scholarly journals Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

2020 ◽  
Vol 89 (325) ◽  
pp. 2113-2139 ◽  
Author(s):  
Yang Yang ◽  
Xiaofeng Cai ◽  
Jing-Mei Qiu
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Space Dimension ◽  
Galerkin Methods ◽  
Optimal Convergence ◽  
Convergence And Superconvergence ◽  
One Space Dimension

scholarly journals Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Julius Witte ◽  
Daniel Arndt ◽  
Guido Kanschat
Keyword(s):  
Finite Element ◽  
Tensor Product ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Space Dimension ◽  
High Order ◽  
Low Rank ◽  
Galerkin Methods ◽  
Low Rank Representation ◽  
Subspace Correction Methods

AbstractWe discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high-order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree k is reduced from {\mathcal{O}(k^{3d})} to {\mathcal{O}(dk^{d+1})} by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations in space dimension d.

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