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

Research output: Contribution to journalArticle

### Abstract

In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form u(4) + f(x,u) = 0. We prove optimal L2 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 p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gauss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

Original language English (US) International Journal of Computational Methods https://doi.org/10.1142/S021987621950035X Published - Jan 1 2019

### Fingerprint

Local Discontinuous Galerkin Method
Fourth-order Boundary Value Problem
Nonlinear Boundary Value Problems
Galerkin methods
Boundary value problems
Discontinuous Galerkin
Piecewise Polynomials
Derivatives
Derivative
Gauss
Exact Solution
Polynomials
Projection
Optimal Rate of Convergence
Optimal Error Estimates
Superconvergence
Auxiliary Variables
Order of Convergence
Numerical Experiment
Boundary conditions

### Keywords

• Fourth-order boundary-value problems
• a priori error estimates
• local discontinuous Galerkin method
• superconvergence

### ASJC Scopus subject areas

• Computer Science (miscellaneous)
• Computational Mathematics

### Cite this

In: International Journal of Computational Methods, 01.01.2019.

Research output: Contribution to journalArticle

title = "A Superconvergent Local Discontinuous Galerkin Method for Nonlinear Fourth-Order Boundary-Value Problems",
abstract = "In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form u(4) + f(x,u) = 0. We prove optimal L2 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 p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gauss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.",
keywords = "Fourth-order boundary-value problems, a priori error estimates, local discontinuous Galerkin method, superconvergence",
author = "Mahboub Baccouch",
year = "2019",
month = "1",
day = "1",
doi = "10.1142/S021987621950035X",
language = "English (US)",
journal = "International Journal of Computational Methods",
issn = "0219-8762",
publisher = "World Scientific Publishing Co. Pte Ltd",

}

TY - JOUR

T1 - A Superconvergent Local Discontinuous Galerkin Method for Nonlinear Fourth-Order Boundary-Value Problems

AU - Baccouch, Mahboub

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form u(4) + f(x,u) = 0. We prove optimal L2 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 p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gauss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

AB - In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form u(4) + f(x,u) = 0. We prove optimal L2 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 p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gauss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

KW - Fourth-order boundary-value problems

KW - a priori error estimates

KW - local discontinuous Galerkin method

KW - superconvergence

UR - http://www.scopus.com/inward/record.url?scp=85066831689&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066831689&partnerID=8YFLogxK

U2 - 10.1142/S021987621950035X

DO - 10.1142/S021987621950035X

M3 - Article

AN - SCOPUS:85066831689

JO - International Journal of Computational Methods

JF - International Journal of Computational Methods

SN - 0219-8762

ER -