It is a well known fact that finite element solutions of convection dominated problems can exhibit spurious oscillations in the vicinity of boundary layers. One way to overcome this numerical instability is to use schemes that satisfy the discrete maximum principle. There are monotone methods for piecewise linear elements on simplices based on the upwind techniques or artificial diffusion. In order to satisfy the discrete maximum principle for the local projection scheme, we add an edge oriented shock capturing term to the bilinear form. The analysis of the proposed stabilisation method is complemented with numerical examples in 2D.

The Generalized Polarization Tensors (GPT) are a series of tensors which contain information on the shape of a domain and its material parameters. The aim of this paper is to provide a method of constructing GPT-vanishing structures using shape derivative for two-dimensional conductivity or anti-plane elasticity problem. We assume a multi-coating geometry as a candidate of GPT-vanishing structure. We define a cost functional to minimize GPT and compute the shape derivative of this functional deriving an asymptotic expansion of the perturbations of the GPTs due to a small deformation of interfaces of the structure. We present some numerical examples of GPT-vanishing structures for several different shaped inclusions.

We consider the inverse backscattering problem for scalar waves in one dimension. We analyze the convergence of the inverse Bremmer series in this context and study its use in numerical simulations.

The modified Hermitian and skew-Hermitian splitting (MHSS) iteration method and preconditioned MHSS (PMHSS) iteration method were introduced respectively. In the paper, on the basis of the MHSS iteration method, we present a PMHSS iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite ⁄ semi-definite matrices. Under suitable conditions, we prove the convergence of the PMHSS iteration method and discuss the spectral properties of the preconditioned matrix. Moreover, to reduce the computing cost, we establish an inexact variant of the PMHSS iteration method and analyze its convergence property in detail. Numerical results show that the PMHSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

In this paper, we present two alternating direction methods for the solution and best approximate solution of the Sylvester-type matrix equation $AXB+CX^⊤D=E$ arising in the control theory, where $A,B,C,D$ and $E$ are given matrices of suitable sizes. If the matrix equation is consistent (inconsistent), then the solution (the least squares solution) can be obtained. Preliminary convergence properties of the proposed algorithms are presented. Numerical experiments show that the proposed algorithms tend to deliver higher quality solutions with less iteration steps and CPU time than some existing algorithms on the tested problems.

In this article, a topological sensitivity framework for far-field detection of a diametrically small electromagnetic inclusion is established. The cases of single and multiple measurements of the electric far-field scattering amplitude at a fixed frequency are taken into account. The performance of the algorithm is analyzed theoretically in terms of its resolution and sensitivity for locating an inclusion. The stability of the framework with respect to measurement and medium noises is discussed. Moreover, the quantitative results for signal-to-noise ratio are presented. A few numerical results are presented to illustrate the detection capabilities of the proposed framework with single and multiple measurements.

A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of $k-1$ or $k$ parameters. This construction includes all methods of maximal order ($p=k$ for stiff, and $p=k+1$ for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included.