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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]

• 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 consider a stationary convection–diffusion–reaction model problem in a two- or three-dimensional bounded domain. We approximate this model by a non-stationary problem and propose a numerical method that combines the method of characteristics with an augmented mixed finite element procedure. We show that this scheme has a unique solution. We also derive a residual-based a posteriori error indicator and prove it is reliable and locally efficient. Finally, we provide some numerical experiments that illustrate the performance of the adaptive algorithm. • Adaptive mixed finite element method for convection–diffusion–reaction model. • Based on reliable and locally efficient a posteriori error estimates. • Combination with the method of characteristics allows to treat dominant convection. • Stabilization of mixed finite element method allows to use a wide set of finite elements. • Satisfactory numerical results. [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]

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]

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]

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]

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]

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]

We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in R d. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included. • Stabilized cut finite element method for stationary convection–diffusion. • Streamline diffusion and surface normal gradient penalization. • Optimal order convergence irrespective of Péclet number. • Optimal condition number of the discrete system. [ABSTRACT FROM AUTHOR]