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

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.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Lan, Yu-Hsiang

Contributors dc:contributor

• Fischer, Paul

• Preconditioner
• Advection Diffusion
• Spectral Element Method
• Overlapping Schwarz

Statement dc:rights

• Copyright 2023 Yu-Hsiang Lan

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120444