The least squares spectral element method for the Cahn–Hilliard equation

Abstract

a b s t r a c tThe problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn-Hilliard (C-H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C-H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the CahnHilliard equation. The LS-SEM is combined with a time-space coupled formulation and a high order continuity approximation by employing C 11 p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn-Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C-H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.

Phase-Field Models for Multi-Component Fluid Flows

“…And the scheme preserved the mass conservation and energy dissipation properties of the original problem. In [34], the least squares spectral element method was used to solve the Cahn-Hilliard equation.…”

The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods

“…Some of these methods are Galerkin finite element method [29,30], secondorder splitting method [31], nonconforming finite element method [32], numerical analysis with a logarithmic free energy [18], unconditionally gradient stable scheme [33], semi-implicit Fourier-spectral method [15,96], stable and conservative finite difference technique [37], conservative multigrid method [56], discontinuous Galerkin method [86], moving mesh method [35], adaptive mesh refinement idea [88], boundary integral method [19], local discontinuous Galerkin method [90], large time-stepping method [43], isogeometric analysis procedure [41], strongly anisotropic CH equation by an adaptive nonlinear multigrid method [87], conservative scheme with contact angle boundary condition [58], conservative scheme with Neumann and Dirichlet boundary conditions in complex domain [61,74], parallel multigrid method [73]. Also a class of stable spectral methods for the CH equation [44], the least squares spectral element method [36], a multigrid finite element solver [52] are applied for solving the one, two and three-dimensional CH equations.…”

Numerical Solution of Cahn-Hilliard System by Adaptive Least-Squares Spectral Element Method

“…The same basis functions and construction approach have been used in our previous study [4]. For more details we also refer to [5] and [6]. Together with integration by the Gaussian quadrature based on the GLLroots, the discretization of the least-squares formulation (10) can be expressed on an element-level as…”