Research on the multigrid method for solving partial differential equations using the discontinuous Galerkin method

Title	Research on the multigrid method for solving partial differential equations using the discontinuous Galerkin method
Authors	M. S. Nefedov¹, R. V. Zhalnin¹, S. H. Zinina¹ ¹Ufa University of Science and Technology
Annotation	This paper investigates the application of the Full Approximation Scheme (FAS) multigrid method combined with the Discontinuous Galerkin (DG) method for solving nonlinear partial differential equations. The study focuses on a transport problem with periodic boundary conditions, developing an algorithm that incorporates projection operators between grids of varying resolution ($V$- and $W$-cycles). Computational experiments were conducted for polynomial bases of degrees $p=2$ and $p=3$, demonstrating convergence rates close to theoretical values. Spectral error analysis confirmed the methods effectiveness in suppressing high-frequency residual components. The results show that the multigrid approach accelerates the convergence of iterative methods for implicit DG schemes while maintaining approximation accuracy.
Keywords	multigrid method, Discontinuous Galerkin, partial differential equations, convergence order, spectral analysis, projection operators, implicit schemes.
Citation	Nefedov M. S., Zhalnin R. V., Zinina S. H. ''Research on the multigrid method for solving partial differential equations using the discontinuous Galerkin method'' [Electronic resource]. Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, July 29-31, 2025). Saransk: SVMO Publ, 2025. - pp. 201-205. Available at: https://conf.svmo.ru/files/2025/papers/paper40.pdf. - Date of access: 30.08.2025.

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Research on the multigrid method for solving partial differential equations using the discontinuous Galerkin method

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