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Title: | The Block-Hexagonal Grid Method for Laplace’s Equation with Singularities |
Authors: | Dosiyev, Adıgüzel Çeliker, Emine Eastern Mediterranean University, Faculty of Arts and Science, Department of Mathematics |
Keywords: | Mathematics Applied Mathematics and Computer Science Hexagonal grids Laplace’s equation singularity problem block-grid method |
Issue Date: | Dec-2014 |
Publisher: | Eastern Mediterranean University (EMU) - Doğu Akdeniz Üniversitesi (DAÜ) |
Citation: | Çeliker, Emine. (2014).The Block-Hexagonal Grid Method for Laplace’s Equation with Singularities. Thesis (Ph.D.), Eastern Mediterranean University, Institute of Graduate Studies and Research, Dept. of Mathematics, Famagusta: North Cyprus. |
Abstract: | A fourth order accurate matching operator is constructed on a hexagonal grid, for the
interpolation of the mixed boundary value problem of Laplace’s equation, by using
the harmonic properties of the solution. With the application of this matching operator
for the connection of the subsystems, the Block-Grid method (BGM), which is
a difference-analytical method, has been analysed on a hexagonal grid, for the solution
of both the Dirichlet and mixed boundary value problems of Laplace’s equation
with singularities. First of all, BGM is considered on staircase polygons and it is justified
that when the boundary functions outside the finite neighbourhood of the singular
points are from the Hölder classes C6;l ; 0 < l < 1; the error of approximation has
an accuracy of O
���
h4
; where h is the mesh size. The analysis of this method is extended
to special polygons whose interior angles are a jp; a j 2
1
3 ; 2
3 ;1;2
; and for
the Dirichlet problem of Laplace’s equation it is proved that, with the application of
BGM, it is possible to lower the smoothness requirement on the boundary functions
to C4;l ; 0 < l < 1; outside the finite neighbourhood of the singular points, in order
to obtain an accuracy of O
���
h4
. For the demonstration of the theoretical results on
staircase polygons, BGM has been applied on an L-shaped domain for two examples,
which has a singularity at the vertex with an interior angle of 3p
2 ; where Dirichlet and
mixed boundary conditions are assumed respectively. The slit problem, which has the
strongest singularity due to the interior angle of 2p at the vertex of the slit, has been
considered on a parallelogram with a slit, in order to illustrate the results obtained on
polygons with interior angles of a jp; a j 2
1
3 ; 2
3 ;1;2
: The second example on a parallelogram
demonstrates the application of BGM on a domain with two singularities as
it is assumed that the vertices with interior angles of 2p
3 are singular points. Solutions
of the numerical examples are consistent with the theoretical results obtained.
Keywords: Hexagonal grids, Laplace’s equation, singularity problem, block-grid method. ÖZ
Laplace denklemi sınır problemleri için, dördüncü derece hata payı olan birless¸tirme (matching)
operatörü petek düg˘ ümleri üzerinde kurulmuss¸tur. Bu enterpolasyon oper- atörünün kurulumu için
çözümün harmonik özellikleri kullanılmıss¸tır. Alt sistemlerin birless¸tirilmesinde uygulanan
matching operatörü ile Block-Grid metodu (BGM), petek ag˘ lar üzerinde analiz edilmiss¸tir. Bu
metod, tekillig˘ i olan Laplace denkleminin Dirich- let ve karıss¸ık (mixed) sınır problemlerine
uygulanmıss¸tır.
˙Ilk önce BGM, iç açıları α j π , α j ∈ { 1 , 1, 3 , 21 olan çokgenler üzerinde incelenmiss¸tir.
2 2
Tekil noktalardan belli bir uzaklıkta olan sınır üzerindeki fonksiyonlar C6,λ , 0 < λ < 1,
Hölder gruplarından oldug˘ u zaman yakınsaklık hatasının O (h4) oldug˘ u kanıtlanmıs¸tır
(h ag˘ aralıg˘ ıdır).
˙Il˙lalaveten, BGM’nin analizi özel çokgenler üzerine genisss¸letilmisss¸tir. Bu özel çokgenlerin
iç açıları α
j
j , α
j ∈
{
{ 3 , 2 , 1, 2
21
iç açıları α j , α j ∈ { 3 , 2 , 1, 21 , olarak verilmisssssssssss¸tir. Laplace’ın Dirichlet
probleminin
yaklasssssssssss¸ık çözümü için, bu çokgenler üzerinde, tekil noktalardan belli bir uzaklıkta olan
sınır fonksiyonlarının C4,λ , 0 < λ < 1, Hölder grubundan olması ve BGM metodunun
uygulanması ile hata payının yine O (h4) oldug˘ u kanıtlanmısssssssssss¸tır.
Teorik sonuçların nümerik çözümlemesi için BGM, iç açılarından biri 3π
olan L-
s¸ekilli (L-shaped) çokgende uygulanmıs¸tır. Açıları α j π , α j ∈ { 1 , 2 , 1, 21 , olan çok-
3 3
genler üzerinde BGM’nin uygulanmasını göstermek üzere, iç açısı 2π oldug˘ undan dolayı en güçlü
tekillig˘ e sahip olan kesik problemi (slit problem), paralelkenar üz- erinde çözülmüs¸tür. Yine
paralelkenar üzerinde, 2π iç açılı kenarların ikisinde de tekil- lik oldug˘ u varsayılarak BGM ile
Laplace sınır problemi çözümlenmis¸tir. Elde edilen
v
sayısal çözümlerin teorik sonuçlarla uyumlu oldug˘ u sergilenmis¸tir.
Anahtar Kelimeler: Laplace denklemi, tekil problemi, Block-Grid metodu, petek
ag˘ lar. |
Description: | Doctor of Philosophy in Applied Mathematics and Computer Science. Thesis (Ph.D.)--Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics, 2014. Supervisor: Prof. Dr. Adıgüzel Dosiyev. |
URI: | http://hdl.handle.net/11129/3245 |
Appears in Collections: | Theses (Master's and Ph.D) – Mathematics
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