Metode dekompozicije domene

master's thesis

• Title (english): Domain decomposition methods
• Author: Patricia Nužda
• Mentor: Nela Bosner (mentor)
• Committee member: Nela Bosner (predsjednik povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Na samom početku ovog rada cilj je bio definirati osnovne jednadžbe i odnose iz parcijalnih diferencijalnih jednadžbi koje ćemo koristiti u particiji na poddomene. Pri tome smo se dotakli osnovih podijela tehnika dekompozicija domena, te smo spomenuli koje ćemo sve metode proučiti u ovom radu. Promatrali smo direktno rješenje i kakav je odnos matrice A te Schurovog komplementa S i promatrali slučaj kada je A, odnosno S simetrična pozitivno definitna matrica. Kod samog Schurovog komplementa koncentrirali smo se na particije koje su se bazirale na vrhovima. Multiplikativna Schwarzova metoda na svakoj poddomeni sekvencijalno dolazi do rješenja i pri tome koristi rubne uvjete za trenutni korak koje dobiva iz posljednje vrijednosti rješenja. Aditivna Schwarzova metoda ne mijenja komponente u svakoj poddomeni dok nisu dovršene sve promijene kroz sve domene. Promatrali smo uvjete konvergencije Schwarzove izmjenične metode u slučaju kada je A simetrična pozitivno definitna matrica. Inducirano prekondicioniranje se bazira na Schurovom komplementu i u samoj suštini koristi ILU faktorizaciju polazne matrice A, a prekondicioniranje Schurovog komplementa bazirano na vrhovima koristi LU faktorizaciju lokalne matrice A. Na posljetku, dotakli smo se i metoda punih matrica. Te metode se baziraju na iteriranju na originalnom sustavu gdje se prekondicioniranje matrice A dobiva iz aproksimiranih graničnih vrijednosti.

Abstract (english)

At the beginning of this thesis our main goal was to define fundamental equations and relations from Partial Differential Equations which we will use here. Essential classification of domain decomposition techniques we mentioned here, and we specified which ones are going to be in our focus. We studied direct solution and checked relations between matrix A and its Schur complement S , and analized those when A and S were symmetric positive definite matrices. Regarding the Schur complement we concentrated on vertex-based partitions. Multiplicative Schwarz procedure on each subdomain sequentially gets solution while using boundary conditions for current step which we get from last value of the solution. Additive Schwarz procedure doesn’t update components in each subdomain until a whole cycle of updates through all domains are completed. When A is symmetric positive definite matrix we checked what are convergence conditions for Schwarz alternating methods. Induced preconditioners are based on Schur complement and it basically uses ILU factorization of original matrix A, while preconditioning of vertex-based Schur complements is based on using LU factorization of local A matrix. At the end we checked full matrix method, these methods are based on iterations on the original system, here we also use approximating interface values to get preconditioners of A while for Schur complement we use LU factorization.

• Keywords: matrica A; Schurov komplement S; Schwarzova metoda; metoda punih matrica
• Keywords (english): matrix A; Schur complement S; Schwarz procedure; full matrix method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:195301
• 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: 2021-02-24 10:05:18