scholarly journals An optimal error bound for a finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

1999 ◽  
Vol 33 (5) ◽  
pp. 971-987 ◽  
Author(s):  
John W. Barrett ◽  
James F. Blowey
Keyword(s):  
Finite Element ◽  
Free Energy ◽  
Phase Separation ◽  
Error Bound ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Optimal Error ◽  
Optimal Error Bound

FINITE ELEMENT APPROXIMATION OF A MODEL FOR PHASE SEPARATION OF A MULTI-COMPONENT ALLOY WITH NONSMOOTH FREE ENERGY AND A CONCENTRATION DEPENDENT MOBILITY MATRIX

1999 ◽  
Vol 09 (05) ◽  
pp. 627-663 ◽  
Author(s):  
JOHN W. BARRETT ◽  
JAMES F. BLOWEY
Keyword(s):  
Finite Element ◽  
Free Energy ◽  
Phase Separation ◽  
Space Dimension ◽  
Piecewise Linear ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Linear Finite Element ◽  
Mobility Matrix ◽  
Two Space Dimensions

We consider a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration dependent mobility matrix. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally numerical experiments with three components in one space dimension are presented.

The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis

1992 ◽  
Vol 3 (2) ◽  
pp. 147-179 ◽  
Author(s):  
J. F. Blowey ◽  
C. M. Elliott
Keyword(s):  
Finite Element ◽  
Free Energy ◽  
Phase Separation ◽  
Variational Inequality ◽  
Numerical Analysis ◽  
Finite Element Approximation ◽  
Gradient Theory ◽  
Element Approximation ◽  
Energy Part ◽  
Two Space Dimensions

In this paper we consider the numerical analysis of a parabolic variational inequality arising from a deep quench limit of a model for phase separation in a binary mixture due to Cahn and Hilliard. Stability, convergence and error bounds for a finite element approximation are proven. Numerical simulations in one and two space dimensions are presented.

A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains

2018 ◽  
Vol 52 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Yinnian He ◽  
Jun Zou
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Error Estimates ◽  
Mhd Flow ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Discrete Velocity ◽  
Mhd System

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

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