Luo, Zhendong; Wang, Lieheng Nonlinear Galerkin mixed element methods for the non stationary conduction-convection problems. II: The backward one-step Euler fully discrete format. (English. Chinese original) Zbl 0959.76044 Chin. J. Numer. Math. Appl. 21, No. 1, 86-105 (1999); translation from Math. Numer. Sin. 20, No. 4, 431-448 (1998). Summary: [For part I see the authors, Chin. J. Numer. Math. Appl. 20, No. 4, 71–94 (1998; Zbl 0928.76059); translation from Math. Numer. Sin. 20, No. 3, 305–324 (1998).]Summary: We present a fully discrete format of nonlinear Galerkin mixed element method with backward one-step Euler time discretization for non-stationary conduction-convection problems. The scheme is based on two finite element spaces \(X_H\) and \(X_h\) for the approximation of velocity, defined respectively on a coarse grid with grid size \(H\), and on another fine grid with grid size \(h\ll H\), a finite element space \(M_h\) for the approximation of pressure, and two finite element spaces \(W_H\) and \(W_h\) for the approximation of temperature, also defined respectively on the coarse grid with grid size \(H\) and on another fine grid with grid size \(h\). We demonstrate the existence and the convergence of fully discrete mixed element solution. The scheme consists in using standard backward one-step Euler-Galerkin fully discrete format for the first \(L_0\) steps \((L_0\geq 2)\) on the fine grid with grid size \(h\), but using nonlinear Galerkin mixed element method of backward one-step Euler-Galerkin fully discrete format through the (\(L_0+1\))th step to the end step. We prove that for the fully discrete nonlinear Galerkin mixed element procedure with respect to the coarse grid spaces with grid size \(H\) holds superconvergence. Cited in 1 ReviewCited in 2 Documents MSC: 76M10 Finite element methods applied to problems in fluid mechanics 76R99 Diffusion and convection 76D05 Navier-Stokes equations for incompressible viscous fluids 80A20 Heat and mass transfer, heat flow (MSC2010) Keywords:fully discrete method; nonlinear Galerkin mixed element method; backward one-step Euler time discretization; non-stationary conduction-convection problems; coarse grid; fine grid; existence; convergence; superconvergence Citations:Zbl 0928.76059 PDFBibTeX XMLCite \textit{Z. Luo} and \textit{L. Wang}, Chin. J. Numer. Math. Appl. 21, No. 1, 131--448 (1998; Zbl 0959.76044); translation from Math. Numer. Sin. 20, No. 4, 431--448 (1998)