Non-smooth optimization problems based on $L^1$ norms are investigated for smoothing of signals with noise or functions with sharp gradients. The use of $L^1$ norms allows to reduce the blurring introduced by methods based on $L^2$ norms. Numerical methods based on over-relaxation and augmented Lagrangian algorithms are proposed. Applications to free surface flows and image denoising are presented.

Numerical simulation of fluid flow and transport processes in the subsurface must account for the presence of wells. The pressure at a gridblock that contains a well is different from the average pressure in that block and different from the flowing bottom hole pressure for the well [17]. Various finite difference well models have been developed to account for the difference. This paper presents a systematical derivation of well models for other numerical methods such as standard finite element, control volume finite element, and mixed finite element methods. Numerical results for a simple well example illustrating local grid refinement effects are given to validate these well models. The well models have particular applications to groundwater hydrology and petroleum reservoirs.

This paper concerns with the finite element approximation of a nonlinear second order parabolic system which describes the $L^2$-gradient flow for a class of linear growth energy functionals. Besides their appeals in differential geometry and calculus of variations, linear growth energy functionals and their gradient flows also arise naturally from emerging applications of image processing such as color image denoising. In this paper, we introduce a family of variational models for color image denoising which minimize linear growth energy functionals of maps into the unit sphere in $\rm{R}^3$. These models generalize the popular 1-harmonic map model which has been studied intensively in recent years. To compute the solutions of the variational models, we first derive their $L^2$-gradient flow equations and then introduce some fully discrete implicit finite element method for the gradient flow equations. It is proved that the proposed finite element method is uniquely solvable and absolutely stable, and the finite element solution converges to the PDE solution as the mesh sizes tend to zero. Numerical experiments are presented to demonstrate the effectiveness of the proposed variational models for color image denoising and to show the efficiency of the proposed finite element method. A numerical comparison of the proposed models with the channel-by-channel model is also presented.

A non-smooth constrained minimization problem arising in the stress analysis of a plastic body is considered. A numerical method for the computation of the load capacity ratio is presented to determine if the elastic body fractures under external traction. In the scalar case, the maximum principle allows one to reduce the problem to a convex one under linear constraints. An augmented Lagrangian method, together with an approximation by finite elements is advocated for the computation of the load capacity ratio and the corresponding elastic stress. The generalized eigenvalues and eigenvectors of the corresponding operator are computed for various two-dimensional bodies and fractures are discussed.

In this paper, we propose a dimension splitting method for Navier-Stokes equations (NSEs). The main idea is as follows. The domain of flow in 3D is decomposed into several thin layers. In each layer, The 3D NSEs can be represented as the sum of a membrane operator and a normal (bending) operator on the boundary of layer. And The Euler central difference is used to approximate the bending operator. When restricting the 3D NSEs on the boundary in each layer, we obtain a series of two dimensional-three components NSEs (called as 2D-3C NSEs). Then we construct an approximate solution of 3D NSES by solutions of those 2D-3C NSEs.

In this article, we investigate the application of an operator-splitting/finite element method to the numerical simulation of a liquid crystal  flow. The operator-splitting is achieved through three stages, so that each stage is simpler and easier to deal with than the step of any un-split implicit scheme. The first stage deals with the system coupling a Stokes equation for velocity with an equation modeling the diffusion of the liquid crystal director field. The second stage deals with the convection of both the velocity and director field; a wave-like equation approach is used to treat this advection part and proves being quite efficient. Finally, the third stage deals with the nonlinear terms; a (quasi) closed form solution can be derived for this stage. Overall, with this type of splitting, the nonlinear terms in the liquid crystal model can be treated quite easily. The results of several numerical experiments show the good performances of the three-stage splitting method discussed in this article.

A spring model is applied to simulate the skeleton structure of the red blood cell (RBC) membrane and to study the red blood cell (RBC) rheology in microvessels. The biconcave RBC shape in static plasma and tank-treading behavior of single cell in shear flows have been successfully captured in this model. The behavior of the RBC in a Poiseuille flow and the lateral migration of the cells in a shear flow have been investigated. It is found that the RBCs exhibit parachute shape in a Poiseuille flow with the curvature closely related to the deformability of the cell membrane and the hematocrit (Hct) of the blood. With this spring model, RBCs can recover their initial shapes associated with the minimal elastic energy when the flow stops. The simulation results also show that the RBCs migrate to the center of the domain in the radial direction in a shear flow, which clearly indicates the Fahraeus-Lindqvist effect in microvessels. The rate of migration toward the center depends on the shape of the RBC; the biconcave shape enhances this migration.

The details of melting of DNA immobilized on a chip or nanoparticle determines the sensitivity and operating characteristics of many analytical and synthetic biotechnological devices. Yet, little is known about the differences in how the DNA melting occurs between a homogeneous solution and that on a chip. We used molecular dynamics simulations to explore possible pathways for DNA melting on a chip. Simulation conditions were chosen to ensure that melting occurred in a submicrosecond timescale. The temperature was set to 400 K and the NaCl concentration was set to 0.1 M. We found less symmetry than in the solution case where for oligomeric double-stranded nucleic acids both ends melted with roughly equal probability. On a prepared silica surface we found melting is dominated by fraying from the end away from the surface. Strand separation was hindered by nonspecific surface adsorption at this temperature. At elevated temperatures the melted DNA was attracted to even uncharged organically coated surfaces demonstrating surface fouling. While hybridization is not the simple reverse of melting, this simulation has implications for the kinetics of hybridization.

Numerical simulation in aluminum production involves many different topics, this paper is dedicated to some aspects of two of them. In the first part, a numerical approach for the computation of the motion of the fluids in an electrolytic cell is presented, with an emphasis on a mesh deformation technique to track the free interface during the time. In a second part, a model for the simulation of alumina dissolution and repartition is exposed.

This paper is concerned with the development and study of a stabilized finite volume method for the transient Stokes problem in two and three dimensions. The stabilization is based on two local Gauss integrals and is parameter-free. The analysis is based on a relationship between this new finite volume method and a stabilized finite element method using the lowest equal-order pair (i.e., the $P_1$-$P_1$ pair). An error estimate of optimal order in the $H^1$-norm for velocity and an estimate in the $L^2$-norm for pressure are obtained. An optimal error estimate in the $L^2$-norm for the velocity is derived under an additional assumption on the body force.

In this paper the mechanism and the character of the formation of singularities caused by eigenvalues or (and) eigenvectors, respectively, will be discussed for 1-D quasilinear hyperbolic systems.