High Order Finite Element Calculations for the Cahn-Hilliard Equation

Abstract

In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient avoiding difficult computations or strategies like C 1 elements, adaptive mesh refinement, multi-grid resolution or isogeometric analysis. Beyond the classical benchmarks and comparisons with other existing methods, a numerical study has been carried out to investigate the influence of a polynomial approximation of the logarithmic free energy.Keywords Cahn-Hilliard · Partial differential equation · Bi-Laplacian · p-version of the finite element method · High order · Logarithm · Polynomial energy · Spinodal decomposition

An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations

“…We take the inner product in H(ω * h ) of this equation and µ and owing to formula (19) and the second boundary condition (33) get…”

“…Other polynomial approximations can be obtained by applying the Taylor expansion of Ψ sep (C) [14,19].…”

“…Now we take the inner product in H(ω hk,l * ,m * ) of the momentum balance equation (21) and u k , replace k with i and sum up the results in i = 1, 2, 3. Using the formula 2 , the first formula (19) two times together with the first boundary condition (31) and (32) as well as formula (18) together with (32) we obtain…”

“…We take the inner product in H(ω * h ) of the equation and µ, see (29), apply both formulas (19) together with the boundary conditions (33) and get…”

The Cahn-Hilliard Equation with Logarithmic Potentials

“…We refer the reader to, e.g., [8], [9], [10], [11], [12], [13], [14], [30], [31], [32], [33], [34], [50], [60], [77], [78], [79], [86], [87], [88], [89], [94], [111], [122], [124],…”