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).     

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 stabilizing augmented grid for rectangular discretizations of the convection–diffusion–reaction problems

We propose a numerical method for approximate solution of the convection–diffusion–reaction problems in the case of small diffusion. The method is based on the standard Galerkin finite element method on an extended space defined on the original grid plus a subgrid, where the original grid consists of rectangular elements. On each rectangular elements, we construct a subgrid with few points whose locations are critical for the stabilization of the problem, therefore they are chosen specially depending on some specific conditions that depend on the problem data. The resulting subgrid is combined with the initial coarse mesh, eventually, to solve the problem in the framework of Galerkin method on the augmented grid. The results of the numerical experiments confirm that the proposed method shows similar stability features with the well-known stabilized methods for the critical range of problem parameters.     

Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations

In this paper, we study the convergence behavior of the local discontinuous Galerkin (LDG) methods when applied to one-dimensional time dependent convection–diffusion equations. We show that the LDG solution will be superconvergent towards a particular projection of the exact solution, if this projection is carefully chosen based on the convection and diffusion fluxes. The order is observed to be at least $k+2$ when piecewise $P^k$ polynomials are used. Moreover, the numerical traces for the solution are also superconvergent, sometimes, of higher-order. This is a continuation of our previous work [Cheng Y, Shu C-W. Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J Comput Phys 2008;227:9612–27], in which superconvergence of DG schemes for convection equations is discussed.     

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.     

FOM – a MATLAB toolbox of first-order methods for solving convex optimization problems

This paper presents the FOM MATLAB toolbox for solving convex optimization problems using first-order methods. The diverse features of the eight solvers included in the package are illustrated through a collection of examples of different nature.     

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.     

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.     

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 web-based platform for the simulation–optimization of industrial problems

This study presents a platform for industrial, real-world simulation–optimization based on web techniques. The design of the platform is intended to be generic and thereby make it possible to apply the platform in various problem domains. In the implementation of the platform, modern web techniques, such as Ajax, JavaScript, GWT, and ProtoBuf, are used. The platform is tested and evaluated on a real industrial problem of production optimization at Volvo Aero Corporation, a company that develops and manufactures high-technology components for aircraft and gas turbine engines. The results of the evaluation show that while the platform has several benefits, implementing a web-based system is not completely straightforward. At the end of the paper, possible pitfalls are discussed and some recommendations for future implementations are outlined.     

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 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.     

Stabilized finite element methods with shock capturing for advection–diffusion problems

Stabilized FEM of streamline-diffusion type for advection–diffusion problems may exhibit local oscillations in crosswind direction(s). As a remedy, a shock-capturing variant of such stabilized schemes is considered as an additional consistent (but nonlinear) stabilization. We prove existence of discrete solutions. Then we present some a priori and a posteriori estimates. Finally we address the efficient solution of the arising nonlinear discrete problems.     

Solution verification,goal-oriented adaptive methods for stochastic advection–diffusion problems

A goal-oriented analysis of linear, stochastic advection–diffusion models is presented which provides both a method for solution verification as well as a basis for improving results through adaptation of both the mesh and the way random variables are approximated. A class of model problems with random coefficients and source terms is cast in a variational setting. Specific quantities of interest are specified which are also random variables. A stochastic adjoint problem associated with the quantities of interest is formulated and a posteriori error estimates are derived. These are used to guide an adaptive algorithm which adjusts the sparse probabilistic grid so as to control the approximation error. Numerical examples are given to demonstrate the methodology for a specific model problem.     

Topological optimization of continuum structures with design-dependent surface loading – Part I: new computational approach for 2D problems

This paper describes a new computational approach for optimum topology design of 2D continuum structures subjected to design-dependent loading. Both the locations and directions of the loads may change as the structural topology changes. A robust algorithm based on a modified isoline technique is presented that generates the appropriate loading surface which remains on the boundary of potential structural domains during the topology evolution. Issues in connection with tracing the variable loading surface are discussed and treated in the paper. Our study indicates that the influence of the variation of element material density is confined within a small neighbourhood of the element. With this fact in mind, the cost of the calculation of the sensitivities of loads may be reduced remarkably. Minimum compliance is considered as the design problem. There are several models available for such designs. In the present paper, a simple formulation with weighted unit cost constraints based on the expression of potential energy is employed. Compared to the traditional models (i.e., the SIMP model), it provides an alternative way to implement the topology design of continuum structures. Some 2D examples are tested to show the differences between the designs obtained for fixed, design-independent loading, and for variable, design-dependent loading. The general and special features of the optimization with design-dependent loads are shown in the paper, and the validity of the algorithm is verified. An algorithm dealing with 3D design problems is described in Part II, which is developed from the 2D algorithm in the present Part I of the paper.     

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.     

Weighted Superposition Attraction (WSA): A swarm intelligence algorithm for optimization problems – Part 1: Unconstrained optimization

This paper is the first one of the two papers entitled “Weighted Superposition Attraction (WSA)”, which is based on two basic mechanisms, “superposition” and “attracted movement of agents”, that are observable in many systems. Dividing this paper into two parts raised as a necessity because of their individually comprehensive contents. If we wanted to write these papers as a single paper we had to write more compact as distinct from its current versions because of the space requirements. So, writing them as a single paper would not be as effective as we desired.In many natural phenomena it is possible to compute superposition or weighted superposition of active fields like light sources, electric fields, sound sources, heat sources, etc.; the same may also be possible for social systems as well. An agent (particle, human, electron, etc.) may be supposed to move towards superposition if it is attractive to it. As systems status changes the superposition also changes; so it needs to be recomputed. This is the main idea behind the WSA algorithm, which mainly attempts to realize this superposition principle in combination with the attracted movement of agents as a search procedure for solving optimization problems in an effective manner. In this current part, the performance of the proposed WSA algorithm is tested on the well-known unconstrained continuous optimization functions, through a set of computational study. The comparison with some other search algorithms is performed in terms of solution quality and computational time. The experimental results clearly indicate the effectiveness of the WSA algorithm.     

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.     

Weighted Superposition Attraction (WSA): A swarm intelligence algorithm for optimization problems – Part 2: Constrained optimization

This paper is the second one of the two papers entitled “Weighted Superposition Attraction (WSA) Algorithm”, which is about the performance evaluation of the WSA algorithm in solving the constrained global optimization problems. For this purpose, the well-known mechanical design optimization problems, design of a tension/compression coil spring, design of a pressure vessel, design of a welded beam and design of a speed reducer, are selected as test problems. Since all these problems were formulated as constrained global optimization problems, WSA algorithm requires a constraint handling method for tackling them. For this purpose we have selected 6 formerly developed constraint handling methods for adapting into WSA algorithm and analyze the effect of the used constraint handling method on the performance of the WSA algorithm. In other words, we have the aim of producing concluding remarks over the performance and robustness of the WSA algorithm through a set of computational study in solving the constrained global optimization problems. Computational study indicates the robustness and the effectiveness of the WSA in terms of obtained results, reached level of convergence and the capability of coping with the problems of premature convergence, trapping in a local optima and stagnation.     

A novel hybrid PSO–GWO algorithm for optimization problems

Engineering with Computers - In this study, we propose a new hybrid algorithm fusing the exploitation ability of the particle swarm optimization (PSO) with the exploration ability of the grey wolf...