[Paper Review] Multiscale finite elements through advection-induced coordinates for transient advection-diffusion equations

Konrad Simon, Jörn Behrens|arXiv (Cornell University)|Feb 21, 2018

Advanced Mathematical Modeling in Engineering3 citations

TL;DR

This paper proposes a novel multiscale finite element method using advection-induced coordinates to improve upscaling in transient advection-diffusion equations, particularly for climate simulations with coarse grids. By transforming coordinates to mitigate transport effects near coarse element boundaries, the method achieves high accuracy in one-dimensional problems with oscillatory velocity and diffusion, demonstrating optimal convergence rates and robustness under scale separation.

ABSTRACT

Long simulation times in climate sciences typically require coarse grids due to computational constraints. Nonetheless, unresolved subscale information significantly influences the prognostic variables and can not be neglected for reliable long term simulations. This is typically done via parametrizations but their coupling to the coarse grid variables often involves simple heuristics. We explore a novel up-scaling approach inspired by multi-scale finite element methods. These methods are well established in porous media applications, where mostly stationary or quasi stationary situations prevail. In advection-dominated problems arising in climate simulations the approach needs to be adjusted. We do so by performing coordinate transforms that make the effect of transport milder in the vicinity of coarse element boundaries. The idea of our method is quite general and we demonstrate it as a proof-of-concept on a one-dimensional passive advection-diffusion equation with oscillatory background velocity and diffusion.

Motivation & Objective

• To address the challenge of unresolved subgrid-scale processes in long-term climate simulations using coarse grids.
• To develop a mathematically consistent upscaling framework that avoids heuristic parametrizations.
• To adapt multiscale finite element methods (MsFEM) for advection-dominated problems common in climate modeling.
• To improve stability and accuracy in transient advection-diffusion equations by transforming coordinates to reduce transport-induced distortions.
• To demonstrate the method's effectiveness in one-dimensional problems with oscillatory background velocity and diffusion.

Proposed method

• The method introduces a coordinate transformation based on advection characteristics to smooth transport effects near coarse element boundaries.
• It constructs multiscale basis functions in the transformed coordinate system that reflect fine-scale behavior while preserving stability.
• The approach uses a Galerkin formulation with non-polynomial basis functions derived from solving local problems in the advection-induced coordinates.
• The method is applied to a one-dimensional transient advection-diffusion equation with oscillatory velocity and diffusion coefficients.
• It ensures mass conservation and stability by aligning the basis functions with the dominant flow direction through the coordinate transformation.
• The method is tested numerically with varying grid resolutions and diffusion frequencies to assess convergence and accuracy.

Experimental results

Research questions

• RQ1 Can coordinate transformations based on advection characteristics improve the stability and accuracy of multiscale finite element methods in advection-dominated transient problems?
• RQ2 How does the proposed method perform in comparison to standard finite element methods when the diffusion coefficient oscillates rapidly relative to the coarse grid?
• RQ3 What are the limitations of the method when the background velocity lacks a dominant mean or contains time-independent zeros?
• RQ4 Can the method be generalized to higher-dimensional or conservative forms of the advection-diffusion equation?
• RQ5 What is the convergence behavior of the method under increasing grid refinement and varying oscillation frequency of the diffusion coefficient?

Key findings

• The method achieves optimal convergence rates in the L² and L∞ norms, with relative errors decreasing by an order of magnitude when grid resolution is doubled.
• For k=10, the L² error decreased from 2.732×10⁻⁴ at N=24 to 3.939×10⁻⁷ at N=384, indicating second-order convergence.
• The method performs well even with high-frequency diffusion oscillations, showing robustness under scale separation.
• Stability issues arise when the background velocity has time-independent zeros, as they create attractors that cause characteristic collapse.
• The method fails for equations in conservation form due to decomposition of the flux term into advective and reactive parts that require different upscaling mechanisms.
• Despite limitations, the method demonstrates strong performance in relevant scenarios and provides a foundation for future extensions using hybrid approaches like semi-Lagrangian methods.