We use a small positive parameter to change the totalvariation function for unconstrainedMRimage reconstruction to a strictly convex perturbed function. Bregman iteration is applied to solve the modified total-variation MR image (TVMRI) reconstruction problem. A lagged diffusivity fixed-point algorithm is applied to solve the minimization problem in the Bregman iteration. We use the periodic boundary condition and a Fourier transform to accelerate TVMRI reconstruction.RealMR images are used to test the approach in numerical experiments. The experimental results demonstrate that the proposed method is very efficient for TVMRI reconstruction.

For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection–diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in Liu and Yu (2014) [23]when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.

In this paper, we consider an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. In Huang and Li (J. Sci. Comput., 42 (2009), 321–340), for both semi and fully discrete DG schemes, we proved error estimates which are optimal in the energy norm, but sub-optimal in the L2-norm. Here by filling this gap, we show that these schemes are optimally convergent in the L2-norm on quasi-uniform tetrahedral meshes if the solution is sufficiently smooth.

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We propose and study a new parallel one-shot Lagrange--Newton--Krylov--Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible Navier--Stokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems iteratively solve the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally are not easy to converge as they are basically nonlinear Gauss--Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization