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Variational Multiscale Finite-Element Methods for a Nonlinear Convection–Diffusion–Reaction Equation

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Abstract

This paper is devoted to developing finite-element methods for solving a two-dimensional boundary value problem for a singularly perturbed time-dependent convection–diffusion–reaction equation. The solution to this problem may vary rapidly in thin layers. As a result, spurious oscillations occur in the solution if the standard Galerkin method is used. In the multiscale finite-element methods, the original problem is split into the grid-scale and subgrid-scale problems, which allows capturing the problem features at a scale smaller than the element mesh size. In this study two methods are considered: the variational multiscale method with algebraic sub-scale approximation (VMM-ASA) and the residual-free bubbles (RFB) method. In the first method the subgrid-scale problem is simulated by the residual of the grid-scale equation and intrinsic time scales. In the second method the subgrid-scale problem is approximated by special functions. The grid-scale and subgrid-scale problems are formulated via the linearization procedure on the subgrid component applied to the original problem. The computer implementation of the methods was carried out using a commercial finite-element package. The efficiency of the developed methods is evaluated by solving a test boundary value problem for the nonlinear equation. Cases with different values of the diffusion coefficient have been analyzed. Based on the numerical investigation, it is shown that the multiscale methods enable improving the stability of the numerical solution and decreasing the quantity and the amplitude of oscillations compared to the standard Galerkin method. In the case of a small diffusion coefficient, the developed methods can yield a satisfactory numerical solution on a sufficiently coarse mesh.

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Funding

This work was supported by the Council for Grants of the President of the Russian Federation for Young Russian Scientists (project no. MK-4174.2018.1).

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Authors and Affiliations

  1. Institute of Continuous Media Mechanics, Ural Branch, Russian Academy of Sciences, 614018, Perm, Russia

    M. S. Zhelnin, A. A. Kostina & O. A. Plekhov

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  1. M. S. Zhelnin
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  2. A. A. Kostina
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  3. O. A. Plekhov
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Correspondence to M. S. Zhelnin, A. A. Kostina or O. A. Plekhov.

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Translated by E. Oborin

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Zhelnin, M.S., Kostina, A.A. & Plekhov, O.A. Variational Multiscale Finite-Element Methods for a Nonlinear Convection–Diffusion–Reaction Equation. J Appl Mech Tech Phy 61, 1128–1139 (2020). https://doi.org/10.1134/S0021894420070226

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