On the choice of stabilizing sub-grid for convection–diffusion problem on rectangular grids

A stabilizing sub-grid which consists of a single additional node in each rectangular element is analyzed for solving the convection–diffusion problem, especially in the case of small diffusion. We provide a simple recipe for spotting the location of the additional node that contributes a very good stabilizing effect to the overall numerical method. We further study convergence properties of the method under consideration and prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfies the same type of a priori error estimates that are typically obtained with the SUPG method. Some numerical experiments that confirm the theoretical findings are also presented.     

An assessment of discretizations for convection-dominated convection–diffusion equations

The performance of several numerical schemes for discretizing convection-dominated convection–diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov–Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.     

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.     

Analysis of lumped approximations in the finite-element method for convection–diffusion problems

The Petrov–Galerkin finite-element method with a lumped mass matrix is analyzed. It is stated that in some cases it excessively smoothes the solutions and causes large errors. It is shown that weight functions can be chosen, which eliminate the above-mentioned drawbacks. The corresponding approximations are constructed in the form of systems of ordinary differential equations and finite-difference schemes. The theoretical results are confirmed by calculated data.     

A family of fourth-order difference schemes on rotated grid for two-dimensional convection–diffusion equation

We derive a family of fourth-order finite difference schemes on the rotated grid for the two-dimensional convection–diffusion equation with variable coefficients. In the case of constant convection coefficients, we present an analytic bound on the spectral radius of the line Jacobi’s iteration matrix in terms of the cell Reynolds numbers. Our analysis and numerical experiments show that the proposed schemes are stable and produce highly accurate solutions. Classical iterative methods with these schemes are convergent with large values of the convection coefficients. We also compare the fourth-order schemes with the nine point scheme obtained from the second-order central difference scheme after one step of cyclic reduction.     

High-order compact boundary value method for the solution of unsteady convection–diffusion problems

In this paper, we propose a new class of high-order accurate methods for solving the two-dimensional unsteady convection–diffusion equation. These techniques are based on the method of lines approach. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulted linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Also this method is unconditionally stable due to the favorable stability property of boundary value methods. Numerical results obtained from solving several problems include problems encounter in many transport phenomena, problems with Gaussian pulse initial condition and problems with sharp discontinuity near the boundary, show that the compact finite difference approximation of fourth order and a boundary value method of fourth order give an efficient algorithm for solving such problems.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

Parameterized ABD preconditioning technique for time-periodic convection–diffusion problems

In this paper, a parameterized additive block diagonal (PABD) preconditioning technique is present for solving the nine-point approximations of the time-periodic convection–diffusion problems. The explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices are presented. The range of the optimal parameters is derived. Numerical experiments show that the generalized minimal residual method preconditioned by the PABD preconditioner with the experimental optimal parameters or some special values is effective for a wide range of problem sizes.     

A posteriori optimization of parameters in stabilized methods for convection–diffusion problems – Part I

Stabilized finite element methods for convection-dominated problems require the choice of appropriate stabilization parameters. From numerical analysis, often only their asymptotic values are known. This paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional. Exemplarily, this framework is applied to the SUPG finite element method and the minimization of a residual-based error estimator, an error indicator, and a functional including the crosswind derivative of the computed solution. Benefits of the basic approach are demonstrated by means of numerical results.     

On the robustness and optimality of algebraic multilevel methods for reaction–diffusion type problems

This paper is on preconditioners for reaction–diffusion problems that are both, uniform with respect to the reaction–diffusion coefficients, and optimal in terms of computational complexity. The considered preconditioners belong to the class of so-called algebraic multilevel iteration (AMLI) methods, which are based on a multilevel block factorization and polynomial stabilization. The main focus of this work is on the construction and on the analysis of a hierarchical splitting of the conforming finite element space of piecewise linear functions that allows to meet the optimality conditions for the related AMLI preconditioner in case of second-order elliptic problems with non-vanishing zero-order term. The finite element method (FEM) then leads to a system of linear equations with a system matrix that is a weighted sum of stiffness and mass matrices. Bounds for the constant \(\gamma \) in the strengthened Cauchy–Bunyakowski–Schwarz inequality are computed for both mass and stiffness matrices in case of a general \(m\) -refinement. Moreover, an additive preconditioner is presented for the pivot blocks that arise in the course of the multilevel block factorization. Its optimality is proven for the case \(m=3\) . Together with the estimates for \(\gamma \) this shows that the construction of a uniformly convergent AMLI method with optimal complexity is possible (for \(m \ge 3\) ). Finally, we discuss the practical application of this preconditioning technique in the context of time-periodic parabolic optimal control problems.     

Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems

This paper deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection–diffusion problems exhibiting a regular boundary layer. For discretizing the time derivative, we use the classical backward-Euler method and for the spatial discretization the simple upwind scheme is used on a piecewise-uniform Shishkin mesh. We show that the use of Richardson extrapolation technique improves the ε-uniform accuracy of simple upwinding in the discrete supremum norm from O (N ⁻¹ ln N + Δt) to O (N ⁻² ln² N + Δt ²), where N is the number of mesh-intervals in the spatial direction and Δt is the step size in the temporal direction. The theoretical result is also verified computationally by applying the proposed technique on two test examples.     

Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method.

The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.

For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.

At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.     

Uniform supercloseness of Galerkin finite element method for convection–diffusion problems with characteristic layers

In this paper, we consider a singularly perturbed convection–diffusion equation posed on the unit square, where the solution has two characteristic layers and an exponential layer. A Galerkin finite element method on a Shishkin mesh is used to solve this problem. Its bilinear forms in different subdomains are carefully analyzed by means of a series of integral inequalities; a delicate analysis for the characteristic layers is needed. Based on these estimations, we prove supercloseness bounds of order $3∕2$ (up to a logarithmic factor) on triangular meshes and of order $2$ (up to a logarithmic factor) on hybrid meshes respectively. The result implies that the hybrid mesh, which replaces the triangles of the Shishkin mesh by rectangles in the exponential layer region, is superior to the Shishkin triangular mesh. Numerical experiments illustrate these theoretical results.     

A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.     

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

Partial semi-coarsening multigrid method based on the HOC scheme on nonuniform grids for the convection–diffusion problems

A partial semi-coarsening multigrid method based on the high-order compact (HOC) difference scheme on nonuniform grids is developed to solve the 2D convection–diffusion problems with boundary or internal layers. The significance of this study is that the multigrid method allows different number of grid points along different coordinate directions on nonuniform grids. Numerical experiments on some convection–diffusion problems with boundary or internal layers are conducted. They demonstrate that the partial semi-coarsening multigrid method combined with the HOC scheme on nonuniform grids, without losing the high-order accuracy, is very efficient and effective to decrease the computational cost by reducing the number of grid points along the direction which does not contain boundary or internal layers.     

Local projection stabilisation on S-type meshes for convection–diffusion problems with characteristic layers

Singularly perturbed convection–diffusion problems with exponential and characteristic layers are considered on the unit square. The discretisation is based on layer-adapted meshes. The standard Galerkin method and the local projection scheme are analysed for bilinear and higher order finite element where enriched spaces were used. For bilinears, first order convergence in the ε-weighted energy norm is shown for both the Galerkin and the stabilised scheme. However, supercloseness results of second order hold for the Galerkin method in the ε-weighted energy norm and for the local projection scheme in the corresponding norm. For the enriched ${\mathcal{Q}_p}$ -elements, p ≥ 2, which already contain the space ${\mathcal{P}_{p+1}}$ , a convergence order p + 1 in the ε-weighted energy norm is proved for both the Galerkin method and the local projection scheme. Furthermore, the local projection methods provides a supercloseness result of order p + 1 in local projection norm.     

Finite elements on locally uniform meshes for convection–diffusion problems with boundary layers

The layer-adapted meshes used to achieve robust convergence results for problems with layers are not locally uniform. We discuss concepts of almost robust convergence and some realizations of locally-uniform meshes.     

A posteriori error estimates for non-stationary non-linear convection–diffusion equations

Motivated by stochastic convection–diffusion problems we derive a posteriori error estimates for non-stationary non-linear convection–diffusion equations acting as a deterministic paradigm. The problem considered here neither fits into the standard linear framework due to its non-linearity nor into the standard non-linear framework due to the lacking differentiability of the non-linearity. Particular attention is paid to the interplay of the various parameters controlling the relative sizes of diffusion, convection, reaction and non-linearity (noise).