Unconditional stability and optimal error estimates of discontinuous Galerkin methods for the second-order wave equation

2019 ◽  
pp. 1-15 ◽  
Author(s):  
Limin He ◽  
Weimin Han ◽  
Fei Wang ◽  
Wentao Cai
Keyword(s):  
Wave Equation ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Unconditional Stability ◽  
Second Order ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Optimal Error ◽  
Second Order Wave Equation

Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements

2020 ◽  
Vol 54 (2) ◽  
pp. 705-726
Author(s):  
Yong Liu ◽  
Chi-Wang Shu ◽  
Mengping Zhang
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Hyperbolic Equations ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Optimal Error ◽  
Piecewise Polynomials ◽  
Cartesian Meshes ◽  
Theoretical Results

In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using Pk elements on uniform Cartesian meshes, and prove that the error in the L2 norm achieves optimal (k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.

scholarly journals DISCONTINUOUS GALERKIN METHODS FOR THE MULTI-DIMENSIONAL VLASOV–POISSON PROBLEM

2012 ◽  
Vol 22 (12) ◽  
pp. 1250042 ◽  
Author(s):  
BLANCA AYUSO DE DIOS ◽  
JOSÉ A. CARRILLO ◽  
CHI-WANG SHU
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Mixed Finite Element ◽  
Galerkin Approximation ◽  
Mixed Finite Element Method ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Poisson System ◽  
Optimal Error ◽  
Poisson Problem

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov–Poisson system. The schemes are constructed by combining a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

scholarly journals Mixed-hybrid and mixed-discontinuous Galerkin methods for linear dynamical elastic-viscoelastic composite structures

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio Márquez ◽  
Salim Meddahi
Keyword(s):  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Variational Formulation ◽  
Composite Structures ◽  
Discontinuous Galerkin Methods ◽  
Energy Estimate ◽  
Well Posedness ◽  
Galerkin Methods ◽  
Optimal Error ◽  
Viscoelastic Composite

Abstract We introduce and analyze a stress-based formulation for Zener’s model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition. We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness of the problem through a standard Galerkin procedure. We propose and analyze mixed continuous and discontinuous Galerkin space discretizations of the problem and derive optimal error bounds for each semidiscrete solution in the corresponding energy norm. Finally, we discuss full discretization strategies for both Galerkin methods.

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