Superconvergence of discontinuous Galerkin solutions for higher-order ordinary differential equations

Temimi, Helmi

doi:10.1016/j.apnum.2014.09.009

Cited by 4 publications

(2 citation statements)

References 11 publications

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“…Moreover, their investigation unveiled superconvergence within each space-time element, particularly at the intersection points of the Lobatto polynomials in space and the Jacobi polynomials in time. Subsequently, Temimi in [24] addressed one-dimensional second-order boundary value problems using the DG scheme, revealing that the leading term of the discretization error in each element is generated by specific combination of Jacobi polynomials. He proved that for a DG solution of degree p, there's a superconvergence rate of O(h p+2 ) at the polynomial roots.…”

Section: Introductionmentioning

confidence: 99%

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Temimi

2024

J. Math. Computer Sci.

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This paper introduces a novel approach for solving systems of boundary value problems (BVPs) by employing the recently developed Discontinuous Galerkin (DG) method, which removes the necessity for auxiliary variables. This marks the initial installment in a sequence of publications dedicated to exploring DG methods for solving partial differential equations (PDEs). In fact, through a systematic application of the DG method to each spatial variable within the PDE, employing the method of lines, we convert the initial problem into a system of ordinary differential equations (ODEs). In the current study, we developed a global error analysis of the DG method applied to systems of ODEs. Our analysis shows that using p-degree piecewise polynomials and h-mesh step size, the DG solutions achieve optimal O(h p+1 ) convergence rates in the L 2 -norm.

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Section: Introductionmentioning

confidence: 99%

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Temimi

2024

J. Math. Computer Sci.

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“…Furthermore, the study revealed the superconvergence of the DG solution within each space-time element, specifically at the intersecting points of the Lobatto polynomials in space and the Jacobi polynomials in time. Later, Temimi [19] solved the one-dimensional second-order BVPs using the developed DG scheme and demonstrated that a specific combination of Jacobi polynomials generates the leading term of the discretization error in each element. He proved that the p-degree DG solution achieves an O(h p+2 ) superconvergence rate at the roots of these particular polynomials.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence Analysis of Discontinuous Galerkin Methods for Systems of Second-Order Boundary Value Problems

Temimi

2023

Computation

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In this paper, we present an innovative approach to solve a system of boundary value problems (BVPs), using the newly developed discontinuous Galerkin (DG) method, which eliminates the need for auxiliary variables. This work is the first in a series of papers on DG methods applied to partial differential equations (PDEs). By consecutively applying the DG method to each space variable of the PDE using the method of lines, we transform the problem into a system of ordinary differential equations (ODEs). We investigate the convergence criteria of the DG method on systems of ODEs and generalize the error analysis to PDEs. Our analysis demonstrates that the DG error’s leading term is determined by a combination of specific Jacobi polynomials in each element. Thus, we prove that DG solutions are superconvergent at the roots of these polynomials, with an order of convergence of O(hp+2).

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A highly accurate discontinuous Galerkin method for solving nonlinear Bratu's problem

Temimi,

Ben-Romdhane

2024

Alexandria Engineering Journal

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Superconvergence of discontinuous Galerkin solutions for higher-order ordinary differential equations

Cited by 4 publications

References 11 publications

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Superconvergence Analysis of Discontinuous Galerkin Methods for Systems of Second-Order Boundary Value Problems

A highly accurate discontinuous Galerkin method for solving nonlinear Bratu's problem

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