TY - JOUR
T1 - A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems
AU - Baccouch, Mahboub
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2023/4
Y1 - 2023/4
N2 - In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form − u(4) = f(x, u). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the L2-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order p + 1 in the L2-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most p and mesh size h are used. We then show that the UWLDG solutions are superconvergent with order p + 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.
AB - In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form − u(4) = f(x, u). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the L2-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order p + 1 in the L2-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most p and mesh size h are used. We then show that the UWLDG solutions are superconvergent with order p + 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.
KW - A priori error estimate
KW - Nonlinear fourth-order boundary-value problems
KW - Superconvergence
KW - Ultra-weak local discontinuous Galerkin method
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U2 - 10.1007/s11075-022-01374-z
DO - 10.1007/s11075-022-01374-z
M3 - Article
AN - SCOPUS:85135704673
SN - 1017-1398
VL - 92
SP - 1983
EP - 2023
JO - Numerical Algorithms
JF - Numerical Algorithms
IS - 4
ER -