Multigrid metoda za numeričko rješavanje Poissonove jednadžbe

master's thesis

• Author: Ivan Lužanac
• Mentor: Nela Bosner (mentor)
• Committee member: Nela Bosner (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Zvonimir Bujanović (član povjerenstva)
• Committee member: Nikola Sarapa (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Osnovna ideja ovoga rada nam je bila Multigrid metoda, tj. njezina usporedba sa ostalim iterativnim metodama. Testiranje metoda smo provodili isključivo koristeći Poissonovu jednadžbu sa homogenim rubnim uvjetom, premda to nije bilo nužno. Također, koristili smo legitimnu pretpostavku da nam je promatrani prostor dimenzije 1D, zbog složenosti postavljanja problema u višim dimenzijama. Radi lakšeg snalaženja, rad smo podijelili na tri osnovne cjeline: Uvod, Poissonova jednadžba, te Multigrid. U prvoj cjelini smo iskazali osnovne matematičke pojmove koje smo koristili, dok smo u drugoj klasificirali Poissonovu jednažbu. Posljednja, glavna cjelina se sastoji od motivacije i općenitih razmatranja koja su dovela do razvijanja Multigrida, osnovnih elemenata Multigrida koje smo morali u konačnici povezati, te naposljetku od pseudokoda algoritma i grafičkog prikaza rezultata. Poseban naglasak smo stavili na rezultate, koji su nam sa pragmatičnog stajališta, najbitniji dio rada. Nismo vršili detaljnu interpretaciju rezultata, radi jasnoće MATLAB-ovskih grafičkih prikaza. Kod uz pomoć kojega su dobiveni rezultati, je pisan posebno u MATLABU, te nam je omogućio usporedbu Multigrida sa osnovnim metodama poput Jacobijeve.

Abstract (english)

Basic idea of this work is Multigrid method, i.e. comparison with other basic iterative methods. Method testing was carried out exclusively using Poisson equation with homogeneous boundary condition, although that was not necessary. Also, we used legitimate assumption that our space has dimension 1D, due to complex set up in higher dimensions. Conveniently, our work is split in three basic units: Introduction, Poisson equation, and Multigrid. In first unit, we have expressed basic mathematical terms which we used, while in second unit we classified Poisson equation. Last, the most significant unit consists of motivation and general considerations which led us to develop Multgrid, basic elements of Multigrid, which we had to connect in the end, and finally algorithm pseudocode and graphical display of results. Special emphasis is made on the results, which are from pragmatical viewpoint, most important part of our work. We did not performed detailed interpretation of our results, because of distinct clearness of MATLAB graphical displays. Code which obtained our results, is written in MATLAB, and allow us comparison of Multigrid with basic iterative methods like Jacobi.

• Keywords: Multigrid metoda; iterativne metode; Poissonova jednadžba; MATLAB; Jacobijeva metoda
• Keywords (english): Multigrid method; iterative methods; Poisson equation; MATLAB; Jacobi method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:400565
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-28 13:22:48