The aim of this paper is to establish the convergence of a fully discrete Crank-Nicolson type Galerkin scheme for the Cauchy problem associated to the KdV equation. The convergence is achieved for initial data in $L^2$, and we show that the scheme converges strongly in $L^2(0, T; L^2_{loc}(\mathbb{R}))$ to a weak solution for some $T >0$. Finally, the convergence is illustrated by a numerical example.

Based on two-grid discretization, a new parallel finite element algorithm for the generalized Stokes problem is proposed and analyzed. Motivated by the observation that for a solution to the generalized Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid, this algorithm first solves the generalized Stokes problem on a coarse grid, and then corrects the resulted residual by standard additive Schwarz method on a fine grid. Under some regular assumptions, error estimates of the approximate solutions are provided. Numerical results are also given to illustrate the effectiveness of the algorithm.

The main novelty of this note is the approach which is used to show that Forward Time and Centred in Space (FTCS) scheme is data dependent stable for scalar hyperbolic conservation laws. Note that FTCS is well known to be unconditionally unstable in von-Neumann sense. In this new approach, the ratio of consecutive gradients is used to classify the initial data region where FTCS is non-oscillatory and stable. Numerical results for 1D scalar and system test problems are given to verify the claim.

We consider the nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be a subset of $\boldsymbol{R^3}$, bounded with two concentric spheres. In the thermodynamical sense the fluid is perfect and polytropic. The homogeneous boundary conditions for velocity, microrotation, heat flux and spherical symmetry of the initial data are proposed. Due to the assumption of spherical symmetry, the problem can be considered as one-dimensional problem in Lagrangian description on the domain that is a segment. We define the approximate equations system by using the finite difference method and construct the sequence of approximate solutions for our problem. By analyzing the properties of these approximate solutions we prove their convergence to the generalized solution of our problem globally in time and establish the convergence of the defined numerical scheme, which is the main result of the paper. The practical application of the proposed numerical scheme is performed on the chosen test example.

For a class of diffusion schemes not satisfying the property of positivity-preserving, we propose an enforcing positivity algorithm. It is locally conservative and easy to be implemented in existing codes. Moreover, this algorithm can be performed on both structured and unstructured meshes. Numerical experiments demonstrate that in terms of $L_2$ error and conservation this algorithm is much better than the trick of directly enforcing the negative values to zero (ENZ), which has been used in applications, meanwhile, in terms of $L_∞$ error it is approximate to ENZ and CEPA repairing algorithms.

This report proves that under the time step condition  $\bigtriangleup t|\Lambda|<1$(| $\cdot$ | = Euclidean norm) suggested by root condition analysis and necessary for stability, all modes of the Crank-Nicolson Leap-Frog (CNLF) approximate solution to the system
                              $\frac{du}{dt}+ Au + \Lambda u = 0$, for $t > 0$ and $u(0) = u_0$,
where $A + A^T$ is symmetric positive definite and $\Lambda$ is skew symmetric, are asymptotically stable. This result gives a sufficient stability condition for non-commutative $A$ and $\Lambda$, and is proven by energy methods. Thus, the growth, often reported in the unstable mode, is not due to systems effects and its explanation must be sought elsewhere.

The tractions that cells exert on a gel substrate from the observed displacements is an increasingly attractive and valuable information in biomedical experiments. The computation of these tractions requires in general the solution of an inverse problem. Here, we resort to the discretisation with finite elements of the associated direct variational formulation, and solve the inverse analysis using a least square approach. This strategy requires the minimisation of an error functional, which is usually regularised in order to obtain a stable system of equations with a unique solution. In this paper we show that for many common three-dimensional geometries, meshes and loading conditions, this regularisation is unnecessary. In these cases, the computational cost of the inverse problem becomes equivalent to a direct finite element problem. For the non-regularised functional, we deduce the necessary and sufficient conditions that the dimensions of the interpolated displacement and traction fields must preserve in order to exactly satisfy or yield a unique solution of the discrete equilibrium equations. We apply the theoretical results to some illustrative examples and to real experimental data. Due to the relevance of the results for biologists and modellers, the article concludes with some practical rules that the finite element discretisation must satisfy.

We consider two second-order variational models in the image inpainting problems. The aim is to obtain in the restored region some fine features of the initial image, e.g. corners, edges, .... The first model is a linear weighted harmonic method well suited for binary images and the second one is its extension to the complex Ginzburg-Landau equation for the inpainting of multi-gray level images. The approach that we introduce consists of constructing a family of regularized functionals and to select locally and adaptively the regularization parameters in order to capture fine geometric features of the image. The parameters selection is performed, at the discrete level, with a posteriori error indicators in the framework of the finite element method. We perform the mathematical analysis of the proposed models and show that they allow us to reconstruct accurately the edges and the corners. Finally, in order to make some comparisons with well established models, we consider the nonlinear anisotropic diffusion and we present several numerical simulations to test the efficiency of the proposed approach.

A kind of non-overlapping domain decomposition preconditioner was proposed to solve the systems generated by the plane wave least-squares (PWLS) method for discretization of Helmholtz equation and Maxwell equations respectively in [13] and [14]. In this paper we introduce overlapping variants of this kind of preconditioner and give some comparison among these domain decomposition preconditioners. The main goal of this paper is to implement in parallel these domain decomposition preconditioners for the system with large wave numbers. The numerical results indicate that the preconditioners are highly scalable and are effective for solving Helmholtz equation and Maxwell's equations with large wave numbers.

We establish the equivalence between the vanishing three-dimensional Riemann- Christoffel curvature tensor of a shell and the two-dimensional Gauss-Codazzi-Mainardi compatibility conditions on its middle surface. Additionally, we produce a new proof of Gauss theorema egregium and Bonnet theorem (reconstructing a surface from its two fundamental forms). This is performed in the very elegant framework of Cartan's moving frames.