Superconvergence of finite difference approximations for convection–diffusion problems

An Erratum has been published for this article in Numerical Linear Algebra with Applications 8 (4) 2001, iii–iv. We are concerned with numerical solutions of convection–diffusion equations. The convergence behaviour of numerical solutions is considered by using the finite difference approximation with respect to spatial variables and implicit method with respect to time variable. It is shown that superconvergence occurs near a part of the boundary which has Dirichlet's data. Copyright © 2001 John Wiley & Sons, Ltd.     

On preconditioned Krylov subspace methods for discrete convection–diffusion problems

We study nonstationary iterative methods for solving preconditioned systems arising from discretizations of the convection–diffusion equation. The preconditioners arise from Gauss–Seidel methods applied to the original system. It is shown that the performance of the iterative solvers is affected by the relationship of the ordering of the underlying grid and the direction of the fow associated with the differential operator. Specifically, only those orderings that follow the fow give fast iterative solvers. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13 :321–330     

Adaptive characteristic finite element approximation of convection–diffusion optimal control problems

In this article, a characteristic finite element approximation of quadratic optimal control problems governed by linear convection–diffusion equations is given. We derive some a posteriori error estimates for both the control and the state approximations, where the control variable is constrained by pointwise inequality. The derived error estimators are then used as an error indicator to guide the mesh refinement. In this sense, they are very important in developing adaptive finite element algorithm for the optimal control problems. Finally, a numerical example is given to validate the efficiency and reliability of the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection–diffusion equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods.     

A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Robust hierarchical a posteriori error estimators for stabilized convection–diffusion problems

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection‐diffusion equations with height Péclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Critical Fujita exponents for a class of nonlinear convection–diffusion equations

In this paper, we establish the blow‐up theorems of Fujita type for a class of homogeneous Neumann exterior problems of quasilinear convection–diffusion equations. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow‐up case under any nontrivial initial data. Copyright © 2010 John Wiley & Sons, Ltd.     

Parallel characteristic finite difference method for convection–diffusion equations

Based on the overlapping domain decomposition, an efficient parallel characteristic finite difference scheme is proposed for solving convection‐diffusion equations numerically. We give the optimal convergence order in error estimate analysis, which shows that we just need to iterate once or twice at each time level to reach the optimal convergence order. Numerical experiments also confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 854–866, 2011     

Patch‐wise local projection stabilized finite element methods for convection–diffusion–reaction problems

In this article, we develop patch‐wise local projection‐stabilized conforming and nonconforming finite element methods for the convection–diffusion–reaction problems. It is a composition of the standard Galerkin finite element method, the patch‐wise local projection stabilization, and weakly imposed Dirichlet boundary conditions on the discrete solution. In this paper, a priori error analysis is established with respect to a patch‐wise local projection norm for the conforming and the nonconforming finite element methods. The numerical experiments confirm the efficiency of the proposed stabilization technique and validate the theoretical convergence rates.     

An exponential high‐order compact ADI method for 3D unsteady convection–diffusion problems

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A control volume technique based on integrated RBFNs for the convection–diffusion equation

This article reports a new high‐order control‐volume discretization for the convection–diffusion equation in one and two dimensions. Diffusive fluxes at the faces of a control volume and other terms embracing the unknown field variable are all approximated using one‐dimensional integrated radial‐basis‐function networks; line integrals involving these fluxes and other integrals are evaluated using a high‐order numerical integration scheme. The accuracy of the proposed technique is investigated numerically through the solution of several linear and nonlinear test problems, including a benchmark thermally driven cavity flow. High‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

A supercloseness result for the discontinuous Galerkin stabilization of convection–diffusion problems on Shishkin meshes

We consider a convection–diffusion problem with Dirichlet boundary conditions posed on a unit square. The problem is discretized using a combination of the standard Galerkin FEM and an h–version of the nonsymmetric discontinuous Galerkin FEM with interior penalties on a layer–adapted mesh with linear/bilinear elements. With specially chosen penalty parameters for edges from the coarse part of the mesh, we prove uniform convergence (in the perturbation parameter) in an associated norm. In the same norm we also establish a supercloseness result. Numerical tests support our theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Parallel characteristic finite element method for time‐dependent convection–diffusion problem

Based on the overlapping‐domain decomposition and parallel subspace correction method, a new parallel algorithm is established for solving time‐dependent convection–diffusion problem with characteristic finite element scheme. The algorithm is fully parallel. We analyze the convergence of this algorithm, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration times and sub‐domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to optimal accuracy at each time step. Copyright © 2010 John Wiley & Sons, Ltd.     

Global a posteriori error estimates for convection–reaction–diffusion problems

In this note we propose a nonstandard technique for constructing global a posteriori error estimates for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in appropriate weighted energy norms, which measures the overall quality of the approximations, the underlying bilinear form is decomposed into several terms which can be directly computed or easily estimated from above using elementary tools of functional analysis. Several auxiliary parameters are introduced to construct such a splitting and tune the resulting upper error bound. It is demonstrated how these parameters can be chosen in some natural and convenient way for computations so that the weighted energy norm of the error is almost recovered, which shows that the estimates proposed are, in fact, quasi-sharp. The presented methodology is completely independent of numerical techniques used to compute approximate solutions. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g., due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors etc. Moreover, the only constant that appears in the proposed error estimates is of global nature and comes from the Friedrichs–Poincaré inequality.     

A note on constraint preconditioners for nonsymmetric saddle point problems

A class of constraint preconditioners for solving two‐by‐two block linear equations with the (1,2)‐block being the transpose of the (2,1)‐block and the (2,2)‐block being zero was investigated in a recent paper of Cao (Numer. Math. 2006; 103 :47–61). In this short note, we extend his idea by allowing the (1,2)‐block to be not equal to the transpose of the (2,1)‐block. Results concerning the spectrum, the form of the eigenvectors and the convergence behaviour of a Krylov subspace method, such as GMRES are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Algebraic multigrid preconditioners for the bidomain reaction–diffusion system

The so-called bidomain system is possibly the most complete model for the cardiac bioelectric activity. It consists of a reaction–diffusion system, modeling the intra, extracellular and transmembrane potentials, coupled through a nonlinear reaction term with a stiff system of ordinary differential equations describing the ionic currents through the cellular membrane. In this paper we address the problem of efficiently solving the large linear system arising in the finite element discretization of the bidomain model, when a semiimplicit method in time is employed. We analyze the use of structured algebraic multigrid preconditioners on two major formulations of the model, and report on our numerical experience under different discretization parameters and various discontinuity properties of the conductivity tensors. Our numerical results show that the less exercised formulation provides the best overall performance on a typical simulation of the myocardium excitation process.     

Solving linear systems using wavelet compression combined with Kronecker product approximation

Discrete wavelet transform approximation is an established means of approximating dense linear systems arising from discretization of differential and integral equations defined on a one-dimensional domain. For higher dimensional problems, approximation with a sum of Kronecker products has been shown to be effective in reducing storage and computational costs. We have combined these two approaches to enable solution of very large dense linear systems by an iterative technique using a Kronecker product approximation represented in a wavelet basis. Further approximation of the system using only a single Kronecker product provides an effective preconditioner for the system. Here we present our methods and illustrate them with some numerical examples. This technique has the potential for application in a range of areas including computational fluid dynamics, elasticity, lubrication theory and electrostatics. AMS subject classification 65F10, 65T60, 65F30 Judith M. Ford: This author was supported by EPSRC Postdoctoral Research Fellowship ref: GR/R95982/01. Current address: Royal Liverpool Children's NHS Trust, Liverpool, L12 2AP. Eugene E. Tyrtyshnikov: This author was supported by the Russian Fund of Basic Research (grant 02-01-00590) and Science Support Foundation.     

On a convection–diffusion problem with a weak layer

In this article we consider a model linear convection–diffusion problem with a weak layer. We analyze the singular-perturbation nature of the problem and show that no special precautions are required to cope with the weak layer: a standard upwind scheme on a (quasi-)uniform mesh is sufficient. We give a simple analysis for the method. Thus highlighting that not all problems with a small parameter multiplying the highest-order derivative are suitable for studying boundary-layer phenomena.     

Multilevel a posteriori error analysis for reaction–convection–diffusion problems

We present a new approach to the a posteriori error analysis of stable Galerkin approximations of reaction–convection–diffusion problems. It relies upon a non-standard variational formulation of the exact problem, based on the anisotropic wavelet decomposition of the equation residual into convection-dominated scales and diffusion-dominated scales. The associated norm, which is stronger than the standard energy norm, provides a robust (i.e., uniform in the convection limit) control over the streamline derivative of the solution. We propose an upper estimator and a lower estimator of the error, in this norm, between the exact solution and any finite dimensional approximation of it. We investigate the behaviour of such estimators, both theoretically and through numerical experiments. As an output of our analysis, we find that the lower estimator is quantitatively accurate and robust.