Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $p\geq 1$ on meshes with granularity $h$ along with a backward Euler time-stepping scheme with time-step $\Delta t$, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $h^p + \Delta t$. The sharpness of the theoretical estimates are verified through several numerical experiments

Isogeometric analysis of the Cahn-Hilliard equation - a convergence study

Herein, we present a numerical convergence study of the Cahn-Hilliard phase-field model within an isogeometric finite element analysis framework. Using a manufactured solution, a mixed formulation of the Cahn-Hilliard equation and the direct discretisation of the weak form, which requires a C1-continuous approximation, are compared in terms of convergence rates. For approximations that are higher than second-order in space, the direct discretisation is found to be superior. Suboptimal convergence rates occur when splines of order p=2 are used. This is validated with a priori error estimates for linear problems. The convergence analysis is completed with an investigation of the temporal discretisation. Second-order accuracy is found for the generalised-α method. This ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the Cahn-Hilliard equation. The isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition

The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper, we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with arXiv:1912.0712

Discontinuous Galerkin finite element approximation of the Cahn--Hilliard equation with convection

The paper is concerned with the construction and convergence analysis of a discontinuous Galerkin finite element method for the Cahn-Hilliard equation with convection. Using discontinuous piecewise polynomials of degree $p\geq1$ and backward Euler discretization in time, we show that the order-parameter $c$ is approximated in the broken ${\rm L}^\infty({\rm H}^1)$ norm, with optimal order ${\cal O}(h^p+\tau)$; the associated chemical potential $w=\Phi'(c)-\gamma^2\Delta c$ is shown to be approximated, with optimal order ${\cal O}(h^p+\tau)$ in the broken ${\rm L}^2({\rm H}^1)$ norm. Here $\Phi(c)=\frac{1}{4}(1-c^2)^2$ is a quartic free-energy function and \gamma>0 is an interface parameter. Numerical results are presented with polynomials of degree $p=1,2,3$