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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]

We present a procedure for coupling the fluid and transport equations to model the distribution of a pollutant in a street canyon, in this case, black carbon (BC). The fluid flow is calculated with a stabilized finite element method using the Quasi-Static Variational Multiscale (QS-VMS) technique. For the temperature and pollutant transport we use a semi-Lagrangian procedure, based on the Particle Finite Element Method (PFEM) combined with an Eulerian method based on a Finite Increment Calculus (FIC) formulation. Both methods are implemented on the open-source KRATOS Multiphysics platform. The coupled numerical formulation is applied to the prediction of the transport of BC in a street canyon, which can be a useful tool to lessen the impact of pollutants on pedestrians. Two test cases have been studied: a 2D simplified case and a more complex 3D one. The main goal of this study is to propose a useful tool to study the effect of pollution on pedestrians in a street-level scale. Good comparison with experimental results is obtained. • A coupled procedure for fluid and transport equations is presented. • The distribution of black carbon (BC) in a street canyon is assessed. • A semi-Lagrangian finite element formulation has been used. • Results have been validated with experimental data. • 2D and 3D simulations have been run. [ABSTRACT FROM AUTHOR]

We present a numerical method for solving advective–diffusive–absorptive problems with high values of advection and absorption. A Lagrangian approach based on the updated version of the classical Particle Finite Element Method (PFEM) is used to calculate advection, while a Eulerian strategy based on the Finite Element Method (FEM) is adopted to compute diffusion and absorption. The Eulerian FEM procedure is based on a Finite Increment Calculus (FIC) stabilized formulation recently developed by the authors. The most relevant features of each computational approach are outlined and the coupling scheme is explained. Several problems are solved to validate the method: the evolution of a localized concentration field in two dimensions (2D), the evolution of a spherical field in 3D and three benchmark problems from the literature with high absorption. [ABSTRACT FROM AUTHOR]

In this work, a new cubic B-spline-based semi-analytical algorithm is presented for solving 3D anisotropic convection-diffusion-reaction (CDR) problems in the inhomogeneous medium. The mathematical model is expressed by the quasi-linear second-order elliptic partial differential equations (EPDE) with mixed derivatives and variable coefficients. The final approximation is obtained as a sum of the rough primary solution and the modified spline interpolants with free parameters. The primary solution mathematically satisfies boundary conditions. Thus, the free parameters of interpolants are chosen to satisfy the governing equation in the solution domain. The numerical examples demonstrate the high accuracy of the proposed method in solving 3D CDR problems in single- and multi-connected domains. [ABSTRACT FROM AUTHOR]