TY - JOUR

T1 - The local discontinuous galerkin method for the fourth-order euler-bernoulli partial differential equation in one space dimension. Part II

T2 - A posteriori error estimation

AU - Baccouch, Mahboub

N1 - Funding Information:
Acknowledgments The author would also like to thank the anonymous referees for their constructive comments and remarks which helped improve the quality and readability of the paper. This research was partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha.

PY - 2014/7

Y1 - 2014/7

N2 - In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler-Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to (k+1) -degree Radau polynomials, when polynomials of total degree not exceeding k are used. These results allow us to prove that the k -degree LDG solution and its derivatives are O(h k+3/2) superconvergent at the roots of (k+1) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time tconverge to the true errors at O(hk+5/4 rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the L2 -norm converge to unity at O(h1/2) rate. Our proofs are valid for arbitrary regular meshes and for Pk polynomials with k ≥ 1, and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.

AB - In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler-Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to (k+1) -degree Radau polynomials, when polynomials of total degree not exceeding k are used. These results allow us to prove that the k -degree LDG solution and its derivatives are O(h k+3/2) superconvergent at the roots of (k+1) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time tconverge to the true errors at O(hk+5/4 rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the L2 -norm converge to unity at O(h1/2) rate. Our proofs are valid for arbitrary regular meshes and for Pk polynomials with k ≥ 1, and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.

KW - A posteriori error estimates

KW - Adaptive mesh method

KW - Fourth-order Euler-Bernoulli equation

KW - Local discontinuous Galerkin method

KW - Radau points

KW - Superconvergence

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U2 - 10.1007/s10915-013-9783-z

DO - 10.1007/s10915-013-9783-z

M3 - Article

AN - SCOPUS:84902296036

VL - 60

SP - 1

EP - 34

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 1

ER -