Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution

2021 ◽  
Vol 182 ◽  
pp. 838-857
Author(s):  
Changpin Li ◽  
Zhen Wang
Keyword(s):  
Discontinuous Galerkin ◽  
Regular Solution ◽  
Galerkin Approximation ◽  
Convection Equation

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

2021 ◽  
Vol 36 (6) ◽  
pp. 313-336
Author(s):  
Ronald H. W. Hoppe ◽  
Youri Iliash
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Interior Penalty ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Flux Function

Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

A local discontinuous Galerkin approximation for systems with p-structure

2013 ◽  
Vol 34 (4) ◽  
pp. 1447-1488 ◽  
Author(s):  
L. Diening ◽  
D. Kroner ◽  
M. R  i ka ◽  
I. Toulopoulos
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Local Discontinuous Galerkin

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.

A two-energies principle for the biharmonic equation and ana posteriorierror estimator for an interior penalty discontinuous Galerkin approximation

2018 ◽  
Vol 52 (6) ◽  
pp. 2479-2504 ◽  
Author(s):  
Dietrich Braess ◽  
R.H.W. Hoppe ◽  
Christopher Linsenmann
Keyword(s):  
Discontinuous Galerkin ◽  
Biharmonic Equation ◽  
Moment Tensor ◽  
Galerkin Approximation ◽  
Direct Consequence ◽  
Mixed Formulation ◽  
Error Estimator ◽  
A Posteriori ◽  
Reliability Estimate ◽  
Interior Penalty

We consider ana posteriorierror estimator for the Interior Penalty Discontinuous Galerkin (IPDG) approximation of the biharmonic equation based on the Hellan-Herrmann-Johnson (HHJ) mixed formulation. The error estimator is derived from a two-energies principle for the HHJ formulation and amounts to the construction of an equilibrated moment tensor which is done by local interpolation. The reliability estimate is a direct consequence of the two-energies principle and does not involve generic constants. The efficiency of the estimator follows by showing that it can be bounded from above by a residual-type estimator known to be efficient. A documentation of numerical results illustrates the performance of the estimator.

Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods

2020 ◽  
Vol 31 (03) ◽  
pp. 2050041
Author(s):  
Dipty Sharma ◽  
Paramjeet Singh
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Strong Stability ◽  
Weno Scheme ◽  
Transmission Delays ◽  
Weighted Essentially Nonoscillatory ◽  
Refractory Periods ◽  
Dg Method ◽  
Analysis Of Stability ◽  
Inhibitory Networks

In this study, we consider the network of noisy leaky integrate-and-fire (NNLIF) model, which governs by a second-order nonlinear time-dependent partial differential equation (PDE). This equation uses the probability density approach to describe the behavior of neurons with refractory states and the transmission delays. A numerical approximation based on the discontinuous Galerkin (DG) method is used for the spatial discretization with the analysis of stability. The strong stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. Finally, some test examples and numerical simulations are given to examine the behavior of the solution. The execution of the constructed scheme is measured by the quantitative comparison with the existing finite difference technique, namely weighted essentially nonoscillatory (WENO) scheme.

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