Diskontinuirana Galerkinova metoda jednadžbe konvekcije difuzije

master's thesis

• Author: Domagoj Bošnjak
• Mentor: Mladen Jurak (mentor)
• Committee member: Mladen Jurak (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavali smo diskontinuiranu Galerkinovu metodu te njenu primjenu na linearnu nestacionarnu jednadžbu konvekcije-difuzije. Iznešeni su rezultati ocjena greške metode u različitim varijantama jednadžbe i rubnih uvjeta. Pokazano je da u slučaju linearne zadaće konvekcije-difuzije-reakcije postoji uniformna ocjena u pogledu difuzijskog člana \(\epsilon\rightarrow 0_+.\) Dan je opis SUPG metode kojom se stabiliziraju diskontinuirani elementi u slučaju velikog Pécletovog broja. Predstavili smo dva dijela koda koji su opisivali implementaciju dva važna koraka u rješavanju jednadžbe unutar DUNE okoline: definiciju i formiranje mreže koja predstavlja diskretiziranu domenu te inicijalizaciju objekata potrebnih za numerizacijski proces. U većini fizikalnih modela konvektivni član iz jednadžbe je dominantan u odnosu na difuzijski. To dovodi do situacije u kojoj je kontinuirani problem dobro postavljen i ima jedinstveno rješenje na osnovu Lax-Milgramove leme, ali numeričko rješenje, dobiveno standardnom metodom konačnih elemenata, nije stabilno. Pojavljuju se oscilacije rješenja koje se propagiraju kroz vremenske slojeve, štoviše, amplitude im rastu. Numeričkim eksperimentima smo potvrdili stabilnost diskontinuirane Galerkinove metode za velike Pécletove brojeve za razliku od metode konačnih elemenata koja proizvodi rubne slojeve i zahtijeva dodatne napredne stabilizacijske tehnike.

Abstract (english)

In this paper, we studied discontinuous Galerkin method as well as its application of linear evolutionary convection diffusion equation. We shown error estimate results under different variances of equation and boundary conditions. Furthermore, it was stated that there exists uniform error estimate with respect to the diffusion coefficient \(\epsilon\rightarrow 0_+\) for linear convection diffusion reaction equation. We introduced SUPG method used for stabilization of discontinuous elements for large Péclet numbers. We presented two code snippets which implement two crucial steps when solving equations using DUNE framework: grid definition, which is used for representing discretization of a domain, and instantiation of objects needed for numerization process. Within most models describing some physical phenomena, convective part is dominant compared to diffusion part. It leads to a well posed continuous problem which, according to Lax-Milgram lemma, has a unique solution. Numerical solutions obtained using conforming method of finite elements, however, suffered from severe instabilities. They often contain oscillations which only amplify through time. Quality of discontinuous Galerkin method was confirmed by numerical experiment for large Péclet numbers, contrary to finite element method which yields boundary layers in solution and requires extra sophisticated stabilization techniques.

• Keywords: Galerkinova metoda; SUPG metoda; Pécletov broj; DUNE okolina; Lax-Milgramova lema
• Keywords (english): Galerkin method; SUPG method; Péclet number; DUNE framework; Lax-Milgram lemma
• Language: croatian
• URN:NBN: urn:nbn:hr:217:699001
• 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: 2020-01-13 11:07:34