A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

• Published in: Numerical algorithms 2023-04, Vol.92 (4), p.1983-2023
• Main Author: Baccouch, Mahboub
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Approximation; Boundary conditions; Boundary value problems; Computer Science; Estimates; Exact solutions; Finite element analysis; Finite volume method; Galerkin method; Numeric Computing; Numerical Analysis; Original Paper; Partial differential equations; Polynomials; Theory of Computation; Variables
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-022-01374-z

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 L 2 -norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order p + 1 in the L 2 -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 P p polynomials with p ≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.