Convergence of Multi-level Algorithms for a Class of Nonlinear Problems

• Published in: Journal of scientific computing Vol. 84; no. 2; p. 34
• Main Authors: Kim, Dongho, Park, Eun-Jae, Seo, Boyoon
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2020; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Error analysis; Estimates; Grid refinement (mathematics); Hypotheses; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Norms; Theoretical; Navier–Stokes equations; 65N15; Error estimates; 65N50; Nonlinear problems; 76D05; Two-grid algorithm; Pseudostress-velocity; Multi-level Mesh refinement; 65N30
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-020-01287-w

In this article, we develop and analyze two-grid/multi-level algorithms via mesh refinement in the abstract framework of Brezzi, Rappaz, and Raviart for approximation of branches of nonsingular solutions. Optimal fine grid accuracy of two-grid/multi-level algorithms can be achieved via the proper scaling of relevant meshes. An important aspect of the proposed algorithms is the use of mesh refinement in conjunction with Newton-type methods for system solution in contrast to the usual Newton’s method on a fixed mesh. The pseudostress-velocity formulation of the stationary, incompressible Navier–Stokes equations is considered as an application and the Raviart–Thomas mixed finite element spaces are used for the approximation. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.