Lagrangian Mixed Finite Element Methods for Nonlinear Thin Shell Problems

2020 ◽  
Vol 20 (4) ◽  
pp. 631-642 ◽  
Author(s):  
Mikhail M. Karchevsky
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Elastic Shell ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Mixed Finite Element Methods ◽  
Systems Of Nonlinear Equations ◽  
Discrete Problem ◽  
Second Derivatives

AbstractA class of Lagrangian mixed finite element methods is constructed for an approximate solution of a problem of nonlinear thin elastic shell theory, namely, the problem of finding critical points of the functional of potential energy according to the Budiansky–Sanders model. The proposed numerical method is based on the use of the second derivatives of the deflection as auxiliary variables. Sufficient conditions for the solvability of the corresponding discrete problem are obtained. Accuracy estimates for approximate solutions are established. Iterative methods for solving the corresponding systems of nonlinear equations are proposed and investigated.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Dongyang Shi ◽  
Zhiyun Yu
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Approximate Solutions ◽  
Mixed Finite Element Methods ◽  
Optimal Error Estimates ◽  
Stationary Incompressible Magnetohydrodynamics ◽  
Friedrichs Inequality ◽  
Incompressible Magnetohydrodynamics ◽  
Low Order

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.

Sign in / Sign up

Export Citation Format

Share Document