An efficient and reliable a-posteriori error estimator is developed for a characteristic-Galerkin finite element method for time-dependent convection-dominated problems. An adaptive algorithm with variable time and space steps is proposed and studied. At each time step in this algorithm grid coarsening occurs solely at the final iteration of the adaptive procedure, meaning that only time and space refinement is allowed before the final iteration. It is proved that at each time step this adaptive algorithm is capable of reducing errors below a given tolerance in a finite number of iteration steps. Numerical results are presented to check the theoretical analysis.

In this paper, we consider the initial data recovery and the solution update based on the local measured data that are acquired during simulations. Each time new data is obtained, the initial condition, which is a representation of the solution at a previous time step, is updated. The update is performed using the least squares approach. The objective function is set up based on both a measurement error as well as a penalization term that depends on the prior knowledge about the solution at previous time steps (or initial data). Various numerical examples are considered, where the penalization term is varied during the simulations. Numerical examples demonstrate that the predictions are more accurate if the initial data are updated during the simulations.

We propose and analyze some itérative algonthms for mixed finite element methods for second-order elhptic équations The method is based on some multilevel décompositions for the finite element space and is related to the standard multigrid and hierarchical basis multigrid methods We show that the algonthms converge wit h rate bounded by 1 (o (2 (o )/(C ƒ), where J is the number of levels and 0 < <o < 2 is a relaxation parameter No regularity assumption beyond that necessary to define the weakform is assumed Hence the result holds for problems withjump coefficients or rough solutions We also estabhsh a uniform convergence rate, independent of the number of levels, by additionally assuming full regularity for the second-order elhptic équations

In this paper we discuss upscaling of immiscible multiphase and miscible multicomponent flow and transport in heterogeneous porous media. The discussion presented in the paper summarizes the results of in Upscaled Modeling in Multiphase Flow Applications by Ginting et al. (2004) and in Upscaling of Multiphase and Multicomponent Flow by Ginting et al. (2006). Perturbation approaches are used to upscale the transport equation that has hyperbolic nature. Our numerical results show that these upscaling techniques give an improvement over the existing upscaled models which ignore the subgrid terms.

The intent of the minisymposium was to discuss the state of the art and the perspectives in approximation and solution strategies related to Domain Decomposition methods for coupled phenomena in physics and engineering that involve multiple models and/or multiple space and time scales. Six from the presented eight talks were devoted to various aspects of the algorithms for solving multiscale problems. These works reflected the common roots of domain decomposition methods and some of the recent approaches for solving multiscale problems, such as Multiscale Finite Element Method, Multiscale Finite Volume Method, etc. The talks covered a wide spectrum of problems related to domain decomposition algorithms for coupled problems, construction and theoretical analysis of a number of algorithm for multiscale problems, and their applications to engineering and industrial problems. The last two talks were devoted to Discontinuous Galerkin Method, which is considered to be suitable for multiphysics problems because of its potential for coupling different discretizations PDE or system of PDEs, as well as for coupling different types of physical models.

In this paper we discuss numerical techniques involved in dynamic data driven application simulations (DDDAS). We present an interpolation technique and update procedures. A multiscale interpolation technique is designed to map the sensor data into the solution space. In particular we show that frequent updating of the sensor data in the simulations can significantly improve the prediction results and thus important for applications. The frequency of sensor data updating in the simulations is related to streaming capabilities and addressed within DDDAS framework (Douglas et al., 2003). We discuss the update of permeability and initial data.

We present an overview of an ongoing project to build a DDDAS for identifying and tracking chemicals in water. The project involves a new class of intelligent sensor, building a library to optically identify molecules, communication techniques for moving objects, and a problem solving environment. We are developing an innovative environment so that we can create a symbiotic relationship between computational models for contaminant identification and tracking in water bodies and a new instrument, the Solid-State Spectral Imager (SSSI), to gather hydrological and geological data and to perform chemical analyses. The SSSI is both small and light and can scan ranges of up to about 10 meters. It can easily be used with remote sensing applications.

The main objective of this report is construction and justification of the new mathematical models for anisotropic nonhomogeneous visco-poro-elastic, piezo-electric and electrically conductive binary mixture and their application in case of thin-walled structures with variable thickness in thermodynamic and stationary nonlinear problems of definition of stress-strain states for thin-walled structures [1]. This investigation could have interesting applications in the areas of pseudo-xsantoma, medical tomography and land mine detection and possible could have an impact in the fields of geophysics, energy exploration, composite manufacturing, earthquake engineering, biomechanics, and many other areas. For the relevant applications it would be necessary to develop and justify new projective numerical-analytical methods. These new methods will be compared with existing methods for problems of that kind and used for recomputating of Basic Elements of Aircrafts. Above proposed models in abstract settings may be presented by the operator equation () [] T

In this paper we consider an integrated model for single-phase fluid flow in elastic porous media. The model and mathematical formulation consist of mass and momentum balance equations for both fluid and porous media. We propose a mixed finite element scheme to solve simultaneously for the porous media displacement, fluid mass flux, and pore pressure. A prototype simulator for solving the integrated problem has been built based on a finite element object library that we have developed. We will present numerical and sensitivity results for the solution algorithm.

Three models for compositional ow in porous media, varying in physical and mathematical complexity, are considered. Various formulations of the governing equations that describe these models, including phase and global pressure-saturation formulations with the total velocity and ux, are presented. Finite element methods for solving these diierential formulations are identiied.

In this article we rst provide a priori estimates of the solution for the nonsta-tionary two-dimensional viscoelastic uid motion equations with periodic boundary condition. We then present an modiied nonlinear Galerkin method for solving such equations. By comparing the convergence rates of the proposed method with the standard Galerkin method, we conclude that the modiied nonlinear Galerkin method is better than the standard Galerkin method because the former can save a large amount of computational work and maintain the convergence rate of the latter.

Tracking characteristics on unstructured meshes is an important part of many numerical methods in computational fluid mechanics. In this paper, we propose an efficient algorithm for characteristic tracking on two-dimensional unstructured triangular meshes. Numerical experiments, including an example for applying this algorithm with the Eulerian-Lagrangian localized adjoint method (ELLAM) to solve a convection-dominated convection-diffusion problem, are presented to demonstrate the efficiency of this algorithm.

Abstract Subgrid effects can have a strong influence on flow and transport in oil reservoirs. In this work a new model for the representation of subgrid terms is introduced and applied to two-phase reservoir flows. The model entails a construction of pseudo relative permeabilities using accurate inlet boundary conditions for the saturation field. Motivation for the form of the model is provided through a consideration of asymptotic techniques for two-phase flow. The numerical computation of the subgrid terms and the implementation of the overall method are described. The accuracy of the new subgrid representation is compared to that of coarse scale models with no subgrid treatment and to coarse models based on traditional pseudo relative permeabilities. In essentially all cases the new model provides more accurate coarse scale predictions relative to reference fine scale results. These are preliminary results.

In this paper we propose a modified multiscale finite element method for two-phase flow simulations in heterogeneous porous media. The main idea of the method is to use the global fine-scale solution at initial time to determine the boundary conditions of the basis functions. This method provides a significant improvement in two-phase flow simulations in porous media where the long-range effects are important. This is typical for some recent benchmark tests, such as the SPE comparative solution project [M. Christie, M. Blunt, Tenth spe comparative solution project: a comparison of upscaling techniques, SPE Reser. Eval. Eng. 4 (2001) 308–317], where porous media have a channelized structure. The use of global information allows us to capture the long-range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions. We present some analysis of the proposed method to illustrate that the method can indeed capture the long-range effect in channelized media.

We develop an Eulerian-Lagrangian iterative solution technique for accurate and efficient numerical simulation of single-phase compositional flow through three-dimensional compressible porous media in the presence of multiple injection and production wells. In the numerical simulator, an Eulerian-Lagrangian method is used to solve the convection-diffusion transport equations and a mixed finite-element method is used to solve the pressure equation. Numerical experiments are presented to investigate the performance of the method. Moreover, because of the Lagrangian nature of the algorithm, the simulator is capable of using large time steps and coarse spatial grids to generate accurate and stable results.

To improve the predictions in dynamic data driven simulations (DDDAS) for subsurface problems, we propose the permeability update based on observed measurements. Based on measurement errors and a priori information about the permeability field, such as covariance of permeability field and its values at the measurement locations, the permeability field is sampled. This sampling problem is highly nonlinear and Markov chain Monte Carlo (MCMC) method is used. We show that using the sampled realizations of the permeability field, the predictions can be significantly improved and the uncertainties can be assessed for this highly nonlinear problem.

We report on two ongoing efforts to build dynamic data driven application systems (DDDAS) for (1) short-range forecasting of weather and wildfire behavior from real time weather data, images, and sensor streams, and (2) contaminant identification and tracking in water bodies. Both systems change their forecasts as new data is received. We use one long term running simulation that self corrects using out of order, imperfect sensor data. The DDDAS versions replace codes that were previously run using data only in initial conditions. DDDAS entails the ability to dynamically incorporate additional data into an executing application, and in reverse, the ability of an application to dynamically steer the measurement process

In this paper we discuss a numerical procedure for performing dynamic data driven simulations (DDDAS). In dynamic data driven simulations our goal is to update the solution as well as input parameters involved in the simulation based on local measurements. The updates are performed in time. In the paper we discuss (1) updating the solution using multiscale interpolation technique (2) recovering as well as updating initial conditions based on least squares approach (3) updating the permeability field using Markov Chain Monte Carlo techniques. We test our method on various synthetic examples.

We develop an Eulerian-Lagrangian numerical model for the simulation of fully miscible, highly compressible, multicomponent fluid flow processes through compressible porous media with multiple injection and production wells. We describe the numerical schemes, the treatment of the multiple injection and production wells, problems related to characteristic tracking, and other issues. We perform numerical experiments to investigate the performance of the numerical model. These results show that the numerical model generates robust, stable, and physically reasonable simulations without nonphysical oscillation or excessive numerical diffusion, even in the presence of multiple injection and production wells and the use of large time steps and coarse spatial grids. Finally, numerical experiments to well known test problems show that the numerical model does not generate noticeable grid orientation effect.

In this paper, we present a hybrid model for coupling of biofilm growth and hydrodynamic flow in a pipe. A cellular automata model, which is a discrete model, is used to describe the growth of biofilm. This stochastic discrete model is coupled with a continuum model of the fluid flow in a pipe. The potential applications of the proposed model are the flow in drinking-water pipe systems or in aquifers, where the porous media can be represented using pore-network models. The proposed model can be used to perform realistic upscaling of cellular automata models to the continuum scale. Numerical results and extensions of the proposed hybrid model are discussed in the paper.