Chen, Zhangxin Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems. (English) Zbl 0932.65126 East-West J. Numer. Math. 4, No. 1, 1-33 (1996). Summary: The purpose of this paper is to establish an equivalence between mixed and nonconforming finite element methods for second-order elliptic problems on both triangular and rectangular finite elements in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) without using any bubbles. We show that the linear system arising from the mixed method can be algebraically condensed to a symmetric and a positive definite system for Lagrange multipliers, using features of mixed finite element spaces, and the system for the Lagrange multipliers is identical to the system arising from the nonconforming method. We also provide an analysis of multigrid methods for both methods based on equivalence. We prove that optimal order multigrid algorithms can be developed for both methods.Two types of multigrid methods are considered. The first one makes use of the coarse-grid correction on nonconforming finite element spaces, while the second has the coarse-grid correction step established on conforming finite element spaces. Finally, numerical examples are given to illustrate this theory. Cited in 26 Documents MSC: 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs Keywords:mixed methods; nonconforming methods; multigrid methods; convergence; finite element methods; second-order elliptic problems; numerical examples PDFBibTeX XMLCite \textit{Z. Chen}, East-West J. Numer. Math. 4, No. 1, 1--33 (1996; Zbl 0932.65126)