith vertices

r1,r2,r3and its star Z(T). Select the fragment

MZ(T)of the matrix MTrelated to to the star

Z(T). In view of Remark 2 above, without loss

of generality, we have

MZ(T)= r1r1r2r1

r2r2r3r3

r3r5r6r4!T

.(33)

Consider vector functions of the variable ξ∈[0,1]

r12(ξ)def

=ξr1+ (1 −ξ)r2,r23(ξ)def

=ξr2+ (1 −ξ)r3,

(34)

r32(ξ)def

=ξr3+ (1 −ξ)r1.(35)

Theorem 3. If Tis an interior triangle

with vertices r1,r2,r3, then for a ﬁxed parameter

ξ∈(0,1) triangulation T(ξ)deﬁned by incidence

matrix

M(ξ)

def

= (V1, V2, V3, V4, V5, V6, V7, V8, V9, V10)T,

(36)

where

V1= r12(ξ)

r23(ξ)

r31 !, V2= r2

r23(ξ)

r12(ξ)!,

V3= r3

r31(ξ)

r23(ξ)!, V4= r1

r12(ξ)

r31(ξ)!,

V5= r1

r12(ξ)!, V6= r12(ξ)

r2!,

V7= r2

r23(ξ)!, V8= r3

r23(ξ)

r6!,

V9= r3

r31(ξ)!, V10 = r1

r31(ξ)

r4!.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

547

Volume 22, 2023

M(ξ)is a reﬁnement with a stable boundary for

triangulation Z(T). In addition, the relations

lim

ξ→+0 T(ξ)= lim

ξ→1−0T(ξ)=Z(T).(37)

Proof. For ﬁxed ξ∈(0,1) formulas (34) – (35)

deﬁne points lying inside the edges (sides) of tri-

angle T. Taking into account the structure of the

matrix (36), we arrive at the conclusion that the

simplicial complex T(ξ)is the subdivision of the

star Z(T). Relation (32) is obvious, and therefore

the subdivision T(ξ)is the reﬁnement with a sta-

ble boundary for Z(T). Using Remark 1 and the

formulas r12(0) = r2,r12(1) = r1,r23(0) = r3,

r23(1) = r2,r32(0) = r2,r32(1) = r3, we derive

at relations (37).

Remark 3. Similar assertions hold for bound-

ary triangles.

5 Energy reﬁnement for the ﬁnite

elements in a two-dimensional

problem

Triangulation reﬁnement while maintaining less

correctness can be obtained by the successive

application of elementary grinding. Elementary

grinding is the grinding in which one vertex with

corresponding edges is added. Two types of ele-

mentary grindings exist.

The ﬁrst type of elementary grinding is as fol-

lows. A pair of adjacent triangles with vertices r1,

r2,r3,r4after adding rξto the edge (r1,r3) and

after adding edges (r2,rξ) and (r4,rξ) becomes

four triangles according to display

r1r1

r2r3

r3r4!T

−→ r1rξrξr1

r2r2r3rξ

rξr3r4r4!T

Let us describe the second type of elementary

grinding. In this case the vertex rξis placed in-

side the triangle and is connected to the edges

with its vertices r1,r2,r3according to the map-

ping

(r1r2r2)−→ r1r2rξ

rξr2r3

r1rξr3!.

Both types of elemental reﬁnements retain tri-

angulation correctness.

Let Ωr0(r) be the Courant function for the ver-

tex r0= (x0, y0),

Ωr0(r) = 









1−∆T(r)/∆0

for r∈ |T|,∀T∈ Z(r0),r0∈ |T|,

for r/∈ |Z(r0)|.

Here r0= (x0, y0), r1= (x1, y1), r2= (x2, y2) are

vertices of triangle T,r= (x, y),

∆T(r) = x−x0y−y0

x2−x1y2−y1,

∆0

T=x1−x0y1−y0

x2−x0y2−y0.

We associate each vertex rifrom the zero-

dimensional spanning tree T0with the Courant

function Ωri(r). These functions are a linear inde-

pendent system of functions. Consider the space

Sof the linear combinations of these functions,

Sdef

={eu|eu=X

ri∈T 0

viΩri, vi∈R1}.

{Ωri}ri∈T 0system we call the standard basis

of Courant spaces.

Elementary triangulation reﬁnement adds ver-

tex rξin triangulation, which allows us to con-

sider a new function Ωrξchime corresponding to

the added top. It is clear that the added function

is linearly independent with the previous ones.

Consider the linear span S1of the functions

{Ωri}ri∈T 0and Ωrξ. It is clear that the result

will be the extension of the space S,S ⊂ S1.

It is easy to see that the set of the features just

listed are not the standard Courant basis for the

space S1. To go to the standard basis, you need

to take into account the change structure of the

barycentric stars, which is local in nature.

Consider an elementary reﬁnement of trian-

gulation. Let us add a new vertex rξ(together

with the corresponding edges). We introduce the

function $rξchime corresponding to the added

vertex. It is clear that this function is linearly

independent with respect to system {$ri}ri∈T 0.

Considering the linear span of the set of functions

{$ri}ri∈T 0∪ {$rξ}, we obtain a linear space S1

containing the space S,S ⊂ S1.

It is easy to see that the set of functions

{$ri}ri∈T 0∪ {$rξ}are not the standard Courant

basis for the space S1. The transition to the stan-

dard basis is obtained by a linear transformation

that takes into account the change structures of

the barycentric stars. The said transition is lo-

cal in the sense that it aﬀects only those Courant

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

548

Volume 22, 2023

functions whose support has a non-empty inter-

section with the support of the added function.

Later on, the transition to a standard basis is not

required.

6 Energy reﬁnement for the ﬁnite

elements in a two-dimensional

problem

Consider the quadratic functional

F(u)def

=ZΩ2

i,j=1

pij (x)∂u

∂xi

∂u

∂xj

+q(x)u2−2u(x)f(x)dx,

(38)

where x= (x1, x2), q(x)>0, aij (x) = aji(x),

i,j=1

pij (x)ζiζj≥γX

i∈{1,2}

ζ2

i, γ =const > 0

∀ζs∈R1, s = 1,2.

(39)

The functions pij (x), q(x) are measurable and

bounded. Functional F(u) is deﬁned on the space

2(Ω). As is known, the solution of a number

of boundary value problems for diﬀerential equa-

tions

−

i,j=1

∂

∂xipij (x)∂u

∂xj+q(x)u=f(x), x ∈Ω

(40)

is reduced to the minimization of the functional

on the corresponding linear manifold of the space

2(Ω). Its subspace is often called the energy

space and is denoted by HA, and the bilinear form

[u, v]def

=ZΩ2

i,j=1

pij (x)∂u

∂xi

∂v

∂xj

+q(x)uvdx.

(41)

turns out to be an inner product in HA. Denote

u∗the solution of the problem

min

u∈HA

F(u).(42)

Let Sbe a subspace in HA,PSbe orthopro-

jector of the space HAonto the subspace S, and

eu∗

def

=PSu∗(43)

be the approximate solution of problem (39).

Let us assume that the next condition is ful-

ﬁlled (A). The closure of Ω contains a polygonal

domain Mwith triangulation T. The approx-

imate solution (42) of problem (38) on polygon

Mis a linear combination Courant functions,

eu∗(x)x∈M=X

ri∈T 0

v∗

i$ri(r).(44)

Under condition (A) we have

F(eu∗) = ZΩ\M

G(x;eu∗(x))dx+

T∈T 2Z|T|

G(x;eu∗(x))dx,

(45)

where

G(x;u) =

i,j=1

pij (x)∂u

∂xi

∂u

∂xj

+q(x)u2−2f(x)u.

(46)

Let T1and T2be two neighboring triangles,

T1, T2∈ T 2deﬁned by the matrix

r1r2r3

r1r3r4.(47)

Let us perform an elementary reﬁnement (5.1).

Consider the linear span of the space Sand the

coordinate span of the Courant function $rξ(r).

According to formulas (25) we have N= 1

A0= [ϕrξ, ϕrξ], B0= [˜u∗, ϕrξ]−(f, ϕrξ),(48)

so that the decrease in energy in this case has the

form

E=|[˜u∗, ϕrξ]−(f, ϕrξ)|2

[ϕrξ, ϕrξ].

In this case we have

$rξ(r) = 









∆e

Ti,rξ(r)/∆0

Ti,rξ

for r∈ | e

Ti|, i = 1,2,3,4,

for r/∈ |Z(rξ)|,

(49)

where

rξ= (xξ, yξ),ri= (xi, yi),ri+1 = (xi+1, yi+1)

are vertices of the triangle e

Ti, and r= (x, y),

∆e

Ti,rξ(r) = xi−x yi−y

xi+1 −xiyi+1 −yi,

∆0

Ti,rξ=xi−xξyi−yξ

xi+1 −xξyi+1 −yξ.

(50)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

549

Volume 22, 2023

Hence, for (x, y)∈ | e

Ti|,i= 1,2,3,4, we get

∂$rξ

∂x =−yi+1 −yi

∆0

Ti,rξ

∂$rξ

∂y =xi+1 −xi

∆0

Ti,rξ

∆0

Ti,rξ=xi−xξyi−yξ

xi+1 −xiyi+1 −yi.

(51)

According to formulas (25) we have N= 1

A0= [$rξ, $rξ], B0= [˜u∗, $rξ]−(f, $rξ),(52)

so that the decrease in energy in this case has the

form

E=|[˜u∗, $rξ]−(f, $rξ)|2

[$rξ, $rξ].

Our task is to evaluate the expressions

[$rξ, $rξ],[˜u∗, $rξ],(f, $rξ).

For brevity, we conﬁne ourselves to the case

q(x, y)(x,y)∈|T1|∪|T2|= 0.

So we have

[$rξ, $rξ] =

i=1

4mes2|e

Ti|Z|

Ti|np11(yi+1 −yi)2+

+p22(xi+1 −xi)2odxdy.

(53)

For r∈ |Tj|,j= 1,2, we have

$r1(r) = ∆Tj,r1(r)/∆0

Tj,r1,(54)

where r1= (x1, y1), rj+1 = (xj+1, yj+1), rj+2 =

(xj+2, yj+2) are vertices of triangle Tj,j= 1,2,

r= (x, y),

∆Tj,r1(r) = xj+1 −x yj+1 −y

xj+2 −xj+1 yj+2 −yj+1,

∆0

Tj,r1=xj+1 −x1yj+1 −y1

xj+2 −x1yj+2 −y1.

(55)

Therefore, for r∈ |Tj|,j= 1,2, we get

∂$r1

∂x (r) = −(yj+2 −yj+1)/∆0

Tj,r1,

∂$r1

∂y (r)=(xj+2 −xj+1)/∆0

Tj,r1.

(56)

Similarly for r∈ |T1|, we have

$r2(r) = ∆T1,r2(r)/∆0

T1,r2.(57)

Let r1= (x1, y1), r2= (x2, y2), r3= (x3, y3)

be vertex of the triangle T1,r= (x, y), where

r1= (x1, y1), r2= (x2, y2), r3= (x3, y3) are

vertices of triangle T1,r= (x, y),

∆T1,r2(r) = x−x1y−y1

x3−x1y3−y1,

∆0

T1,r2=x2−x1y2−y1

x3−x2y3−y2.

(58)

Therefore, for r∈ |T1|, we get

∂$r2

∂x (r)=(y3−y1)/∆0

T1,r2,

∂$r2

∂y (r) = −(x3−x1)/∆0

T1,r2.

(59)

For r∈ |Tj|,j= 1,2, we have

$r3(r)=∆Tj,r3(r)/∆0

Tj,r3,(60)

where r1= (x1, y1), r2= (x2, y2), r3= (x3, y3)

are vertices of T1,r1= (x1, y1), r3= (x3, y3),

r4= (x4, y4) are vertices of T2,r= (x, y),

∆T1,r3(r) = x1−x y1−y

x2−x1y2−y1,

∆0

T1,r3=x2−x1y2−y1

x3−x2y3−y2,

(61)

∆T2,r3(r) = x−x1y−y1

x4−x1y4−y1,

∆0

T2,r3=x3−x1y3−y1

x4−x3y4−y3.

(62)

Therefore, for r∈ |T1|we get

∂$r3

∂x (r) = −(y2−y1)/∆0

T1,r3,

∂$r3

∂y (r) = (x2−x1)/∆0

T1,r3,

(63)

and for r∈ |T2|we ﬁnd

∂$r3

∂x (r)=(y4−y1)/∆0

T2,r3,

∂$r3

∂y (r) = −(x4−x1)/∆0

T2,r3.

(64)

Similarly for r∈ |T2|, we have

$r4(r) = ∆T2,r4(r)/∆0

T2,r4,(65)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

550

Volume 22, 2023

where r1= (x1, y1), r3= (x3, y3), r4= (x4, y4)

are vertices of triangle T2,r= (x, y),

∆T2,r4(r) = x1−x y1−y

x3−x1y3−y1,

∆0

T2,r4=x3−x1y3−y1

x4−x3y4−y3.

(66)

Therefore, for r∈ |T2|, we get

∂$r4

∂x (r) = −(y3−y1)/∆0

T2,r4,

∂$r4

∂y (r)=(x3−x1)/∆0

T2,r4.

(67)

Let as proceed to the calculation of the expres-

sion [˜u∗, $rξ]. We have

[˜u∗, $rξ] =

j=1

vj[$rj, $rξ],(68)

[$rj, $rξ] =

i=1

[$rj, $rξ]e

Ti.(69)

Let us introduce the notation

Aji(ξ)def

= [$rj, $rξ]e

Ti,

ep(i)

def

=Z|

Ti|

pkkdxdy,

xij

def

=xi−xj,

yij

def

=yi−yj.

From relations (68) – (69) we have

[˜u∗, $rξ] =

j=1

i=1

Aji(ξ).(70)

Using formulas (51), (56), (59), (63) – (64), (67),

we successively ﬁnd

A11(ξ)=(mT1me

T1)−1[y32y21 ep(1)

1+x32x21 ep(1)

2],

A12(ξ)=(mT1me

T2)−1[y2

32 ep(2)

1+x2

32 ep(2)

2],

A13(ξ)=(mT2me

T3)−1[y2

43 ep(3)

1+x2

43 ep(3)

2],

A14(ξ)=(mT2me

T4)−1[y43y14 ep(4)

1+x43x14 ep(4)

2],

A21(ξ)=(mT1me

T1)−1[y13y21 ep(1)

1+x13x21 ep(1)

2],

A22(ξ)=(mT1me

T2)−1[y13y32 ep(2)

1+x13x32 ep(2)

2],

A23(ξ) = A24(ξ) = 0.

7 Conclusion

A reﬁnement of the approximate solution in the

projection method is usually obtained by an or-

thogonal projection in HAonto a wider space.

This requires signiﬁcant computer resources. The

approach proposed in this paper leads to signif-

icant resource savings due to the use of a re-

ﬁned approximate solution. The peculiarity of

this work lies in a fairly simple clariﬁcation of

an obtained solution in a small subdomain of the

considered area.

This paper also gives an energy estimate clar-

iﬁcation. This paper is devoted to the posteriori

improvement of the approximation in the projec-

tion method for solving a linear equation with

self-adjoint positive deﬁnite operator. Improve-

ment was obtained by expanding of the projec-

tion space.

The mentioned expanding is a linear span for

original projection space and added element of

energy space. Consideration of a priori given

parameterized class of such elements allows the

construction of an adaptive method for the men-

tioned improvement. In this work, the proposed

approach is applied to the method ﬁnite elements

for a two-dimensional boundary value problem of

the second order. The use of this approach allows

you to optimize the process reﬁnement of the nu-

merical solution of the boundary value problem

without signiﬁcant increase in computation time

and resource requirements computing system.

References:

[1] Ullah, A., Dral, P. O., Predicting the

future of excitation energy transfer in

light-harvesting complex with artiﬁcial

intelligence-based quantum dynamics,

Communications, 13 (1), 2022, pp. 1930.

[2] Dinh, C. T., Tran, N. H., Nguyen, T. D.,

Zhou, B. B., Zomaya, A. Y., DONE: Dis-

tributed Approximate Newton-type Method

for Federated Edge Learning, IEEE Trans-

actions on Parallel and Distributed Systems,

33 (11), 2022, pp. 2648-2660.

[3] Jiao, Y., Gu, Y., Communication-Eﬃcient

Decentralized Subspace Estimation, IEEE

Journal on Selected Topics in Signal Process-

ing, Vol. 16 (3), 2022, pp. 516-531.

[4] De Berardinis, J., Cangelosi, A., Coutinho,

E., Measuring the Structural Complexity

of Music: From Structural Segmentations

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

551

Volume 22, 2023

to the Automatic Evaluation of Models for

Music Generation, IEEE/ACM Transactions

on Audio Speech and Language Processing,

Vol. 30, 2022, pp. 1963-1976.

[5] Huang, B., Salgia, S., Zhao, Q.,

Disagreement-Based Active Learning in

Online Settings, IEEE Transactions on

Signal Processing, Vol. 70, 2022, pp.

1947-1958.

[6] Rahman, M. H., Xie, C., Sha, Z., Pre-

dicting sequential design decisions using the

function-behavior-structure design process

model and recurrent neural networks, Jour-

nal of Mechanical Design, Transactions of

the ASME, Vol. 143 (8), 2022, Art. number

081706.

[7] Cern, D., Wavelet Method for Sensitivity

Analysis of European Options under Mer-

ton Jump-Diﬀusion Model, AIP Conference

Proceedings, Vol. 2425, 2022, Art. number

110004.

[8] Erfanian, M., Zeidabadi, H., Solving of Non-

linear Volterra Integro-Diﬀerential Equa-

tions in the Complex Plane with Periodic

Quasiwavelets, International Journal of Ap-

plied and Computational Mathematics, Vol.

7 (6), 2021, Art. number 221.

[9] Priyadarshi, G., Rathish Kumar, B. V., Pa-

rameter identiﬁcation in multidimensional

hyperbolic partial diﬀerential equations us-

ing wavelet collocation method, Mathemat-

ical Methods in the Applied Sciences, Vol.

44 (11), 2021, pp. 9079-9095.

[10] Burova, I. G., Application Local Polynomial

and Non-polynomial Splines of the Third Or-

der of Approximation for the Construction of

the Numerical Solution of the Volterra Inte-

gral Equation of the Second Kind, WSEAS

Transactions on Mathematics, Vol. 20, 2021,

pp. 9-23.

[11] Menshov, E. N., Approximation of Fami-

lies of Characteristics of Energy Objects in

the Class of Linear Diﬀerential Equations,

WSEAS Transactions on Computers, Vol.

20, 2021, pp. 239-246.

Contribution of Individual Authors to the

Creation of a Scientiﬁc Article (Ghostwriting

Policy)

The authors contributed in the present re-

search, at all stages from the formulation of

the problem to the ﬁnal ﬁndings and solution.

Sources of Funding for Research Presented in a

Scientiﬁc Article or Scientiﬁc Article Itself

The authors are highly gratefully and indebted to

St. Petersburg University for ﬁnancial supporting

the preparation of this paper (Pure ID 94029567).

Conﬂicts of Interest

The authors have no conﬂict of interest to declare

that is relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.60

Yuri K. Dem'yanovich, Irina G. Burova

E-ISSN: 2224-2880

552

Volume 22, 2023