A high-order discontinuous Galerkin approximation to ordinary differential equations with applications to elastodynamics

2017 ◽  
Vol 38 (4) ◽  
pp. 1709-1734 ◽  
Author(s):  
Paola F Antonietti ◽  
Ilario Mazzieri ◽  
Niccolò Dal Santo ◽  
Alfio Quarteroni
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
High Order

scholarly journals Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 49 ◽  
Author(s):  
Carlo Garoni ◽  
Mariarosa Mazza ◽  
Stefano Serra-Capizzano
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Discontinuous Galerkin ◽  
Spectral Distribution ◽  
Galerkin Approximation ◽  
Higher Order ◽  
Numerical Discretization

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a GLT sequence or one of its “relatives”, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations.

scholarly journals On Conjugacy of High-Order Linear Ordinary Differential Equations

1994 ◽  
Vol 1 (1) ◽  
pp. 1-8
Author(s):  
T. Chanturia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
High Order ◽  
Summable Function ◽  
Order Linear ◽  
Linear Ordinary Differential Equations

Abstract It is shown that the differential equation u (n) = p(t)u, where n ≥ 2 and p : [a, b] → ℝ is a summable function, is not conjugate in the segment [a, b], if for some l ∈ {1, . . . , n – 1}, α ∈]a, b[ and β ∈]α, b[ the inequalities hold.

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