An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy

1995 ◽  
Vol 72 (1) ◽  
pp. 1-20 ◽  
Author(s):  
John W. Barrett ◽  
James F. Blowey
Keyword(s):  
Finite Element ◽  
Free Energy ◽  
Error Bound ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation

2018 ◽  
Vol 52 (3) ◽  
pp. 827-867 ◽  
Author(s):  
A. Agosti
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Fourth Order ◽  
Finite Element Approximation ◽  
Diffuse Interface ◽  
Degenerate Parabolic Equations ◽  
Element Approximation ◽  
Hilliard Equation ◽  
Degenerate Mobility ◽  
Cahn Hilliard Equation

This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions.

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