PENALTY-FACTOR-FREE DISCONTINUOUS GALERKIN METHODS FOR 2-DIM STOKES PROBLEMS

• Published in: SIAM journal on numerical analysis Vol. 49; no. 5/6; pp. 2165 - 2181
• Main Author: LIU, JIANGGUO
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Algebra; Algebraic geometry; Approximation; Boundary conditions; Estimation methods; Exact sciences and technology; Finite element method; Galerkin methods; Linear systems; Mathematical discontinuity; Mathematics; Navier Stokes equation; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Shape functions; Velocity; Discontinuous Galerkin method; divergence residuals; Error estimation; Stokes problem; nonconforming finite elements; Numerical linear algebra; Non conforming finite element; Iterative method; Optimal estimation; 76D07; Stokes problems; Implementation; Penalty method; Direct method; Finite element method; flux jumps; Numerical analysis; Linear system; locally divergence-free (LDF); Matrix inversion; discontinuous Galerkin (DG) methods; Two-dimensional calculations; 65N30; weak over penalization
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/10079094X

This paper presents two new finite element methods for two-dimensional Stokes problems. These methods are developed by relaxing the constraints of the Crouzeix-Raviart nonconforming P₁ finite elements. Penalty terms are introduced to compensate for lack of continuity or the divergence-free property. However, there is no need for choosing penalty factors, and the formulations are symmetric. These new methods are easy to implement and avoid solving saddle-point linear systems. Numerical experiments are presented to illustrate the proved optimal error estimates.