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The solution of the singular perturbation problems (SPP) of convection-diffusion-reaction type may exhibit regular and corner layers in a rectangular domain. In this work, we construct and analyze a parameter-uniform operator-splitting alternating direction implicit (ADI) scheme to efficiently solve a two-dimensional parabolic singularly perturbed problem with two positive parameters. The proposed model is a combination of the backward-Euler method defined on a uniform mesh in time and a hybrid method in space defined on a special Shishkin mesh. The analysis is presented on a layer adapted piecewise-uniform Shishkin mesh. The developed numerical method is proved to be first-order convergent in time and almost second-order convergent in space. The numerical experiments are performed to validate the theoretical convergence results and illustrate the efficiency of the current strategy. [ABSTRACT FROM AUTHOR]

Generalized eigenvalue problems on the overlap(GenEO) is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers that are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection–diffusion–reaction problems, which may be nonself-adjoint and indefinite, and whose discretizations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalized eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts that are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the nonself-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus, we obtain an iterative solver that is efficient in parallel and very effective for a wide range of convection–diffusion–reaction problems. [ABSTRACT FROM AUTHOR]

The goal of the work is to solve the nonlinear convection-diffusion-reaction problem using the variational iteration method with the combination of the Chebyshev wavelet. This work developed a hybrid iterative technique named as Variational iteration method with the Chebyshev wavelet for the solutions of nonlinear convection-diffusion-reaction problems. The aim of applying the derived algorithm is to achieve fast convergence. During the solution of the given problem, the restricted variations will be mathematically justified. The effects of the scaling and other parameters like diffusion parameter, convection parameter, and reaction parameter on the solution are also focused on by their suitable selection. The approximate results include the error profiles and the simulations. The results of variational iteration with the Chebyshev wavelet are compared with variational iteration method, the Modified variational iteration method, and the Variational iteration method with Legendre wavelet. The error profiles allow us to compare the results with well-known existing schemes. [ABSTRACT FROM AUTHOR]

In order to obtain the numerical results of 3D convection-diffusion-reaction problems with variable coefficients efficiently, we select the improved element-free Galerkin (IEFG) method instead of the traditional element-free Galerkin (EFG) method by using the improved moving least-squares (MLS) approximation to obtain the shape function. For the governing equation of 3D convection-diffusion-reaction problems, we can derive the corresponding equivalent functional; then, the essential boundary conditions are imposed by applying the penalty method; thus, the equivalent integral weak form is obtained. By introducing the IMLS approximation, we can derive the final solved linear equations of the convection-diffusion-reaction problem. In numerical examples, the scale parameter and the penalty factor of the IEFG method for such problems are discussed, the convergence is proved numerically, and the calculation efficiency of the IEFG method are verified by four numerical examples. [ABSTRACT FROM AUTHOR]

In this work, a priori error estimates for finite element approximations to the governing equations of heat and mass transfer in fluidized beds are derived. These equations are time dependent strongly coupled system of five semilinear convection-diffusion-reaction equations. The a priori error estimates for all the five variables are obtained for the error measured in L∞(L2) and L 2 (E) , E is the energy norm. [ABSTRACT FROM AUTHOR]

In order to obtain the numerical results of 3D convection-diffusion-reaction problems with variable coefficients efficiently, we select the improved element-free Galerkin (IEFG) method instead of the traditional element-free Galerkin (EFG) method by using the improved moving least-squares (MLS) approximation to obtain the shape function. For the governing equation of 3D convection-diffusion-reaction problems, we can derive the corresponding equivalent functional; then, the essential boundary conditions are imposed by applying the penalty method; thus, the equivalent integral weak form is obtained. By introducing the IMLS approximation, we can derive the final solved linear equations of the convection-diffusion-reaction problem. In numerical examples, the scale parameter and the penalty factor of the IEFG method for such problems are discussed, the convergence is proved numerically, and the calculation efficiency of the IEFG method are verified by four numerical examples.

• The governing equations are a system of five coupled convection-diffusion-reaction equations. • Error estimates are obtained using residuals due to space discretization, time discretization and linearization. • Fully computable a posteriori error estimates are derived using the total residual. • An adaptive finite element solution of these model equations is computed and the performance is illustrated. The a posteriori error estimates for finite element approximations to the governing equations of heat and mass transfer in fluidized beds are derived in this work. These are a system of five time dependent coupled nonlinear convection-diffusion-reaction equations. Based on the variational formulation, computable residual based a posteriori error estimates are obtained. The time discretization has been done using the implicit Euler method. The a posteriori error estimates for all the five variables are derived using the total residual and error indicators due to spatial discretization, time discretization and linearization. An adaptive finite element solution of these model equations is computed and the performance is illustrated. [ABSTRACT FROM AUTHOR]

We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and interstitial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper. [ABSTRACT FROM AUTHOR]

• The governing equations are five coupled semilinear convection-diffusion-reaction equations. • DG method is used to find the numerical solution which is well suited for convection dominated problems. • Experimental order of convergence of two and three are achieved. • DG methods are easily parallelizable. Parallel efficiency of strong scaling is presented. • A priori error estimates of DG method for all the five primary variables are derived. In this work, the Discontinuous Galerkin (DG) schemes were studied for the solution of convection-diffusion-reaction equations which arise from mathematical modeling of fluidized bed spray granulation process (FBSG). The discontinuous Galerkin method for space discretization is employed to treat the dominated convection behavior in the governing equations. The implicit Euler method for the temporal discretization is used. The mathematical analysis of the governing equations with a priori bounds for all the state variables is demonstrated. The investigation of experimental order of convergence is presented for test functions of degrees one and two. Finally, parallel efficiency of strong scaling is presented for the employed DG schemes. [ABSTRACT FROM AUTHOR]

• Two/three-scale high-order solutions for CDR transport problems in heterogeneous porous media are developed. • A very innovative, efficient, unified framework for developing high-order multiscale models is proposed. • This framework is more convenient to implement compared to other multiscale homogenization models. • The high-order solutions show high accuracy in capturing CDR coupling processes at multiple scales. • The effects of microstructures at each scale on the overall catalyst performance are analyzed. Developing highly accurate models for predicting the convection–diffusion-reaction (CDR) transport in hierarchical porous media with strong heterogeneities on multiple scales is crucial but not yet available. In this work, an innovative high-order multiscale computational framework is developed to capture the local and global variation characteristics of flow fields and reactant concentration at multiple scales. The homogenized solutions and macro-meso high-order solutions are established by the formal two-scale asymptotic analysis. By directly expanding the mesoscopic cell functions to the microscopic levels, the three-scale high-order models are built by assembling the meso-micro high-order expansions of mesoscopic cell functions and macro-meso low/high-order models. The present approaches follow the reverse thought process of the reiteration homogenization method, and provide a very innovative way to develop highly accurate and efficient solutions for the CDR coupling problems in multiscale porous media. The effectiveness and accuracy of the proposed multiscale models are validated by several representative cases. [ABSTRACT FROM AUTHOR]

We consider reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems with proper orthogonal decomposition (POD) in combination with isogeometric analysis. Isogeometric analysis has potential advantages in exact geometry representations, efficient mesh generation, different (h , p , and k) refinements and smooth Bspline/NURBS basis functions. In order to compensate the oscillations caused by the convection-dominated effect, the streamline-upwind Petrov-Galerkin (SUPG) stabilization method is used both in generation of snapshots and POD-Galerkin method. Based on the recent novel and promising discretization method-Isogeometric analysis, we propose a new fully discrete SUPG-stabilized scheme, the associated numerical error features three components due to spatial discretization by isogeometric analysis with SUPG stabilization, time discretization with the Crank-Nicolson scheme, and modes truncation by POD. We show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically. A variety of two and three-dimensional benchmark tests and numerical examples are provided to show the effectiveness, accuracy, and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis. • We consider reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems. • POD is combined with IGA for model order reduction. • SUPG stabilization is used in both offline and online. • A priori error estimates of the fully discrete scheme are given. • Various numerical examples and benchmark tests are presented on single/multi-patch domains in 2D/3D. [ABSTRACT FROM AUTHOR]

This paper presents a dual reciprocity boundary element method (DRBEM) formulation for the solution of steady-state convection-diffusion-reaction problems with variable velocity field at moderately high Péclet number. This scheme is based on utilising the fundamental solution of the convection-diffusion-reaction equation with constant coefficients. In this case, we decompose the velocity field into an average and a perturbation, with the latter being treated using a dual reciprocity approximation to convert the domain integrals arising in the boundary element formulation into equivalent boundary integrals. A proposed approach is implemented to treat the convective terms with variable velocity, for which the concentration is expanded as a series of functions. Four numerical experiments are included with available analytical solutions, to establish the validity of the approach and to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Numerical simulations of chromatography are conventionally performed using reduced-order models that homogenize aspects of flow and transport in the radial and angular dimensions. This enables much faster simulations at the expense of lumping the effects of inhomogeneities into a column dispersion coefficient, which requires calibration via empirical correlations or experimental results. We present a high-definition model with spatially resolved geometry. A stabilized space–time finite element method is used to solve the model on massively parallel high-performance computers. We simulate packings with up to 10,000 particles. The impact of particle size distribution on velocity and concentration profiles as well as breakthrough curves is studied. Our high-definition simulations provide unique insight into the process. The high-definition data can also be used as a source of ground truth to identify and calibrate appropriate reduced-order models that can then be applied for process design and optimization. • Stabilized space–time finite element method used on massively parallel computers. • Packed bed chromatography simulated at unprecedented spatial resolution. • Packings with up to 10.000 particles simulated. • Impact of particle size distribution and wall effects on band broadening quantified. • Calibration data for reduced order models generated. [ABSTRACT FROM AUTHOR]

Lateral flow assays (LFAs) have found widespread applications in biomedical fields, but improving their sensitivity remains challenging mainly due to the unclear convection-diffusion-reaction process. Therefore, we developed a 1D mathematical model to solve this process in LFAs. The model depicts the actual situation that one report particle may combine more than one target, which overcomes the deficiency of existing models where one report particle combines only one target. With this model, we studied the effect of report particle characteristics on LFAs, including binding site density, target analyte and report particle concentration. The model was qualitatively validated by reported experimental data and our designed experiments where the report particle with different accessible binding site (HIV-DP) densities is obtained by changing the ratio of HIV-DP and Dengue-DP in preparing AuNP-DP aggregates. The results indicate that a strong signal intensity can be obtained without consuming excess detector probe with the optimum binding site ( N = 30). A maximum normalized target concentration of 120 is obtained to prevent the false-negative result, while a minimum normalized report particle concentration of 0.015 is recommended to produce a strong signal. The developed model would serve as a powerful tool for designing highly effective LFAs. [ABSTRACT FROM AUTHOR]

This is an introduction to the first eight articles in this volume that contains the special issue on Recent Advances in PDE: Theory, Computations and Applications. These peer-reviewed articles address recent developments in the areas of convection-diffusion-reaction problems, stabilizability of control systems with application to Oseen problems, obstacle problems, multigrid methods for quad-curl problems and discontinuous Petrov–Galerkin methods for spectral approximations. Some of the contributors of these articles were plenary speakers of the conference organized in honor of the numerical analyst Professor Amiya Kumar Pani. The conference was organized to acknowledge his outstanding contribution for the growth of applied mathematics in India. [ABSTRACT FROM AUTHOR]

An affine finite element scheme approximation of a time dependent linear convection–diffusion–reaction problem in 2D and 3D is presented. For these equations which do not satisfy an underlying maximum principle, sufficient conditions are given in terms of the coefficient functions, the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation. Numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation. [ABSTRACT FROM AUTHOR]

A numerical scheme for the convection–diffusion–reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving CDR problems particularly designed to treat the most interesting case of small diffusion. We use the subgrid nodes in the Link-cutting bubble (LCB) strategy [5] to construct a numerical algorithm that can easily be extended to the higher dimensions. The method adapts very well to all regimes with continuous transitions from one regime to another. We also compare the performance of the present method with the Streamline-upwind Petrov–Galerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method. [ABSTRACT FROM AUTHOR]

Dissolution fingers (or wormholes) are formed during the dissolution of a porous rock as a result of nonlinear feedback between the flow, transport, and chemical reactions at pore surfaces. We analyze the shapes and growth velocities of such fingers within the thin-front approximation, in which the reaction is assumed to take place instantaneously with reactants fully consumed at the dissolution front. We concentrate on the case when the main flow is driven by a constant pressure gradient far from the finger, and the permeability contrast between the inside and the outside of the finger is finite. Using Ivantsov ansatz and conformal transformations we find the family of steadily translating fingers characterized by a parabolic shape. We derive the reactant concentration field and the pressure field inside and outside of the fingers and show that the flow within them is uniform. The advancement velocity of the finger is shown to be inversely proportional to its radius of curvature in the small Péclet number limit and independent of the radius of curvature for large Péclet numbers. [ABSTRACT FROM AUTHOR]

We present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized Burgers–Huxley equation. Proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given. Under appropriate conditions on the mesh and the initial and boundary data, boundedness and nonnegativity of the finite element approximation are established. An a priori error estimate for the approximation is also derived. Numerical experiments are presented which support the derived theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a novel two-level finite element method for convection–diffusion–reaction problems. The proposed scheme consists of a global problem and an ensemble of local problems. The boundary conditions of the local problems are provided by a global approximation of the same problem. The stability is ensured with an artificial diffusion mechanism that acts on the difference between local and global approximation, and leads to an a priori error estimate for the global solution. For the computation of the solutions, an efficient fixed-point algorithm is proposed as an alternative to the well established static condensation technique. The local solutions can be computed in parallel and enter in a communication step. This step takes place in the entire domain and consists of a mass matrix inversion only. Therefore, the overall algorithm is easy to implement and computationally inexpensive. • A two-level finite element method is introduced and theoretically analyzed. • The solution space is as a group space with continuous and discontinuous parts. • The method is consistent, stable and provides control over the streamline derivative. • A cost-inexpensive iterative solution scheme is proposed to compute the solutions. • The proposed two-level method prevents too much numerical smoothing. [ABSTRACT FROM AUTHOR]

The design of efficient, simple, and easy to code, second-order finite volume methods is an important challenge to solve practical problems in physics and in engineering where complex and very accurate techniques are not required. We propose an extension of the original Frink’s approach based on a cell-to-vertex interpolation to compute vertex values with neighbour cell values. We also design a specific scheme which enables to use whatever collocation point we want in the cells to overcome the mass centre point restrictive choice. The method is proposed for two- and three-dimensional geometries and a second-order extension time-discretization is given for time-dependent equation. A large number of numerical simulations are carried out to highlight the performance of the new method. [ABSTRACT FROM AUTHOR]

Despite recent data that suggest that the overall performance of drug-eluting stents (DES) is superior to that of bare-metal stents, the long-term safety and efficacy of DES remain controversial. The risk of late stent thrombosis associated with the use of DES has also motivated the development of a new and promising treatment option in recent years, namely drug-coated balloons (DCB). Contrary to DES where the drug of choice is typically sirolimus and its derivatives, DCB use paclitaxel since the use of sirolimus does not appear to lead to satisfactory results. Since both sirolimus and paclitaxel are highly lipophilic drugs with similar transport properties, the reason for the success of paclitaxel but not sirolimus in DCB remains unclear. Computational models of the transport of drugs eluted from DES or DCB within the arterial wall promise to enhance our understanding of the performance of these devices. The present study develops a computational model of the transport of the two drugs paclitaxel and sirolimus eluted from DES in the arterial wall. The model takes into account the multilayered structure of the arterial wall and incorporates a reversible binding model to describe drug interactions with the constituents of the arterial wall. The present results demonstrate that the transport of paclitaxel in the arterial wall is dominated by convection while the transport of sirolimus is dominated by the binding process. These marked differences suggest that drug release kinetics of DES should be tailored to the type of drug used. [ABSTRACT FROM AUTHOR]

Abstract: A multidimensional extension of the HRPG method (doi:10.1016/j.cma.2009.10.009) using the lowest order block finite elements is presented. First, we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions. This is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1D reaction–diffusion problem. The nondimensional element number is then defined using this fictitious reaction coefficient, the diffusion coefficient and an appropriate element size. Next, we introduce anisotropic element length vectors l i and the stabilization parameters α i , β i are calculated along these l i . Except for the modification to include the new dimensionless number that quantifies the characteristic layers, the definitions of α i , β i are a direct extension of their counterparts in 1D. Using α i , β i and l i , objective characteristic tensors associated with the HRPG method are defined. The numerical artifacts across the characteristic layers are manifested as the Gibbs phenomenon. Hence, we treat them just like the artifacts formed across the parabolic layers in the reaction-dominant case. Several 2D examples are presented that support the design objective—stabilization with high-resolution. [Copyright &y& Elsevier]

Abstract: We present the design of a high-resolution Petrov–Galerkin (HRPG) method using linear finite elements for the problem defined by the residualwhere . The structure of the method in 1D is identical to the consistent approximate upwind Petrov–Galerkin (CAU/PG) method [A.C. Galeão, E.G. Dutra do Carmo, A consistent approximate upwind Petrov–Galerkin method for the convection-dominated problems, Comput. Methods Appl. Mech. Engrg. 68 (1988) 83–95] except for the definitions of the stabilization parameters. Such a structure may also be attained via the finite-calculus (FIC) procedure [E. Oñate, Derivation of stabilized equations for numerical solution of advective–diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg. 151 (1998) 233–265; E. Oñate, J. Miquel, G. Hauke, Stabilized formulation for the advection–diffusion–absorption equation using finite-calculus and linear finite elements, Comput. Methods Appl. Mech. Engrg. 195 (2006) 3926–3946] by an appropriate definition of the characteristic length. The prefix ‘high-resolution’ is used here in the sense popularized by Harten, i.e. second order accuracy for smooth/regular regimes and good shock-capturing in nonregular regimes. The design procedure embarks on the problem of circumventing the Gibbs phenomenon observed in L 2-projections. Next we study the conditions on the stabilization parameters to circumvent the global oscillations due to the convective term. A conjuncture of the two results is made to deal with the problem at hand that is usually plagued by Gibbs, global and dispersive oscillations in the numerical solution. It is shown that the method indeed reproduces stabilized high-resolution numerical solutions for a wide range of values of . Finally, some remarks are made on the extension of the HRPG method to multidimensions. [Copyright &y& Elsevier]

The article presents the results of a study which investigated the phenomenon known as viscous fingering, focusing on the type of viscous fingering found in miscible systems. An overview of related previous research is provided, along with details of the experimental protocol, which involved two syringes connected to a single syringe pump. It is demonstrated how the development of fingering patterns are determined at the molecular level by factors such as the Péclet number and the initial concentrations of reactants.

Abstract: In this paper, an improved two-level method is presented for effectively solving the incompressible Navier–Stokes equations. This proposed method solves a smaller system of nonlinear Navier–Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier–Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection–diffusion–reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier–Stokes solutions. [Copyright &y& Elsevier]

Linear equilibrium and non-equilibrium models for leaching of solutes in soils give rise to unsteady linear convection–diffusion–reaction problems. We present several numerical schemes to approximate the solution of this kind of problems based on Stabilized Finite Element Methods, including the recent Link-Cutting Bubbles strategy adapted to deal with unsteady problems, which gives the best numerical results. [Copyright &y& Elsevier]

Abstract: We consider the Mizukami–Hughes method for the numerical solution of scalar two-dimensional steady convection–diffusion equations using conforming triangular piecewise linear finite elements. We propose several modifications of this method to eliminate its shortcomings. The improved method still satisfies the discrete maximum principle and gives very accurate discrete solutions in convection-dominated regime, which is illustrated by several numerical experiments. In addition, we show how the Mizukami–Hughes method can be applied to convection–diffusion–reaction equations and to three-dimensional problems. [Copyright &y& Elsevier]

Abstract: The present study aims to develop a new method for obtaining the non-oscillatory incompressible Navier–Stokes solutions on the non-staggered grids. Within the segregated grid framework, the divergence-free equation is chosen to replace one of the momentum equations so as to preserve the fluid incompressibility. For the sake of numerical accuracy, the five-point stencil convection–diffusion–reaction scheme is developed to obtain the nodally exact solution for this chosen momentum equation. The validity of the proposed mass-preserving Navier–Stokes method is justified by solving the three problems which are amenable to analytical solutions. The simulated solution quality is shown to outperform that of the conventional segregated approach, besides gaining a very high spatial rate of convergence. [Copyright &y& Elsevier]

Abstract: In this paper we propose and study a subgrid model for linear convection–diffusion–reaction equations with fractal rough coefficients. The subgrid model is based on scale extrapolation of a modeling residual from coarser scales using a computed solution on a finest scale as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a scale less than h/4. We also present error estimates for the modeling error in terms of modeling residuals. [Copyright &y& Elsevier]

We study a model linear convection–diffusion–reaction problem where both the diffusion term and the convection term are multiplied by small parameters ϵd and ϵc, respectively. Depending on the size of the parameters the solution of the problem may exhibit exponential layers at both end points of the domain. Sharp bounds for the derivatives of the solution are derived using a barrier-function technique. These bounds are applied in the analysis of a simple upwind-difference scheme on Shishkin meshes. This method is established to be almost first-order convergent, independently of the parameters ϵd and ϵc. [Copyright &y& Elsevier]

In this paper we study a subgrid model based on extrapolation of a modeling residual, in the case of a linear convection-diffusion-reaction problem Lu=f in two dimensions. The solution u to the exact problem satisfies an equation L[sub h]u=[f][sup h]+F[sub h](u), where L[sub h] is the operator used in the computation on the finest computational scale h, [f][sup h] is the approximation of f on the scale h, and F[sub h](u) is a modeling residual, which needs to be modeled. The subgrid modeling problem is to compute approximations of F[sub h](u) without using finer scales than h. In this study we model F[sub h](u) by extrapolation from coarser scales than h, where F[sub h](u) is directly computed with the finest scale h as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a mesh of size less than h/4. [ABSTRACT FROM AUTHOR]