In this paper we consider, from the numerical point of view, a thermoelastic diffusion porous problem. This is written as a coupled system of two hyperbolic equations, for the displacement and porosity fields, and two parabolic equations, for the temperature and chemical potential fields. Its variational formulation leads to a coupled system of four parabolic variational equations in terms of the velocity, porosity speed, temperature and chemical potential. The existence and uniqueness of weak solutions, as well as an energy decay property, are recalled. Then, the numerical approximation is introduced by using the finite element method for the spatial approximation and the implicit Euler scheme to discretize the time derivatives. A stability property is proved and some a priori error estimates are obtained, from which the convergence of the algorithm is derived and, under suitable additional regularity conditions, its linear convergence is deduced. Finally, some numerical approximations are presented to demonstrate the accuracy of the algorithm and to show the behaviour of the solution.

A novel Immersed-Finite-Element Particle-in-Cell (IFE-PIC) simulation tool is presented in this paper for plasma surface interaction where charged plasma particles are represented by a number of simulation particles. The Particle-in-Cell (PIC) method is one of the major particle models for plasma simulation, which utilizes a huge number of simulation particles and hence provides a first-principle-based kinetic description of particle trajectories and field quantities. The immersed finite element method provides an accurate approach with convenient implementations to solve interface problems based on structured interface-independent meshes on which the PIC method works most efficiently. In the presented IFE-PIC simulation tool, different geometries can be treated automatically for both PIC and IFE through the geometric information specified in an input file. The set of parameters for plasma properties is also assembled into a single input file which can be easily modified for a variety of plasma environments in different applications. Collisions between particles are also incorporated in this tool and can be switched on/off with one parameter in the input file. Efficient modules are adopted to integrate PIC and IFE together into the final simulation tool. Hence our IFE-PIC simulation package offers a convenient and efficient tool to study the microcosmic plasma features for a wide range of applications. Numerical experiments are provided to demonstrate the capability of this tool.

This paper is concerned with an efficient algorithm for damping optimization in mechanical systems with a prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent to the minimization of the trace of the corresponding Lyapunov equation. Thus, the prescribed structure in our case means that a mechanical system is close to a modally damped system. Although our approach is very efficient (as expected) for mechanical systems close to modally damped system, our experiments show that for some cases when systems are not modally damped, the proposed approach provides efficient approximation of optimal damping.

Compositional grading in hydrocarbon reservoirs caused by the gravity force highly affects the design of production and development strategies. In this paper, we propose a novel mathematical modeling for compositional grading based on the laws of thermodynamics. Different from the traditional modeling, the proposed model can dynamically describe the evolutionary process of compositional grading, and it satisfies the energy dissipation property, which is a key feature that real systems obey. The model is formulated for the two scales of free spaces without solids (laboratory scale) and porous media (geophysical scale). For the numerical simulation, we propose a physically convex-concave splitting of the Helmholtz energy density, which leads to an energy-stable numerical method for compositional grading. Using the proposed methods, we simulate binary and ternary mixtures in the free spaces and porous media, and demonstrate that compared with the laboratory scale, the simulation at large geophysical scales has more advantages in simulating the features of compositional grading.

Phase field models are widely used to describe multiphase systems. Here a smooth indicator function, called phase field, is used to describe the spatial distribution of the phases under investigation. Material properties like density or viscosity are introduced as given functions of the phase field. These parameters typically have physical bounds to fulfil, e.g. positivity of the density. To guarantee these properties, uniform bounds on the phase field are of interest. In this work we derive a uniform bound on the solution of the Cahn-Hilliard system, where we use the double-obstacle free energy, that is relaxed by Moreau-Yosida relaxation.

High-resolution image reconstruction obtains one high-resolution image from multiple low-resolution, shifted, degraded samples of a true scene. This is a typical ill-posed problem and optimization models such as the $\ell^2$/TV model are previously studied for solving this problem. It is based on the assumption that during acquisition digital images are polluted by Gaussian noise. In this work, we propose a new optimization model arising from the statistical assumption for mixed Gaussian and impulse noises, which leads us to choose the Moreau envelop of the $\ell^1$-norm as the fidelity term. The developed env$_{\ell^1}$/TV model is effective to suppress mixed noises, combining the advantages of the $\ell^1$/TV and the $\ell^2$/TV models. Furthermore, a fixed-point proximity algorithm is developed for solving the proposed optimization model and convergence analysis is provided. An adaptive parameter choice strategy for the developed algorithm is also proposed for fast convergence. The experimental results confirm the superiority of the proposed model compared to the previous $\ell^2$/TV model besides the robustness and effectiveness of the derived algorithm.

In this paper we present an improved error analysis for a finite difference scheme for solving the 1-D epitaxial thin film model with slope selection. The unique solvability and unconditional energy stability are assured by the convex nature of the splitting scheme. A uniform-in-time $H^m$ bound of the numerical solution is acquired through Sobolev estimates at a discrete level. It is observed that a standard error estimate, based on the discrete Gronwall inequality, leads to a convergence constant of the form exp($CT\varepsilon^{-m}$), where $m$ is a positive integer, and $\varepsilon$ is the corner rounding width, which is much smaller than the domain size. To improve this error estimate, we employ a spectrum estimate for the linearized operator associated with the 1-D slope selection (SS) gradient flow. With the help of the aforementioned linearized spectrum estimate, we are able to derive a convergence analysis for the finite difference scheme, in which the convergence constant depends on $\varepsilon^{-1}$ only in a polynomial order, rather than exponential.

We develop a simple finite element method for solving second order elliptic equations in non-divergence form by combining least squares concept with discontinuous approximations. This simple method has a symmetric and positive definite system and can be easily analyzed and implemented. Also general meshes with polytopal element and hanging node can be used in the method. We prove that our finite element solution approaches to the true solution when the mesh size approaches to zero. Numerical examples are tested that demonstrate the robustness and flexibility of the method.