We define an error indicator for mixed mortar formulation of flow in porous media. The mixed mortar domain decomposition method for single-phase flow problems was defined by Arbogast et al; it relies on coupling of subdomain problems using mortar Lagrange multipliers defined as continuous piecewise linears on the subdomain interface. The accuracy and efficiency of the resulting interface formulation depends on the number of mortar degrees of freedom which we propose to adapt using error indicators involving jump of the flux across the interface. Rigorous a-posteriori analysis and proof of reliability of the estimator are established for single-phase 2D flow problems with diagonal coefficients for $\rm{RT}_{[0]}$ spaces on rectangular grids. Computational experiments demonstrate the application of the estimator. Next, the algorithm and indicator are extended to the two-phase flow case which is illustrated with numerical examples. We focus on adapting the mortar grid while keeping subdomain grid fixed. Full mortar adaptivity is discussed elsewhere.

The paper discusses a general framework for handling curvilinear geometries in high accuracy Finite Element (FE) simulations, for both elliptic and Maxwell problems. Based on the differential manifold concept, the domain is represented as a union of geometrical blocks prescribed with globally compatible, explicit or implicit parameterizations. The idea of parametric $H^1$-, $H(curl)$- and $H(div)$-conforming elements is reviewed, and the concepts of exact geometry elements and isoparametric elements are discussed. The paper focuses then on isoparametric elements, and two ways of computing FE discretization errors: a popular one, neglecting the geometry approximation, and a precise one, utilizing the exact geometry representation. Presented numerical examples indicate the necessity of accounting for the geometry error in FE error calculations, especially for the $H(curl)$ problems.

On the basis of the estimates for the regularized Green's functions with memory terms, optimal order $L^∞$-error estimates are established for the nonFickian flow of fluid in porous media by means of a mixed Ritz-Volterra projection. Moreover, local $L^∞$-superconvergence estimates for the velocity along the Gauss lines and for the pressure at the Gauss points are derived for the mixed finite element method, and global $L^∞$-superconvergence estimates for the velocity and the pressure are also investigated by virtue of an interpolation post-processing technique. Meanwhile, some useful a-posteriori error estimators are presented for this mixed finite element method.

Emerging diseases in animals and plants have led to much research on questions of evolution and persistence of pathogens. In particular, there have been numerous investigations on the evolution of virulence and the dynamics of epidemic models with multiple pathogens. Multiple pathogens are involved in the spread of many human diseases including influenza, HIV-AIDS, malaria, dengue fever, and hantavirus pulmonary syndrome [9, 15, 16, 23, 24, 27]. Understanding the impact of these various pathogens on a population is particularly important for their prevention and control. W summarize some of the results that have appeared in the literature on multiple pathogen models. Then we study the dynamics of a deterministic and a stochastic susceptible-infected epidemic model with two pathogens, where the population is subdivided into susceptible individuals and individuals infected with pathogen $j$ for $j = 1, 2$. The deterministic model is a system of ordinary differential equations, whereas the stochastic model is a system of stochastic differential equations. The models assume total cross immunity and vertical transmission. The conditions on the parameters for coexistence of two pathogens are summarized for the deterministic model. Then we compare the coexistence dynamics of the two models through numerical simulations. We show that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two pathogens. The probability of coexistence in the stochastic epidemic model is very small. Stochastic variability results in extinction of at least one of the strains. Our results demonstrate the importance of understanding the dynamics of both the deterministic and stochastic epidemic models.

This paper analyzes a nonstandard form of the Stokes problem where the mass conservation equation is expressed in the form of a Poisson equation for the pressure. This problem is shown to be well-posed in the $d$-dimensional torus. A nonconforming approximation is proposed and, contrary to what happens when using the standard saddle-point formulation, the proposed setting is shown to yield optimal convergence for every pairs of approximation spaces.

Global almost sure asymptotic stability of the trivial solution of some nonlinear stochastic difference equations with in-the-arithmetic-mean-sense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in $\rm{IR}^1$. This result can be used to verify asymptotic stability of stochastic-numerical methods such as partially drift-implicit trapezoidal methods for nonlinear stochastic differential equations with variable step sizes.