[Paper Review] A discontinuous Galerkin multiscale method for convection-diffusion problems

Daniel Elfverson|arXiv (Cornell University)|Sep 11, 2015

Advanced Mathematical Modeling in Engineering23 references3 citations

TL;DR

This paper proposes a discontinuous Galerkin local orthogonal decomposition multiscale method for convection-diffusion problems with highly heterogeneous and rough coefficients. By incorporating the convective term into the computation of corrected basis functions on localized patches of size $\mathcal{O}(H\log(H^{-1}))$, the method achieves convergence rates independent of coefficient variation, even in convection-dominated regimes, verified by numerical experiments.

ABSTRACT

We propose an discontinuous Galerkin local orthogonal decomposition multiscale method for convection-diffusion problems with rough, heterogeneous, and highly varying coefficients. The properties of the multiscale method and the discontinuous Galerkin method allows us to better cope with multiscale features as well as interior/boundary layers in the solution. In the proposed method the trail and test spaces are spanned by a corrected basis computed on localized patches of size $\mathcal{O}(H\log(H^{-1}))$, where $H$ is the mesh size. We prove convergence rates independent of the variation in the coefficients and present numerical experiments which verify the analytical findings.

Motivation & Objective

• Address the challenge of solving convection-diffusion problems with highly heterogeneous, rough, and rapidly varying coefficients.
• Overcome the limitations of classical finite element methods in resolving multiscale features and boundary/interior layers.
• Develop a multiscale method that remains stable and convergent under strong convection without requiring scale separation or periodicity.
• Integrate the convective term into the basis function correction process to enhance stability in convection-dominated regimes.
• Establish convergence rates independent of coefficient variation, ensuring robust performance across diverse multiscale settings.

Proposed method

• Formulate a discontinuous Galerkin (DG) variational formulation for convection-diffusion problems with rough diffusion and divergence-free convection coefficients.
• Apply the local orthogonal decomposition (LOD) framework to decompose the solution space into coarse and fine scales.
• Construct corrected basis functions on localized patches of size $\mathcal{O}(H\log(H^{-1}))$ that incorporate both diffusion and convection terms in their construction.
• Use a Petrov-Galerkin approach with test functions derived from the same corrected basis functions to stabilize the method.
• Ensure the method maintains linear complexity and avoids eigenvalue solves or large patch supports.
• Prove convergence in the energy norm with rates independent of the coefficient variation, under assumptions on the convection magnitude.

Experimental results

Research questions

• RQ1 Can a discontinuous Galerkin multiscale method be constructed that remains stable and convergent for convection-diffusion problems with rough, non-periodic coefficients?
• RQ2 How does including the convective term in the basis function correction improve stability in convection-dominated regimes compared to standard LOD?
• RQ3 What is the optimal patch size for the corrected basis functions to ensure robust convergence independent of coefficient variation?
• RQ4 Can the method achieve convergence rates independent of the spectral condition number $C_A = (\beta/\alpha)^{1/2}$?
• RQ5 How does the method perform numerically in comparison to standard finite element and LOD methods for multiscale convection-diffusion problems?

Key findings

• The proposed DG-LOD method achieves convergence rates in the energy norm that are independent of the variation in the diffusion and convection coefficients.
• Inclusion of the convective term in the basis function correction ensures stability and convergence even under strong convection, where standard LOD fails.
• The method uses localized patches of size $\mathcal{O}(H\log(H^{-1}))$, enabling linear computational complexity without requiring scale separation or periodicity.
• Numerical experiments confirm the theoretical convergence rates and demonstrate robust performance across various multiscale coefficient configurations.
• The error estimate shows exponential decay of the fine-scale component in the residual, with a rate dependent on the patch size and convection magnitude.
• The method maintains optimal convergence order in the energy norm, with constants independent of the coefficient variation, as proven via a recursive decay estimate of the residual.