Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation

This paper provides a discrete Poincaré inequality innspace dimensions on a simplexKwith explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero onKand all integrals of jumps zero along all interior sides by its Lebesgue norm times{C(n)\operatorname{diam}(K)}. The explicit constant{C(n)}depends only on the dimension{n=2,3}in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle{\omega_{0}}in the triangulation. The analysis allows the bound of two constants{\Lambda_{1}}and{\Lambda_{3}}in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.

Low-Order Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics Equations

The nonconforming mixed finite element methods (NMFEMs) are introduced and analyzed for the numerical discretization of a nonlinear, fully coupled stationary incompressible magnetohydrodynamics (MHD) problem in 3D. A family of the low-order elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field, and the magnetic field. The existence and uniqueness of the approximate solutions are shown, and the optimal error estimates for the corresponding unknown variables inL2-norm are established, as well as those in a brokenH1-norm for the velocity and the magnetic fields. Furthermore, a new approach is adopted to prove the discrete Poincaré-Friedrichs inequality, which is easier than that of the previous literature.