Local Discontinuous Galerkin and Classical Finite Element Methods: Differences and Similarities

Rates Of Convergence ◽

Spectral Condition ◽

Qualitative Comparison ◽

Approximation Rates ◽

Local Discontinuous Galerkin

In this work a quantitative and qualitative comparison of the Local Discontinuous Galerkin method and classical finite element methods applied to elliptic problems is performed. High order discretizations are considered. The methods are compared with respect to accuracy of the approximation, rates of convergence, asymptotic behavior of the spectral condition number of the stiffness matrix.

Numerical solutions of systems with (p,δ)-structure using local discontinuous Galerkin finite element methods

International Journal for Numerical Methods in Fluids ◽

10.1002/fld.3955 ◽

2014 ◽

Vol 76 (11) ◽

pp. 855-874 ◽

Cited By ~ 4

Author(s):

Dietmar Kröner ◽

Michael Růžička ◽

Ioannis Toulopoulos

Keyword(s):

Finite Element ◽

Discontinuous Galerkin ◽

Finite Element Methods ◽

Numerical Solutions ◽

Galerkin Finite Element ◽

Galerkin Finite Element Methods

Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation

Open Mathematics ◽

10.1515/math-2018-0091 ◽

2018 ◽

Vol 16 (1) ◽

pp. 1091-1103 ◽

Cited By ~ 2

Author(s):

Leilei Wei ◽

Yundong Mu

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Discontinuous Galerkin ◽

Stability And Convergence ◽

Lax Equation ◽

Numerical Performance ◽

Element Method

AbstractIn this paper we develop and analyze the local discontinuous Galerkin (LDG) finite element method for solving the general Lax equation. The local discontinuous Galerkin method has the flexibility for arbitrary h and p adaptivity, and allows for hanging nodes. By choosing the numerical fluxes carefully we prove stability and give an error estimate. Finally some numerical examples are computed to show the convergence order and excellent numerical performance of proposed method.

The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: Numerical analysis

Applied Numerical Mathematics ◽

10.1016/j.apnum.2019.01.007 ◽

2019 ◽

Vol 140 ◽

pp. 1-22 ◽

Cited By ~ 18

Author(s):

Changpin Li ◽

Zhen Wang

Keyword(s):

Finite Element ◽

Numerical Analysis ◽

Partial Differential Equations ◽

Differential Equations ◽

Discontinuous Galerkin ◽

Finite Element Methods ◽

Galerkin Finite Element ◽

Galerkin Finite Element Methods

The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: Mathematical analysis

Applied Numerical Mathematics ◽

10.1016/j.apnum.2019.11.007 ◽

2020 ◽

Vol 150 ◽

pp. 587-606 ◽

Cited By ~ 7

Author(s):

Changpin Li ◽

Zhen Wang

Keyword(s):

Finite Element ◽

Partial Differential Equations ◽

Differential Equations ◽

Discontinuous Galerkin ◽

Finite Element Methods ◽

Mathematical Analysis ◽

Galerkin Finite Element ◽

Galerkin Finite Element Methods

NUMERICAL ANALYSIS OF AN IMPLICIT FULLY DISCRETE LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE FRACTIONAL ZAKHAROV–KUZNETSOV EQUATION

Mathematical Modelling and Analysis ◽

10.3846/13926292.2012.708675 ◽

2012 ◽

Vol 17 (4) ◽

pp. 558-570 ◽

Cited By ~ 3

Author(s):

Zongxiu Ren ◽

Leilei Wei ◽

Yinnian He ◽

Shaoli Wang

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Numerical Analysis ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Methods ◽

Galerkin Methods ◽

Fully Discrete ◽

Local Discontinuous Galerkin Methods

In this paper we develop and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Zakharov–Kuznetsov equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditional stable and L 2 error estimate for the linear case with the convergence rate through analysis.

A combined mixed finite element method and local discontinuous Galerkin method for miscible displacement problem in porous media

Science China Mathematics ◽

10.1007/s11425-014-4879-y ◽

2014 ◽

Vol 57 (11) ◽

pp. 2301-2320 ◽

Cited By ~ 8

Author(s):

Hui Guo ◽

QingHua Zhang ◽

Yang Yang

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Porous Media ◽

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Mixed Finite Element ◽

Mixed Finite Element Method ◽

Local Discontinuous Galerkin

A Superconvergent Local Discontinuous Galerkin Method for Nonlinear Fourth-Order Boundary-Value Problems

International Journal of Computational Methods ◽

10.1142/s021987621950035x ◽

2019 ◽

Vol 17 (07) ◽

pp. 1950035 ◽

Cited By ~ 2

Author(s):

Mahboub Baccouch

Keyword(s):

Exact Solutions ◽

Boundary Value Problems ◽

Discontinuous Galerkin ◽

Fourth Order ◽

Boundary Value ◽

Rates Of Convergence ◽

Piecewise Polynomials ◽

Derivatives Of

In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form [Formula: see text]. We prove optimal [Formula: see text] error estimates for the solution and for the three auxiliary variables that approximate the first, second, and third-order derivatives. The order of convergence is proved to be [Formula: see text], when piecewise polynomials of degree at most [Formula: see text] are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order [Formula: see text] toward the derivatives of Gauss–Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order [Formula: see text] toward Gauss–Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is [Formula: see text]. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree [Formula: see text] and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.