We study the biofilm-flow interaction resulting in biofilm growth and deformation in a water channel in a 3-D setting using the phase field model developed recently [28, 29]. In this biofilm model, the biofilm made up of the EPS, bacteria and solvent is tracked using a biofilm volume fraction which vanishes outside the biofilm region. The interface between the biofilm and the solvent is marked by the zero level surface of the volume fraction measured from the biofilm to the solvent. The growth of the biofilm and the solvent-biofilm interaction with the top nutrient feeding condition is simulated in the viscous regime (growth regime) of the biofilm-solvent mixture flow. In quiescent flows, the model predicts growth patterns consistent with experimental findings for single or multiple adjacent biofilm colonies, in which the known mushroom shape growth pattern is obtained. Shear induced deformation in biofilms is simulated in a shear cell, providing a viable numerical evidence for using simulation tool to study biofilm growth and interaction dynamics in aqueous environment.

We propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

In this paper we use two numerical methods to solve constrained optimal control problems governed by elliptic equations with rapidly oscillating coefficients: one is finite element method and the other is multiscale finite element method. We derive the convergence analysis for those two methods. Analytical results show that finite element method can not work when the parameter $\varepsilon$ is small enough, while multiscale finite element method is useful for any parameter $\varepsilon$.

In this paper, a new type of gradient recovery method based on vertex-edge-face interpolation is introduced and analyzed. This method gives a new way to recover gradient approximations and has the same simplicity, efficiency, and superconvergence properties as those of superconvergence patch recovery method and polynomial preserving recovery method. Here, we introduce the recovery technique and analyze its superconvergence properties. We also show a simple application in the a posteriori error estimates. Some numerical examples illustrate the effectiveness of this recovery method.

In this paper, we propose a new projection method for solving a general minimization problems with two $L^1$-regularization terms for image denoising. It is related to the split Bregman method, but it avoids solving PDEs in the iteration. We employ the fast iterative shrinkage-thresholding algorithm (FISTA) to speed up the proposed method to a convergence rate $O$($k^-$$^2$). We also show the convergence of the algorithms. Finally, we apply the methods to the anisotropic Lysaker, Lundervold and Tai (LLT) model and demonstrate their efficiency.

In this paper, we propose a mixed Fourier-Jacobi spectral method for two dimensional Neumann boundary value problem. This method differs from the classical spectral method. The homogeneous Neumann boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation. For analyzing the numerical error, we establish the mixed Fourier-Jacobi orthogonal approximation. The convergence of proposed scheme is proved. Numerical results demonstrate the efficiency of this approach.