Advanced Engineering Research (Jun 2014)

NUMERICAL SOLUTION TO BOUNDARY PROBLEMS FOR POISSON EQUATION BY POINTSOURCE METHOD

  • Sergey Yuryevich Knyazev,
  • Elena Evgenyevna Shcherbakova,
  • Alesha Aleksandrovich Yengibaryan

DOI
https://doi.org/10.12737/4543
Journal volume & issue
Vol. 14, no. 2
pp. 15 – 20

Abstract

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The aim o f t his p a p e r is t h e e fficie n c y im p r o v e m e n t o f o n e o f t h e m o s t a d v a n c e d t e c h niq u e s o f s olvin g t h e ellip tic b o u n d a r y v alu e p r o ble m s — t h e field p oin t- s o u r c e m e t h o d d e sig n a t e d a s t h e f u n d a m e n t al s olu tio n t e c h niq u e in t h e f o r eig n lit e r a t u r e. N o w it is u s e d p rim a rily f o r s olvin g L a pla c e e q u a tio n . S e v e r al alt e r n a t e n u m e ric al s olu tio n s t o t h e b o u n d a r y v alu e p r o ble m s f o r P ois s o n e q u a tio n u sin g t h e field p oin t - s o u r c e m e t h o d a r e p r o p o s e d . T his m e t h o d a p plic a tio n t o t h e n o n h o m o g e n e o u s e q u a tio n s olu tio n, s u c h a s P ois s o n e q u a tio n, in m o s t c a s e s le a d s t o t h e d r a m a tic in c r e a s e o f t h e n u m e ric al e r r o r d u e t o mis t a k e s in P ois s o n e q u a tio n s p e cific s olu tio n. T h e rig h t m e m b e r o f P ois s o n e q u a tio n is a p p r o xim a t e d b y a t w o - dim e n sio n al trig o n o m e tric p oly n o mial (in t h e s olu tio n o f t w o - dim e n sio n al b o u n d a r y v alu e p r o ble m s ), t h e n it b e c o m e s p o s sible t o o b t ain t h e s p e cific s olu tio n n e c e s s a r y fo r s olvin g a n initial b o u n d a r y v alu e p r o ble m b y t h e field p oin t- s o u r c e m e t h o d. T h e t e s tin g r e s ult s o f t h e p r o p o s e d t e c h niq u e im ply it s e fficie n c y , a s t h e y allo w o b t ainin g t h e s olu tio n wit h a r ela tiv e e r r o r o f 1 0 − 6 a t minim u m m a c hin e tim e s p e n din g . T h e d e v elo p e d t e c h niq u e o f t h e n u m e ric al s olu tio n t o t h e b o u n d a r y v alu e p r o ble m s fo r P ois s o n e q u a tio n c a n b e u s e d f o r m o d elin g p h y sic al field s in t h e e n gin e e rin g d e vic e s o f v a rio u s a p plic a tio n s . K e y w o r d s : P ois s o n e q u a tio n , ellip tic b o u n d a r y v alu e p r o ble m s , field p oin t- s o u r c e m e t h o d, m e t h o d o f f u n d a m e n t al s olu tio n s.

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