A posteriori error analysis of an ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

• Published in: Numerical algorithms 2024-12, Vol.97 (4), p.1895-1933
• Main Author: Baccouch, Mahboub
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Approximation; Boundary conditions; Boundary value problems; Computer Science; Convergence; Differential equations; Error analysis; Estimates; Galerkin method; Mathematical analysis; Methods; Numeric Computing; Numerical Analysis; Original Paper; Partial differential equations; Polynomials; Theory of Computation
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-024-01773-4

In this paper, we present and analyze a posteriori error estimates for the ultra-weak local discontinuous Galerkin (UWLDG) method applied to nonlinear fourth-order boundary-value problems for ordinary differential equations of the form - u ( 4 ) = f ( x , u ) . Building upon the superconvergence results established in Baccouch ( Numer Algor 92(4):1983–2023, 2023), we demonstrate the convergence of the UWLDG solution, in the L 2 -norm, towards a special p -degree interpolating polynomial when piecewise polynomials of degree at most p ≥ 2 are employed. The convergence order is proven to be p + 2 . Additionally, we decompose the UWLDG error on each element into two components. The dominant component is proportional to a special ( p + 1 ) -degree polynomial, represented as a linear combination of Legendre polynomials with degrees p - 1 , p , and p + 1 . The second component converges to zero with an order of p + 2 in the L 2 -norm. These findings enable the construction of computationally efficient a posteriori error estimates for the UWLDG method. These estimates are obtained by solving a local problem on each element without imposing boundary conditions. Furthermore, we establish that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the L 2 -norm as the mesh is refined, with a convergence order of p + 2 . In addition, we prove that the global effectivity index converges to unity at a rate of O ( h ) . Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Numerical results are provided to illustrate the reliability and efficiency of the proposed error estimator.