Semi-algebraic mode analysis for multigrid method on regular rectangular and triangular grids

Document Type : Research Article

Authors

Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

Abstract

In this work, a Semi-Algebraic Mode Analysis (SAMA) technique for multigrid waveform relaxation method applied to the finite element discretization on rectangular and regular triangular grids in two dimensions and cubic and triangular prism elements in three dimensions for the heat equation is proposed.  For all the studied cases especially for the general triangular prism element, both the stiffness and mass stencils are introduced comprehensively. Moreover, several numerical examples are included to illustrate the efficiency of the convergence estimates. Studying this analysis for the finite element method is more involved and more general than that finite-difference discretization since the mass matrix must be considered. The proposed analysis results are a very useful tool to study the behavior of the multigrid waveform relaxation method depending on the parameters of the problem.   

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[1] C. Engwer, R.D. Falgout, U.M. Yang, Stencil computations for PDE-based applications with ex-
amples from DUNE and hypre, Concurr. Comput. Pract. Exp 29 (2017) e4097.
[2] S. Friedhoff, S. MacLachlan, A generalized predictive analysis tool for multigrid methods, Numer.
Linear Algebra Appl. 22 (2015) 618–647.
[3] M.J. Gander, 50 Years of Time Parallel Time Integration, Springer, (2015) 69–113.
[4] F.J. Gaspar, J.L. Gracia, F.J. Lisbona, Fourier analysis for multigrid methods on triangular grids,
SIAM J. Sci. Comput. 31 (2009) 2081–2102.
[5] F.J. Gaspar, C. Rodrigo, Multigrid waveform relaxation for the time-fractional heat equation, SIAM
J. Sci. Comput. 39 (2017) A1201–A1224.
[6] M.S. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006.
[7] X. Hu, C. Rodrigo, F.J. Gaspar, Using hierarchical matrices in the solution of the time-fractional
heat equation by multigrid waveform relaxation., J. Comput. Phys. 416 (2020) 109540.
[8] Ch. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations, BIT Numer.
Math. 27 (1987) 216–234.
[9] M.F. Malacarne, M.A. Villela Pinto, S.R. Franco, Subdomain Method in Time with Waveform Re-
laxation in Space Applied to the Wave Equation Combined with the Multigrid Method, Available at
SSRN 4089078 (2022).
[10] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems,
SIAM J. Sci. Statist. Comput. 8 (1987) 459–482.
[11] C. Rodrigo,F.J. Gaspar, F.J. Lisbona, Geometric multigrid methods on semi-structured triangular
grids, The 7 th GRACM Congress hosts the 1 st ECCOMAS PhD Olympiad (2011) P.175.
[12] U. Trottenberg,C.W. Oosterlee, A. Schuller, Multigrid, Elsevier, 2000.
[13] S. Vandewalle, D. Roose, The parallel waveform relaxation multigrid method, Parallel Processing
for Scientific Computing (1989) 152–156.
[14] R. Wienands, W. Joppich, Practical Fourier Analysis for Multigrid Methods, Chapman and
Hall/CRC, 2004.