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Partial semi-coarsening multigrid method based on the HOC scheme on nonuniform grids for the convection–diffusion problems

Fujun Cao, Yongbin Ge, Hai-Wei Sun
2017 International Journal of Computer Mathematics  
Numerical experiments on some convection-diffusion problems with boundary or internal layers are conducted.  ...  A partial semi-coarsening multigrid method based on the high order compact (HOC) difference scheme on nonuniform grids is developed to solve the two dimensional (2D) convection-diffusion problems with  ...  i for the approximate solution at the fine grid points (big black points) corresponding to the coarse grid points are transferred directly.  ... 
doi:10.1080/00207160.2017.1283408 fatcat:uf6tibnebbg2jh7tw2ceihhkdm

Convergence and performance of iterative methods for solving variable coefficient convection-diffusion equation with a fourth-order compact difference scheme

S. Karaa, Jun Zhang
2002 Computers and Mathematics with Applications  
we conduct convergence analysis on some classical stationary iterative methods for solving the two-dimensional variable coefficient convection-diffusion equation discretized by a fourthorder compact difference  ...  We further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods, multigrid method, and preconditioned GMRES.  ...  convection-diffusion problems.  ... 
doi:10.1016/s0898-1221(02)00162-1 fatcat:65a26yj7djfijn62ovmhsz57qm

Development of semi-coarsening techniques

Paul M. de Zeeuw
1996 Applied Numerical Mathematics  
A sawtooth multi-level algorithm is proposed for the case of multiple semi-coarsening. A hierarchical set of basis functions for finite volumes on sparse grids is briefly discussed. ).  ...  Departing from Mulder's semi-coarsening technique for first order PDEs, the notion of a grid of grids is introduced and a multi-level finite-volume technique for second order elliptic PDEs is developed  ...  Acknowledgement The referees of this article are gratefully acknowledged for their constructive comments.  ... 
doi:10.1016/0168-9274(95)00095-x fatcat:s5mk5fihzrfxzcu3essxmcos4i

A robust algebraic multilevel preconditioner for non-symmetricM-matrices

Y van Notay
2000 Numerical Linear Algebra with Applications  
Stable finite difference approximations of convection-diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M-matrix, which is highly non-symmetric when the  ...  For an efficient iterative solution of such systems, it is proposed to consider in the non-symmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion problems, and  ...  diffusion dominated problems and better for convection dominated problems.  ... 
doi:10.1002/1099-1506(200007/08)7:5<243::aid-nla195>3.0.co;2-y fatcat:jux4qoyyu5bkpc6zndmghzqu7i

Aggregation-Based Algebraic Multilevel Preconditioning

Yvan Notay
2006 SIAM Journal on Matrix Analysis and Applications  
to three-dimensional convection-diffusion problems with high Reynolds number and strongly varying convection.  ...  each level, regardless of the problem at hand.  ...  I thank the referees for their careful reading and helpful suggestions.  ... 
doi:10.1137/04061129x fatcat:utqpirzfmbfnbh5ibsnvlprphu

An algebraic multigrid method for Q_2-Q_1 mixed discretizations of the Navier-Stokes equations [article]

Andrey Prokopenko, Raymond S. Tuminaro
2017 arXiv   pre-print
Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily  ...  co-located at mesh points.  ...  The paper concentrated on the Q 2 -Q 1 approximation due to it simplicity.  ... 
arXiv:1607.02489v2 fatcat:446wrfqs6rgb7jymiykzcom4uu

A multilevel algorithm for inverse problems with elliptic PDE constraints

George Biros, Günay Dogan
2008 Inverse Problems  
We present a multilevel algorithm for the solution Tikhonov-regularized first-kind Fredholm equations.  ...  Our method assumes the availability of an approximate Hessian operator for which, first, the spectral decomposition is known, and second, there exists a fast algorithm that can perform the spectral transform  ...  We perform the tests for variable coefficients α(x) = smooth(x), ellipse(x) and mesh sizes n = 32, 64, 128. We report the number of iterations and the L 2 error in the reconstructions.  ... 
doi:10.1088/0266-5611/24/3/034010 fatcat:gcqzw7cb2bbidaj56uunm2ng6a

A Boundary-Layer Preconditioner for Singularly Perturbed Convection Diffusion [article]

Scott P. MacLachlan, Niall Madden, Thái Anh Nhan
2021 arXiv   pre-print
sub-discipline within the study of the numerical approximation of solutions to differential equations.  ...  using layer-adapted meshes for convection-diffusion equations, proving a strong condition-number bound on the preconditioned system in one spatial dimension, and a weaker bound in two spatial dimensions  ...  For smooth forcing functions, f , the behaviour of the solution, u, is known to be different for the reaction-diffusion case (with c = 0 or c = 0) and the convection-diffusion case (with c = 0 or c = 0  ... 
arXiv:2108.13468v2 fatcat:4tov5l5ssvgmpnsvhcpn7griga

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale [chapter]

Daniel Arndt, Niklas Fehn, Guido Kanschat, Katharina Kormann, Martin Kronbichler, Peter Munch, Wolfgang A. Wall, Julius Witte
2020 Lecture Notes in Computational Science and Engineering  
point [29, 51] .  ...  This text presents contributions to efficient high-order finite element solvers in the context of the project ExaDG, part of the DFG priority program 1648 Software for Exascale Computing (SPPEXA).  ...  Splitting methods separate the solution of the incompressible Navier-Stokes equations into sub-problems such as a Poisson equation for the pressure and a (convection-)diffusion equation for the velocity  ... 
doi:10.1007/978-3-030-47956-5_8 dblp:series/lncse/0003FKK0MWW20 fatcat:tgo4p7knyzeozpgwi5wf7enuuq

A multigrid-based preconditioned Krylov subspace method for the Helmholtz equation with PML

Zhongying Chen, Dongsheng Cheng, Wei Feng, Tingting Wu, Hongqi Yang
2011 Journal of Mathematical Analysis and Applications  
The spectral analysis of the linear system is given, and a new matrix-based interpolation operator is proposed for the multigrid method, which is used to approximately invert the preconditioner.  ...  for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in [10].  ...  The interpolation in [8] was specially proposed for the convection-diffusion problems, while the one in [22] has complex coefficients, which works poorly for our problems.  ... 
doi:10.1016/j.jmaa.2011.05.054 fatcat:szgqouke2vajbgaj4c3nwduwzm

Robust Solution of Singularly Perturbed Problems Using Multigrid Methods

Scott MacLachlan, Niall Madden
2013 SIAM Journal on Scientific Computing  
We consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction-diffusion type.  ...  We investigate the use of standard robust multigrid preconditioners for these linear systems, and we propose and prove optimality of a new block-structured preconditioning approach.  ...  Multigrid methods for convection diffusion problems on Shishkin meshes are discussed in [21, 20] , where a scalable multigrid scheme is introduced.  ... 
doi:10.1137/120889770 fatcat:k6hnndvyxjc4nmtiuqgbuojxda

Discretization and solution of elliptic PDEs-a digital signal processing approach

C.-C.J. Kuo, B.C. Levy
1990 Proceedings of the IEEE  
In the area of discretization, mode-dependent finite-difference schemes for general second-order elliptic PDEs are examined, and are illustrated by considering the Poisson, Helmholtz, and convection-diffusion  ...  Whereas conventional PDE analysis techniques rely on matrix analysis and on a space-domain point of view to study the performance of solution methods, the DSP approach described here relies on frequency  ...  and (c) uniformly distributed modedependent 5-point discretizations of the convection-diffusion equation.  ... 
doi:10.1109/5.60919 fatcat:r7bjwtk6ejhvzl2jancjpputzu

On multigrid solution of the implicit equations of hydrodynamics

K. Kifonidis, E. Müller
2012 Astronomy and Astrophysics  
Beyond a Courant number of nine thousand, even complete multigrid breakdown is observed.  ...  We describe and study a family of new multigrid iterative solvers for the multidimensional, implicitly discretized equations of hydrodynamics.  ...  Cord Rossow (DLR, Braunschweig) for many helpful hints and discussions on multigrid techniques, to Prof. Norbert Kroll for the hospitality of DLR's CASE institute, and to Prof.  ... 
doi:10.1051/0004-6361/201116979 fatcat:vv5lxhulhreo7n34tj7ipc76ia

Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet

Tobin Isaac, Noemi Petra, Georg Stadler, Omar Ghattas
2015 Journal of Computational Physics  
We present efficient and scalable algorithms for this end-to-end, data-to-prediction process under the Gaussian approximation and in the context of modeling the flow of the Antarctic ice sheet and its  ...  efficient and scalable algorithms to (1) infer the model parameters from the data (the deterministic inverse problem), (2) quantify the uncertainty in the inferred parameters (the Bayesian inference problem  ...  Scalability of Bayesian solution of the inverse problem We now discuss the overall scalability of our algorithms for Bayesian solution of the inverse problem.  ... 
doi:10.1016/j.jcp.2015.04.047 fatcat:fpshnibcz5curhq6wtmznqc7vm

Numerical Simulation of Laminar Incompressible Fluid-Structure Interaction for Elastic Material with Point Constraints [chapter]

M. Razzaq, J. Hron, S. Turek
2009 Advances in Mathematical Fluid Mechanics  
By identifying the center of the cylinder with one grid point of the computational mesh we prescribe a Dirichlet type boundary condition for the velocity and the displacement of the structure at this point  ...  We present numerical techniques for solving the problem of fluid structure interaction with a compressible elastic material in a laminar incompressible viscous flow via fully coupled monolithic Arbitrary  ...    =   f u f v f p   (20) which is typical saddle point problem, where S describes the diffusive and convective terms from the governing equations.  ... 
doi:10.1007/978-3-642-04068-9_27 fatcat:dzfqq2dz3nf4vft65b2vm42dmi
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