Convection-dominated Diffusion Problems Research Articles

Super-localized orthogonal decomposition for convection-dominated diffusion problems

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm, the Galerkin projection onto this generalized finite element space even yields ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-independent error bounds, ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.

A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms

This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework’s performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem).

NIPG Finite Element Method for Convection-Dominated Diffusion Problems with Discontinuous Data

This paper presents the nonsymmetric interior penalty Galerkin (NIPG) finite element method for a class of one-dimensional convection dominated diffusion problems with discontinuous coefficients. The solution of the considered class of problem exhibits boundary and interior layers. Piecewise uniform Shishkin-type meshes are used for the spatial discretization. The error estimates in the energy norm have been derived for the proposed schemes. Theoretical results are supported by conducting numerical experiments. It is established that the errors are uniform with respect to the perturbation parameter [Formula: see text]. The uniformness of the error estimates with the perturbation parameter [Formula: see text] has also been established numerically for [Formula: see text]- norm.

A new GPU-based corrected explicit-implicit domain decomposition scheme for convection-dominated diffusion problems

In this study, a class of GPU-based corrected explicit-implicit domain decomposition schemes (GCEIDD) is proposed to accelerate the solution of convection-dominated diffusion problems on GPUs. All of these methods take advantage of the fractional steps and corrected explicit-implicit domain decomposition techniques. In each sweep, the domain is decomposed into many strip-divided subdomains. The aim is to reduce the size and increase the number of tri-diagonal systems to provide enough parallelism to keep the GPU occupied. In different steps of the algorithms, we use low-complexity schemes with a high degree of parallelism. The generic form of GCEIDD allows different schemes for the prediction, correction, and one-dimensional implicit solution. We have proposed two methods based on modified upwind and characteristics finite difference and a combined method that bypasses the limitations of these two. Also, important aspects of CUDA programming, which are critical to the implementation of the GCEIDD algorithm, are discussed in detail. For solving tri-diagonal systems, a three-stage strategy is proposed, which creates a balance between occupancy, cache hit rate, and shared memory usage and reduces cache contentions. Results show that GCEIDD has good accuracy and stability even when the number of subdomains is very large. The proposed method can accelerate the solution by a factor of up to 3 compared with the GPU implementation of the classic fractional step methods.

A higher-order uniformly convergent defect correction method for singularly perturbed convection-diffusion problems on an adaptive mesh

This paper presents a defect correction scheme based on finite-difference discretization for a singularly perturbed convection-dominated diffusion problem. The solution of this class of problems exhibits a multiscale character. There are narrow regions in which the solution grows exponentially and reveals layer behaviour. The defect correction method that we propose improves the efficiency of a numerical solution through iterative improvement and generates a stable higher-order method over an adaptive non-uniform polynomial-Shishkin mesh. An extensive theoretical analysis is presented, which establishes that the method is second-order uniformly convergent and highly stable. The convergence obtained is optimal because it is free from any logarithmic term. The numerical result for two model problems is presented, agreeing with the theoretical estimates. Furthermore, we compare the results with those of other non-uniform mesh found in the literature.

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

We present a Petrov-Galerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al. (2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform h- and p-refinements, and adaptive h-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible, following the notion in Tian and Du (2014).

Upwind Strategy for Localized Method of Approximate Particular Solutions with Applications to Convection Dominated Diffusion Problems

Upwind Strategy for Localized Method of Approximate Particular Solutions with Applications to Convection Dominated Diffusion Problems

Variational multiscale virtual element method for the convection-dominated diffusion problem

In this paper a new stabilized virtual element method for the convection-dominated diffusion problem is proposed. By taking advantage of robustness of variational multiscale method and flexibility of virtual element method, the proposed method is proven to be very easy to implement. The adding stabilized term in the variational formulation can be calculated directly using the stability portion of the virtual element method. There is no user-defined parameter. And the priori error estimate can be easily obtained as there is no change in the consistency portion in the virtual element method. Numerical experiments show good agreement with the theoretical convergence rate and validate the stabilization effect.

Analysis of Lowest-Order Characteristics-Mixed FEMs for Incompressible Miscible Flow in Porous Media

The time discrete scheme of characteristics type is especially effective for convection-dominated diffusion problems. The scheme has been used in various engineering areas with different approximations in spatial direction. The lowest-order mixed method is the most popular one for miscible flow in porous media. The method is based on a linear Lagrange approximation to the concentration and the zero-order Raviart-Thomas approximation to the pressure/velocity. However, the optimal error estimate for the lowest-order characteristics-mixed FEM has not been presented although numerous effort has been made in last several decades. In all previous works, only first-order accuracy in spatial direction was proved under certain time-step and mesh size restrictions. The main purpose of this paper is to establish optimal error estimates, $i.e.$, the second-order in $L^2$-norm for the concentration and the first-order for the pressure/velocity, while the concentration is more important physical component for the underlying model. For this purpose, an elliptic quasi-projection is introduced in our analysis to clean up the pollution of the numerical velocity through the nonlinear dispersion-diffusion tensor and the concentration-dependent viscosity. Moreover, the numerical pressure/velocity of the second-order accuracy can be obtained by re-solving the (elliptic) pressure equation at a given time level with a higher-order approximation. Numerical results are presented to confirm our theoretical analysis.

Goal-oriented error estimation for the automatic variationally stable FE method for convection-dominated diffusion problems

We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS-FE) method (Calo et al., 2020) for scalar-valued convection–diffusion problems. The AVS-FE method is a Petrov–Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C0 or Raviart–Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov–Galerkin (DPG) method by Demkowicz and Gopalakrishnan (Demkowicz and Gopalakrishnan, 2010; Carstensen et al., 2014; Demkowicz and Gopalakrishnan, 2011a; Demkowicz and Gopalakrishnan, 2011b; Demkowicz and Gopalakrishnan, 2012), leading to unconditionally stable FE approximations. Remarkably, by using C0 or Raviart–Thomas trial spaces, the optimal discontinuous test functions can be computed in a completely decoupled element-by-element fashion.To establish the error estimators we present two approaches: (i) following the classical approach of Becker and Rannacher (Becker and Rannacher, 2001), i.e., the dual solution is sought in the (broken) test space, and (ii) introducing an alternative approach in which we seek C0, or Raviart–Thomas, AVS-FE approximations of the dual solution by using the underlying strong form of the dual boundary value problem (BVP). Various numerical verifications for 2D convection-dominated diffusion BVPs show that the estimates of the approximation error by the new alternative method are highly accurate, while the classical approach leads to error estimates of poor quality. Lastly, we present an algorithm for h-adaptive processes based on control of the numerical approximation error via the new alternative approach. Numerical verifications show that the estimator maintains high accuracy as the error converges to zero.

The double adaptivity paradigm: (How to circumvent the discrete inf–sup conditions of Babuška and Brezzi)

We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in the context of the convection-dominated diffusion problem. The presented ideas, however, apply to virtually any well-posed system of first-order partial differential equations, including singular perturbation problems.

An [formula omitted]-DPG method for the convection–diffusion problem

Following Muga and van der Zee (Muga and van der Zee, 2015), we generalize the standard Discontinuous Petrov–Galerkin (DPG) method, based on Hilbert spaces, to Banach spaces. Numerical experiments using model 1D convection-dominated diffusion problem are performed and compared with Hilbert setting. It is shown that Banach-based method gives solutions less susceptible to Gibbs phenomenon. h-adaptivity is implemented with the help of the error representation function as error indicator.

Tailored finite point method for time fractional convection dominated diffusion problems with boundary layers

We propose a tailored finite point method (TFPM) for solving time fractional convection dominated diffusion equations in this paper. The main idea of TFPM is to firstly approximate the diffusion, convection coefficient near each grid by a constant, and then determine the weights of the finite difference scheme by using the exact solution of the convection diffusion equation with constant coefficients. This adaptation perfectly captures the rapid transition of the solutions which contain sharp boundary layers even with coarse meshes. The accuracy and stability of the scheme are rigorously analyzed. Numerical examples are shown to verify the accuracy and reliability of the proposed scheme.

A layers capturing type H-adaptive finite element method for convection–diffusion–reaction equations on surfaces

In this work, we study the finite element approximation of surface convection–diffusion–reaction equations which are important fundamental model problems in simulations of complex physical phenomena on moving interfaces, ultra-thin materials and biological films. The optimal error estimates of finite element method are shown. According to the analyzed results, the solutions of convection-dominated diffusion problems have low accuracy in the regions of layers. To overcome such drawback which may lead to non-physical oscillations, the adaptive mesh refinement method is considered. Due to the numerical characteristic of non-physical oscillations and layers, we present an error estimator which specializes in capturing the non-physical oscillations and layers. The mesh refinement marking strategies which take into account the large curvatures of the surfaces are developed. The proposed method can efficiently capture layers, non-physical oscillations and provide high resolution solutions with fewer degree of freedoms. A series of numerical examples are designed to demonstrate the effectiveness of the proposed H-adaptive method and make exploration of the solution behavior of convection-dominated diffusion problems on surfaces.

A Compact Higher-Order Scheme for Two-Dimensional Unsteady Convection–Diffusion Equations

In this paper, we study a compact higher-order scheme for the two-dimensional unsteady convection–diffusion problems using the nearly analytic discrete method (NADM), especially, focusing on the convection dominated-diffusion problems. The numerical scheme is constructed and the stability analysis is implemented. We find the order of accuracy of scheme is [Formula: see text], where [Formula: see text] is the size of time steps and [Formula: see text] the size of spacial steps, especially, making clear the relation between [Formula: see text] and [Formula: see text] is according to the different values of diffusion parameter [Formula: see text] through von Neumann stability analysis. The obtained numerical results for several benchmark problems show that our method makes progress in the numerical study of NADM for convection–diffusion equation and is to be effective and helpful particularly in computations for the convection dominated-diffusion equations and, furthermore, valuable in the numerical treatment of many real-world problems such as MHD natural convection flow.

Exponential high-order compact finite difference method for convection-dominated diffusion problems on nonuniform grids

In this article, an exponential high-order compact (EHOC) difference method is introduced for solving the convection-dominated convection diffusion problems on nonuniform grid without any coordinate transformation from the physical space to the computational space. The derived EHOC schemes on nonuniform grid can not only preserve nonoscillation property and yield high accuracy approximation solutions, but also efficiently handle the convection diffusion problems with boundary layers by employing a flexible discretized grid that can be adapted to the singularity in the domain. To demonstrate the performances in computational accuracy, efficiency, and stability of the proposed EHOC schemes, several problems with boundary or interior layers where sharp gradients may appear are solved numerically. The results obtained by the present EHOC schemes on nonuniform grid are compared with analytical solutions and those using the EHOC on uniform grid and other methods. The present EHOC schemes on nonuniform grid provide excellent results for all test problems. It is shown that the proposed EHOC schemes on nonuniform grid are accurate, efficient, and stable, as well as have the advantage of better-scale resolution for convection-dominated diffusion problems.

A mixed volume element with upwind multistep mixed volume element and convergence analysis for numerical simulation of nuclear waste contaminant disposal

In this paper the authors discuss a numerical simulation problem of three-dimensional contamination treatment from nuclear waste. The mathematical model is defined by a nonlinear system of an initial–boundary problem of four partial differential equations: an elliptic equation for the fluid pressure, three convection–diffusion equations for the brine, radionuclides and heat. The pressure appears within the concentration equations and heat conduction equation, and the Darcy velocity controls the concentrations and the temperature. The pressure is solved by the conservative mixed volume element method, and the order of accuracy is improved by the Darcy velocity. The concentrations and temperature are treated by a combination of a multistep method, the upwind approximation and a mixed volume element. The partial derivatives to time variable are approximated by a multistep method. A mixed volume element and an upwind scheme are used to discretize the diffusion and the convection, respectively. The composite method can solve the convection-dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy. The mixed volume element has the local conservation of mass and energy, and it can obtain the concentrations and temperature and their adjoint vector functions simultaneously. The conservation nature plays an important role in numerical simulation of underground fluid. By the technique of a priori estimates of differential equations, we derive an optimal second order result in L 2 norm. Numerical examples are given to show the effectiveness and practicability of our approach and the composite method is testified as a powerful tool for solving the challenging benchmark problem.

An anisotropic a priori error analysis for a convection-dominated diffusion problem using the HDG method

This paper deals with the a priori error analysis for a convection-dominated diffusion 2D problem, when applying the HDG method on a family of anisotropic triangulations. It is known that in this case, boundary or interior layers may appear. Therefore, it is important to resolve these layers in order to recover, if possible, the expected order of approximation. In this work, we extend the use of HDG method on anisotropic meshes. In this context, when the discrete local spaces are polynomials of degree k≥0, this approach is able to recover an order of convergence k+12 in L2 for all the variables, under certain assumptions on the stabilization parameter and family of triangulations. Numerical examples confirm our theoretical results.

On the a posteriori error analysis for linear Fokker–Planck models in convection-dominated diffusion problems

This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection-dominated diffusion problems motivated by a linear Fokker–Planck problem appearing in computational neuroscience. We obtain computable error bounds of functional type for the static and time-dependent case and for different boundary conditions (mixed and pure Neumann boundary conditions). Finally, we present a set of various numerical examples including discussions on mesh adaptivity and space-time discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived.

Multiquadric RBF-FD method for the convection-dominated diffusion problems base on Shishkin nodes

In this paper, a new hybrid scheme based multiquadric radial basis function-generated finite difference (RBF-FD) method with Shishkin nodes is proposed to solve stationary convection-dominated diffusion problems, i.e., the boundary layer problems, which combines the midpoint upwind scheme on the coarse grid with a standard central scheme on the fine grid. Numerical results demonstrate the hybrid scheme with Shishkin nodes has an absolute advantage in accurate over standard upwind scheme with equidistant nodes. Moreover, by varying the shape parameter in multiquadric RBF-FD method, the convergence rates of numerical solutions have been further improved compared with the corresponding FD method.