University of Illinois Urbana-Champaign

On the development of preconditioning the advection diffusion operators with spectral element discretization

Lan, Yu-Hsiang

This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.

Permalink

https://hdl.handle.net/2142/120444

Description

Title
On the development of preconditioning the advection diffusion operators with spectral element discretization
Author(s)
Lan, Yu-Hsiang
Issue Date
2023-05-01
Director of Research (if dissertation) or Advisor (if thesis)
Fischer, Paul
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
M.S.
Degree Level
Thesis
Keyword(s)
  • Preconditioner
  • Advection Diffusion
  • Spectral Element Method
  • Overlapping Schwarz
Abstract
We discuss the two-level overlapping Schwarz methods as the main component to build the preconditioner of the advection diffusion operators under the spectral element discretization. During the construction of the Schwarz subdomain problems, several options are consid- ered including additive Schwarz versus multiplicative Schwarz, restricted or not, Dirichlet- Neumann (Dir-Neu) boundary conditions (BCs) and different sizes of overlap. To assemble the local subdomain matrix, we propose the mixed usage of spectral element matrix and linear finite element matrix (SF-matrix) based on the different regions of the subdomains so it can recover the exact BCs, Dir-Neu in particular. We verify our implementation with 1D examples of both Poisson problem and the advection diffusion cases. This is done by a careful derivation from theories to the algorithms and we present the numerical results in the comparison among various combinations of the options. It’s shown that Dir-Neu can indeed reduce the oscillations and the SF-matrix is the key to sustain the approximated BCs. Subdomains for 2D and 3D cases, however, are not well-defined due to the imperfect shape of subdomains with undefined corner regions. This will require extra treatments to recover the exact BCs in the future development. The second component experiments the usage of low-order finite element method (FEM) to build the global operator as the preliminary study of the coarse grid operator using 2D test cases. Among many variants of incomplete LU factorization (ILU), the Crout version seems to be able to produce stable and sparse factorization. We also find the dealiased coarse matrix improve the stability for the deformed elements and it’s needed to have good iteration number at convection dominant scenario. As there is no spectral equivalence of SEM-FEM for advection operator, the future development should focus on a better representation of the convection term especially at a coarse grid. Few directions are discussed in the final conclusion as our future works.
Graduation Semester
2023-05
Type of Resource
Thesis
Copyright and License Information
Copyright 2023 Yu-Hsiang Lan

Owning Collections

Graduate Dissertations and Theses at Illinois PRIMARY
Graduate Theses and Dissertations at Illinois