Finite element approximations of positive weak solutions to a one-phase unidimensional moving-boundary system with kinetic condition describing the penetration of a sharp-reaction interface in concrete are considered. A priori and a posteriori error estimates for the semi-discrete fields of active concentrations and for the position of the moving interface are obtained. The important feature of the system of partial differential equations is that the nonlinear coupling occurs due to the presence of both the moving boundary and the non-linearities of localized sinks and sources by reaction.

The aim of this paper is to study the convergence analysis of three low order Crouzeix-Raviart type nonconforming rectangular finite elements to Maxwell's equations, on a mixed finite element scheme and a finite element scheme, respectively. The error estimates are obtained for one of above elements with regular meshes and the other two under anisotropic meshes, which are as same as those in the previous literature for conforming elements under regular meshes.

Parabolic equations with unbounded coefficients and even generalized functions (in particular Dirac-delta functions) model large-scale of problems in the heat-mass transfer. This paper provides estimates for the convergence rate of difference scheme in discrete Sobolev like norms, compatible with the smoothness of the differential problems solutions, i.e. with the smoothness of the input data.

This paper studies various mathematical methods for image reconstruction in electrical impedance and magnetic induction tomography. Linear, nonlinear and semilinear methods for the inverse problems are studied. Depending on the application, one of these methods can be selected as the image reconstruction algorithm. Linear methods are suitable for low contrast imaging, and nonlinear methods are used when more accurate imaging results are required. A semilinear method can be used to preserve some properties of the nonlinear inverse solver and at the same time can have some advantages in computational time. Methods design specifically for jump in material distribution as well as dynamical imaging have been reviewed.

In this paper, we investigate the $L^∞$-error estimates for the solutions of general optimal control problem by mixed finite element methods. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive $L^∞$-error estimates of optimal order both for the state variables and the control variable.

The cycle index polynomial of a symmetric group is a basic tool in combinatorics and especially in Pόlya enumeration theory. It seems irrelevant to numerical analysis. Through Faá di Bruno's formula, cycle index is connected with numerical analysis. In this work, the Hermite interpolation polynomial is explicitly expressed in terms of cycle index. Applications in Gauss-Turán quadrature formula are also considered.

In this work we present three age-structured models with spatial dependence. We introduce an improved explicit method, namely Super-Time-Stepping (STS) developed for parabolic problems and we use its modification for the numerical treatment of our models. We explain how the acceleration scheme can be adapted to the age-dependent models. We prove convergence of the method in case of Dirichlet boundary conditions and we demonstrate the accuracy and the efficiency of the Modified STS comparing it with other numerical algorithms of same or higher order, namely the explicit, fully implicit and Crank-Nicolson standard schemes.

In this article, we are interested in the mathematical modeling of singular electromagnetic fields, in a non-convex polyhedral domain. We first describe the local trace (i. e. defined on a face) of the normal derivative of an $L^2$ function, with $L^2$ Laplacian. Among other things, this allows us to describe dual singularities of the Laplace problem with homogeneous Neumann boundary condition. We then provide generalized integration by parts formulae for the Laplace, divergence and curl operators. With the help of these results, one can split electromagnetic fields into regular and singular parts, which are then characterized. We also study the particular case of divergence-free and curl-free fields, and provide non-orthogonal decompositions that are numerically computable.

In this paper, we will develop the convergence of the solution of TV-regularization equations with regularized parameter $\varepsilon \rightarrow 0$ in BV($\Omega$) for practical purposes. Originated from the effects of regularized parameter $\varepsilon$, the error rate of finite element approximation for TV-regularization equations will be controlled by the regularized parameter $\varepsilon^{-1}$ polynomially in the energy norm when using linearization technique and duality argument. And in the $L^p$-norm, the effect of regularized parameter $\varepsilon$ will be more extremely.