TY - JOUR
T1 - Optimal error estimates and superconvergence of an ultra weak discontinuous Galerkin method for fourth-order boundary-value problems
AU - Baccouch, Mahboub
AU - Temimi, Helmi
AU - Ben-Romdhane, Mohamed
N1 - Funding Information:
This research was supported by the Kuwait Foundation for the Advancement of Sciences (KFAS), Project Code PR17-16SM-04 . The research of the first author was also supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2017-01-F ) at the University of Nebraska at Omaha.
Publisher Copyright:
© 2018 IMACS
PY - 2019/3
Y1 - 2019/3
N2 - In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.
AB - In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.
KW - A priori error estimates
KW - Fourth-order boundary-value problems
KW - Superconvergence
KW - Ultra weak discontinuous Galerkin method
KW - Upwind and downwind points
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U2 - 10.1016/j.apnum.2018.11.011
DO - 10.1016/j.apnum.2018.11.011
M3 - Article
AN - SCOPUS:85057542659
VL - 137
SP - 91
EP - 115
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
SN - 0168-9274
ER -