In this paper, an iterative approach for constructing immersed finite element spaces is developed for various interface conditions of interface problems involving multiple primary variables. Combining such iteratively constructed immersed finite element spaces with the distributed Lagrange multiplier/fictitious domain (DLM/FD) method, we further present a new discretization method that can uniformly solve general interface problems with multiple primary variables and/or with different governing equations on either side of the interface, including fluid-structure interaction problems. The strengths of the proposed method are shown in the numerical experiments for Stokes- and Stokes/elliptic interface problems with different types of interface conditions, where, the optimal or nearly optimal convergence rates are obtained for the velocity variable in $H^1$, $L^2$ and $L$^∞ norms, and at least 1.5-th order convergence for the pressure variable in $L^2$ norm within few number of iterations. In addition, numerical experiments show that such iterative process uniformly converges and the number of iteration is independent of mesh ratios and jump ratios.

Convexity splitting like schemes with improved accuracy are proposed for a phase field model for surface diffusion. The schemes are developed to enable large scale simulations in three spatial dimensions describing experimentally observed solid state dewetting phenomena. We carefully elaborate the loss in accuracy associated with large time steps in such schemes and show the existence of a maximal numerical timestep to achieve a prescribed accuracy. We demonstrate the increase of this maximal numerical time step by at least one order of magnitude using a Rosenbrock method. This convexity splitting scheme with improved accuracy is used to study the effect of contact angle on solid state dewetting phenomena.

In this paper, we develop the adaptive interior penalty discontinuous Galerkin method based on a new $a$ $posteriori$ error estimate for the second-order elliptic boundary-value problems. The new $a$ $posteriori$ error estimate is motivated from the smoothing iteration of the $m$-time Gauss-Seidel iterative method, and it is used to construct the adaptive finite element method. The efficiency and robustness of the proposed adaptive method is demonstrated by extensive numerical experiments.

Based on convolution quadrature in time and continuous piecewise linear finite element approximation in space, a Crank-Nicolson type method is proposed for solving a partial differential equation involving a fractional time derivative. The method achieves second-order convergence in time without being corrected at the initial steps. Optimal-order error estimates are derived under regularity assumptions on the source and initial data but without having to assume regularity of the solution.

In this paper, we study a new nodal-type trapezoidal rule for approximating Hadamard finite-part integrals, and its application to numerical solution of certain finite-part integral equation. We start with a nodal-type trapezoidal rule discussed in [21], and then establish its error expansion analysis, from which a new nodal-type trapezoidal rule with higher order accuracy is proposed and corresponding error analysis is also obtained. Based on the proposed rule, a new collocation scheme is then constructed to solve certain finite-part integral equation, with the optimal error estimate being rigorously derived. Some numerical experiments are also performed to verify the theoretical results.

In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L$² stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method.

In this paper, a new time second-order characteristic finite element method is proposed and analyzed for solving the advection-diffusion equations involving a nonlinear right side term. In order to obtain a time second-order characteristic scheme, the global derivative term transferred from the time derivative and advection terms is discretized by using difference operator along the characteristics, the diffusion term is discretized by the average operator along the characteristics, while specially a second-order extrapolation along the characteristics is applied to the right side nonlinear function. We analyze and prove that the proposed scheme for the nonlinear advection-diffusion equations has second order accuracy in time step size, which improves the first order accuracy in time of the classical characteristic methods. The proposed characteristic FEM scheme allows to use large time step sizes in computation. Numerical tests are taken to show the accuracy of our proposed scheme, and the case of single-species population dynamics is further simulated and analyzed by using our method and numerical results show its advantage and effectiveness.

In this paper we assess the performance of a selection of load balancing strategies for a parallel, adaptive multigrid solver that has been developed for the implicit solution of phase-field problems. The strategies considered include a number of standard approaches and a new technique that we propose specifically for multigrid solvers. This technique takes account of the sequential nature of the grid correction used in multiplicative multilevel algorithms such as multigrid. The paper focuses on two phase-field example problems which model the rapid solidification of an undercooled binary alloy: using isothermal and non-isothermal models respectively. We undertake a systematic comparison of the different load-balancing strategies for a selection of different adaptive mesh scenarios. We conclude that the optimal choice of load-balancing strategy depends critically on the computation to communication ratio of the parallel multigrid solver, and that in the computation-dominated limit our proposed technique is typically the most effective of those considered.

In this paper, an energy-conservation scheme is derived and analysed for solving Hamiltonian wave equations subject to Neumann boundary conditions in two dimensions. The energy-conservation scheme is based on the blend of spatial discretisation by a fourth-order finite difference method and time integration by the Average Vector Field (AVF) approach. The spatial discretisation via the fourth-order finite difference leads to a particular Hamiltonian system of second-order ordinary differential equations. The conservative law of the discrete energy is established, and the stability and convergence of the semi-discrete scheme are analysed. For the time discretisation, the corresponding AVF formula is derived and applied to the particular Hamiltonian ODEs to yield an efficient energy-conservation scheme. The numerical simulation is implemented for various cases including a linear wave equation and two nonlinear sine-Gordon equations. The numerical results demonstrate the spatial accuracy and the remarkable energy-conservation behaviour of the proposed energy-conservation scheme in this paper.

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [7]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.

Experimental investigation has shown that semiflexible polymers can wrap orderly around a cylinder. Recent Monte Carlo simulations also show that semiflexible polymers can develop linear or helical or random phase structures, depending on the rigidity or length of the polymer. Here, we use wormlike chain model and self-consistent field theory with Onsager interaction to study the micro-phase structure of polymers with local rigidity. We first give the modified diffusion equation for a wormlike chain on cylindrical surface, and then solve the equilibrium equations of the self-consistent field. A time splitting scheme is developed to solve the modified diffusion equation. However, only two kinds of nematic structures (N1 and N2) are detected in our simulation. In N1, the polymers are mainly oriented perpendicular to the axis of the cylinder; while in N2, the polymers are mainly oriented parallel to the axis of the cylinder. N1 is a metastable structure with free energy higher than N2.