A nonlinear and discontinuous multiscale formulation for convection-diffusion-reaction problems
DOI:
https://doi.org/10.35819/remat2024v10iespecialid7088Keywords:
discontinuous Galerkin, artificial diffusion, convection-diffusion-reaction, bubble functions, multiscale methodsAbstract
This work presents a formulation of discontinuous multiscale and nonlinear Galerkin method aiming to solve convection-diffusion-reaction problems. Considering a decomposition of the approximation space into two scales, macro and micro, the new method introduces a nonlinear artificial diffusion operator at both discretization scales, while employing the discontinuous approach only at the macro scale. The micro scale is modeled through bubble functions (polynomial functions that vanish at the boundary of the elements), allowing the application of the static condensation process in each element. The discretization of the numerical model results in a global system of equations associated with nodal points only at the macro scale. To assess the stability and convergence properties of the proposed scheme, several numerical experiments were conducted and compared with the classical discontinuous Galerkin method. The proposed formulation proved to be efficient in eliminating spurious oscillations that appear in regions of high gradients in problems with dominant convection/reaction. Furthermore, the method exhibited optimal convergence rates.
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