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Multigrid Preconditioners for the Discontinuous Galerkin Spectral Element Method : Construction and Analysis

Versbach, Lea Miko LU (2020) In Licentiate Thesis in Mathematical Sciences 2020(2).
Abstract
Discontinuous Galerkin (DG) methods offer a great potential for simulations of turbulent and wall bounded flows with complex geometries since these high-order schemes offer a great potential in handling eddies. Recently, space-time DG methods have become more popular. These discretizations result in implicit schemes of high order in both spatial and temporal directions. In particular, we consider a specific DG variant, the DG Spectral Element Method (DG-SEM), which is suitable to construct entropy stable solvers for conservation laws. Since the size of the corresponding nonlinear systems is dependent on the order of the method in all dimensions, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time... (More)
Discontinuous Galerkin (DG) methods offer a great potential for simulations of turbulent and wall bounded flows with complex geometries since these high-order schemes offer a great potential in handling eddies. Recently, space-time DG methods have become more popular. These discretizations result in implicit schemes of high order in both spatial and temporal directions. In particular, we consider a specific DG variant, the DG Spectral Element Method (DG-SEM), which is suitable to construct entropy stable solvers for conservation laws. Since the size of the corresponding nonlinear systems is dependent on the order of the method in all dimensions, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption.
Currently, there is a lack of good solvers for three-dimensional DG applications, which is one of the major obstacles why these high order methods are not used in e.g. industry. We suggest to use Jacobian-free Newton- Krylov (JFNK) solvers, which are advantageous in memory minimization. In order to improve the convergence speed of these solvers, an efficient preconditioner needs to be constructed for the Krylov sub-solver. However, if the preconditioner requires the storage of the DG system Jacobian, the favorable memory consumption of the JFNK approach is obsolete.
We therefore present a multigrid based preconditioner for the Krylov sub-method which retains the low mem- ory consumption, i.e. a Jacobian-free preconditioner. To achieve this, we make use of an auxiliary first order finite volume replacement operator. With this idea, the original DG mesh is refined but can still be implemented algebraically. As smoother, we consider the pseudo time iteration W3 with a symmetric Gauss-Seidel type approx- imation of the Jacobian. Numerical results are presented demonstrating the potential of the new approach.
In order to analyze multigrid preconditioners, a common tool is the Local Fourier Analysis (LFA). For a space- time model problem we present this analysis and its benefits for calculating smoothing and two-grid convergence factors, which give more insight into the efficiency of the multigrid method. (Less)
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author
supervisor
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Discontinuous Galerkin Method, Finite Volume Method, Implicit Schemes, Local Fourier Analysis, Multigrid Method, Preconditioner, Space-Time Discretization, Spectral Element Method
in
Licentiate Thesis in Mathematical Sciences
volume
2020
issue
2
pages
105 pages
publisher
Lund University
ISSN
1404-028X
ISBN
978-91-7895-592-3
978-91-7895-593-0
project
Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods
language
English
LU publication?
yes
id
e83d751d-034c-42bb-b61b-9d946aa8f996
date added to LUP
2021-06-03 13:50:24
date last changed
2022-01-27 12:25:29
@misc{e83d751d-034c-42bb-b61b-9d946aa8f996,
  abstract     = {{Discontinuous Galerkin (DG) methods offer a great potential for simulations of turbulent and wall bounded flows with complex geometries since these high-order schemes offer a great potential in handling eddies. Recently, space-time DG methods have become more popular. These discretizations result in implicit schemes of high order in both spatial and temporal directions. In particular, we consider a specific DG variant, the DG Spectral Element Method (DG-SEM), which is suitable to construct entropy stable solvers for conservation laws. Since the size of the corresponding nonlinear systems is dependent on the order of the method in all dimensions, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption.<br/>Currently, there is a lack of good solvers for three-dimensional DG applications, which is one of the major obstacles why these high order methods are not used in e.g. industry. We suggest to use Jacobian-free Newton- Krylov (JFNK) solvers, which are advantageous in memory minimization. In order to improve the convergence speed of these solvers, an efficient preconditioner needs to be constructed for the Krylov sub-solver. However, if the preconditioner requires the storage of the DG system Jacobian, the favorable memory consumption of the JFNK approach is obsolete.<br/>We therefore present a multigrid based preconditioner for the Krylov sub-method which retains the low mem- ory consumption, i.e. a Jacobian-free preconditioner. To achieve this, we make use of an auxiliary first order finite volume replacement operator. With this idea, the original DG mesh is refined but can still be implemented algebraically. As smoother, we consider the pseudo time iteration W3 with a symmetric Gauss-Seidel type approx- imation of the Jacobian. Numerical results are presented demonstrating the potential of the new approach.<br/>In order to analyze multigrid preconditioners, a common tool is the Local Fourier Analysis (LFA). For a space- time model problem we present this analysis and its benefits for calculating smoothing and two-grid convergence factors, which give more insight into the efficiency of the multigrid method.}},
  author       = {{Versbach, Lea Miko}},
  isbn         = {{978-91-7895-592-3}},
  issn         = {{1404-028X}},
  keywords     = {{Discontinuous Galerkin Method; Finite Volume Method; Implicit Schemes; Local Fourier Analysis; Multigrid Method; Preconditioner; Space-Time Discretization; Spectral Element Method}},
  language     = {{eng}},
  note         = {{Licentiate Thesis}},
  number       = {{2}},
  publisher    = {{Lund University}},
  series       = {{Licentiate Thesis in Mathematical Sciences}},
  title        = {{Multigrid Preconditioners for the Discontinuous Galerkin Spectral Element Method : Construction and Analysis}},
  url          = {{https://lup.lub.lu.se/search/files/98654604/LicLea.pdf}},
  volume       = {{2020}},
  year         = {{2020}},
}