Wave equations with interface jump conditions have wide applications in engineering and science, for example in acoustics, elastodynamics, seismology, and electromagnetics. In this paper, an efficient non-traditional finite element method with non-body-fitted grids is proposed to solve variable coefficient wave equations with interface jump conditions. Numerical experiments show that this method is approximately second order accurate both in the $L^∞$ norm and $L^2$ norm for piecewise smooth solutions.

Reactive transport problems involve the coupling between the chemical interactions of different species and their transport by advection and diffusion. It leads to the solution of a non-linear systems of partial differential equations coupled to local algebraic or differential equations. Developing software for these two components involves fairly different techniques, so that methods based on loosely coupled modules are desirable. On the other hand, numerical issues such as robustness and convergence require closer couplings, such as simultaneous solution of the overall system. The method described in this paper allows a separation of transport and chemistry at the software level, while keeping a tight numerical coupling between both subsystems. We give a formulation that eliminates the local chemical concentrations and keeps the total concentrations as unknowns, then recall how each individual subsystem can be solved. The coupled system is solved by a Newton–Krylov method. The block structure of the model is exploited both at the nonlinear level, by eliminating some unknowns, and at the linear level by using block Gauss-Seidel or block Jacobi preconditioning. The methods are applied to a 1D case of the MoMaS benchmark.

A posteriori error estimators are studied for discontinuous Galerkin methods for solving a representative elliptic variational inequality of the second kind, known as a simplified frictional contact problem. The estimators are derived by relating the error of the variational inequality to that of a linear problem. Reliability and efficiency of the estimators are theoretically proved.

A general 3D flow and transport model in porous media was derived applying an axiomatic continuum modeling approach, which was implemented using the finite element method to numerically simulate, analyze and interpret microbial enhanced oil recovery (MEOR) processes under laboratory conditions at core scale. From the methodological point of view the development stages (conceptual, mathematical, numerical and computational) of the model are shown. This model can be used as a research tool to investigate the effect on the flow behavior, and consequently the impact on the oil recovery, due to clogging/declogging phenomena by biomass production, and interfacial tension changes because of biosurfactant production. The model was validated and then applied to a case study. The experimental results were accurately predicted by the simulations. Due to its generality, the model can be easily extended and applied to other cases.

In this paper we study the existence and stability of nonconstant positive steady states to a reaction–advection–diffusion system with Rosenzweig–MacArthur kinetics. This system can be used to model the spatial–temporal distributions of predator and prey species . We investigate the effect of prey–taxis on the formation of nonconstant positive steady states in 1D. Stability and instability of these nonconstant steady states are also obtained. We also perform some numerical studies to support the theoretical findings. It is also shown that the Rosenzweig–MacArthur prey–taxis model admits very rich and complicated spatial–temporal dynamics.

We consider a unified variational PDEs model to solve the optic flow problem for large displacements and varying illumination. Although, the energy functional is nonconvex and severely nonlinear, we show that the model offers a well suited framework to extend the efficient methods we used for small displacements. In particular, we resort to an adaptive control of the diffusion and the illumination coefficients which allows us to preserve the edges and to obtain a sparse vector field. We develop a combined space-time parallel programming strategy based on a Schwarz domain decomposition method to speed up the computations and to handle high resolution images, and the parareal algorithm, to enhance the speedup and to achieve a lowest-energy local minimum. This full parallel method gives raise to several iterative schemes and allows us to obtain a good balance between several objectives, e.g. accuracy, cost reduction, time saving and achieving the "best" local minimum. We present several numerical simulations to validate the different algorithms and to compare their performances.

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell–Schrödinger system in the Coulomb gauge. We propose a fully discrete finite element scheme for the system and prove the conservation of energy and the stability estimates of the scheme. By approximating the vector potential A and the scalar potential $ϕ$ respectively in two finite element spaces satisfying certain orthogonality relation, we tackle the mixed derivative term in the discrete system and make the numerical computations and the theoretical analysis more easier. The existence and uniqueness of solutions to the discrete system are also investigated. The (almost) unconditionally error estimates are derived for the numerical scheme without certain restriction like $τ$ ≤ $Ch$^$α$ on the time step $τ$ by using a new technique. Finally, numerical experiments are carried out to support our theoretical analysis.