Evaporation within porous media is both a multiscale and interface-driven process, since the phase change at the evaporating interfaces within the pores generates a vapour flow and depends on the transport of vapour through the porous medium. While homogenised models of flow and chemical transport in porous media allow multiscale processes to be modelled efficiently, it is not clear how the multiscale effects impact the interface conditions required for these homogenised models. In this paper, we derive a homogenised model, including effective interface conditions, for the motion of an evaporation front through a porous medium, using a combined homogenisation and boundary layer analysis. This analysis extends previous work for a purely diffusive problem to include both gas flow and the advective–diffusive transport of material. We investigate the effect that different microscale models describing the chemistry of the evaporation have on the homogenised interface conditions. In particular, we identify a new effective parameter, $\mathcal{L}$, the average microscale interface length, which modifies the effective evaporation rate in the homogenised model. Like the effective diffusivity and permeability of a porous medium, $\mathcal{L}$ may be found by solving a periodic cell problem on the microscale. We also show that the different microscale models of the interface chemistry result in fundamentally different fine-scale behaviour at, and near, the interface.

Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$, and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$-elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments.

This paper presents a new immersed finite volume element method for solving second-order elliptic problems with discontinuous diffusion coefficient on a Cartesian mesh. The new method possesses the local conservation property of classic finite volume element method, and it can overcome the oscillating behaviour of the classic immersed finite volume element method. The idea of this method is to reconstruct the control volume according to the interface, which makes it easy to implement. Optimal error estimates can be derived with respect to an energy norm under piecewise $H^{2}$ regularity. Numerical results show that the new method significantly outperforms the classic immersed finite volume element method, and has second-order convergence in $L^{\infty }$ norm.

The decontamination of hazardous chemical agents from porous media is an important and critical part of the clean-up operation following a chemical weapon attack. Decontamination is often achieved through the application of a cleanser, which reacts on contact with an agent to neutralise it. While it is relatively straightforward to write down a model that describes the interplay of the agent and cleanser on the scale of the pores in the porous medium, it is computationally expensive to solve such a model over realistic spill sizes.

In this paper, we consider the homogenisation of a pore-scale model for the interplay between agent and cleanser, with the aim of generating simplified models that can be solved more easily on the spill scale but accurately capture the microscale structure and chemical activity. We consider two situations: one in which the agent completely fills local porespaces and one in which it does not. In the case when the agent does not completely fill the porespace, we use established homogenisation techniques to systematically derive a reaction–diffusion model for the macroscale concentration of cleanser. However, in the case where the agent completely fills the porespace, the homogenisation procedure is more in-depth and involves a two-timescale approach coupled with a spatial boundary layer. The resulting homogenised model closely resembles the microscale model with the effect of the porous material being incorporated into the parameters. The two models cater for two different spill scenarios and provide the foundation for further study of reactive decontamination.

In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with ε-periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative via a function of order εγ with γ ≤−1. We give the homogenization results which include those obtained by Jose in [Rev. Roum. Math. Pures Appl. 54 (2009) 189–222]. We also get the corrector results.

This article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems. The numerical examples demonstrate the features of the proposed algorithms, including the optimal convergence in both L2 and semi-H1 norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location.

The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of orderεγ . We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.

We investigate time harmonic Maxwell equations in heterogeneous media, where thepermeability μ and the permittivity ε are piecewise constant. Theassociated boundary value problem can be interpreted as a transmission problem. Ina very natural way the interfaces can have edges and corners. We give a detaileddescription of the edge and corner singularities of the electromagnetic fields.