As we know, the variational multiscale element free Galerkin (VMEFG) method may still suffer from nonphysical oscillations near the boundary or interior layers when solving the convection-diffusion-reaction problems w...
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As we know, the variational multiscale element free Galerkin (VMEFG) method may still suffer from nonphysical oscillations near the boundary or interior layers when solving the convection-diffusion-reaction problems with strong convection-dominated. To overcome this shortcoming, we consider incorporating an adaptive algorithm based on the residual error estimations into the VMEFG, which formed the adaptive VMEFG (AVMEFG) method. With respect to the adaptive technique adopted, two residual-based a posteriori error estimators in the H-1-semi norm and energy norm are respectively used to locate and remark the high-gradient numerical solution regions. Several stationary convection-diffusion-reaction problems are solved to verify the effectiveness of the proposed method. Among them, the first example makes a comparison between the proposed method and the adaptive element free Galerkin method, while the other remaining examples compare two different residual-based a posteriori error estimators for the proposed method. Numerical examples illustrate that the proposed method is effective and efficient in solving the convection-dominated problem involving various layers. Moreover, the AVMEFG methods with the two residual-based a posteriori error estimators are generally equivalent except in the case that the solution has an exponential boundary layer, where the energy norm error estimator can achieve significantly better results than the other one.
We consider some (anisotropic and piecewise constant) convection-diffusion-reaction problems in domains of R-2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose two a posterio...
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We consider some (anisotropic and piecewise constant) convection-diffusion-reaction problems in domains of R-2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose two a posteriori error estimators based on gradient recovery by averaging. It is shown that these estimators give rise to an upper bound where the constant is explicitly known up to some additional terms that guarantee reliability. The lower bound is also established, one being robust when the convection term (or the reaction term) becomes dominant. Moreover, the estimator is asymptotically exact when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimators are confirmed by some numerical tests.
We study a class of steady nonlinear convection-diffusion-reaction problems in porous media. The governing equations consist of coupling the Darcy equations for the pressure and velocity fields to two equations for th...
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We study a class of steady nonlinear convection-diffusion-reaction problems in porous media. The governing equations consist of coupling the Darcy equations for the pressure and velocity fields to two equations for the heat and mass transfer. The viscosity and diffusion coefficients are assumed to be nonlinear depending on the temperature and concentration of the medium. Well-posedness of the coupled problem is analyzed and existence along with uniqueness of the weak solution is investigated based on a fixed-point method. An iterative scheme for solving the associated fixed-point problem is proposed and its convergence is studied. Numerical experiments are presented for two examples of coupled convection-diffusion-reaction problems. Applications to radiative heat transfer and propagation of thermal fronts in porous media are also included in this study. The obtained results show good numerical convergence and validate the established theoretical estimates.
Boundary value problems for time-dependent convection-diffusion-reaction equations are basic models of problems in continuum mechanics. To study these problems, various numerical methods are used. With a finite differ...
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Boundary value problems for time-dependent convection-diffusion-reaction equations are basic models of problems in continuum mechanics. To study these problems, various numerical methods are used. With a finite difference, finite element, or finite volume approximation in space, we arrive at a Cauchy problem for systems of ordinary differential equations whose operator is asymmetric and indefinite. Explicit-implicit approximations in time are conventionally used to construct splitting schemes in terms of physical processes with separation of convection, diffusion, and reaction processes. In this paper, unconditionally stable schemes for unsteady convection-diffusion-reaction equations are constructed with explicit-implicit approximations used in splitting the operator reaction. The schemes are illustrated by a model 2D problem in a rectangle.
Algebraically stabilized finite element discretizations of scalar steady-state convection-diffusion-reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). Howe...
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Algebraically stabilized finite element discretizations of scalar steady-state convection-diffusion-reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). However, it was observed that a deterioration of the accuracy and convergence rates may occur for some problems if meshes without local symmetries are used. The paper investigates these phenomena both numerically and analytically and the findings are used to design a new algebraic stabilization called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is proved that the SMUAS method is linearity preserving and satisfies the DMP on arbitrary simplicial meshes. Moreover, numerical results indicate that the SMUAS method leads to optimal convergence rates on general simplicial meshes.
This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized b...
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This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection-)diffusion-reactionproblems.
We present a new discretization method for convection-diffusion-reaction boundary value problems in 3D with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means ...
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We present a new discretization method for convection-diffusion-reaction boundary value problems in 3D with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered as a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments. Our experiments also show an improved resolution of the exponential layer at the outflow boundary for our proposed method when compared to the SUPG method. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
We are promoting the use of variational principles as some generalizing idea that allows us to create consistent approximations of mathematical models. In this paper, we formulate a computing technology to solve direc...
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We are promoting the use of variational principles as some generalizing idea that allows us to create consistent approximations of mathematical models. In this paper, we formulate a computing technology to solve direct and inverse problems of atmospheric dynamics and chemistry. For the implementation, we use functional decomposition methods, splitting techniques, and integrating factors designed as the solutions of some adjoint problems. One of the main purposes of the paper is to relate the theory of approximation with the concept of Eulerian integrating factors that plays the fundamental role in the theory of differential equations. We demonstrate the utility and versatility of the concept applying it to solve the systems of differential equations of convectiondiffusion (with dominant convection) and stiff systems of kinetic equations. To identify the prospects of our approach, we briefly introduce how to use integrating factors in the case of unstructured grids. (C) 2014 Elsevier Ltd. All rights reserved.
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