Construction and application of an algebraic dual basis and the Fine-Scale Greens' Function for computing projections and reconstructing unresolved scales

TL;DR

The paper introduces dual basis functions to explicitly construct Fine-Scale Greens' functions for variational multiscale methods, enabling the reconstruction of unresolved scales for arbitrary projections in any dimension. By encoding the projector with $\boldsymbol{\mu}$ via dual bases, the authors derive a general formula $\mathcal{G}' = \mathcal{G} - \mathcal{G}\boldsymbol{\mu}^T(\boldsymbol{\mu}\mathcal{G}\boldsymbol{\mu}^T)^{-1}\boldsymbol{\mu}\mathcal{G}$ and demonstrate its validity for $L^2$ and $H^1_0$ projections. They provide detailed construction of primal/dual bases, implement the approach in 1D Poisson and advection–diffusion problems, and extend the methodology to 2D Poisson, showing accurate recovery of fine scales missing from the coarse projection. The work suggests broad applicability to Finite/Spectral Element and Isogeometric frameworks and outlines future directions, including robust numerical integration and extensions to nonlinear problems like Navier–Stokes. Overall, the method offers a general, explicit mechanism to incorporate unresolved physics into resolved-scale computations across dimensions and projection choices.

Abstract

In this paper, we build on the work of [T. Hughes, G. Sangalli, VARIATIONAL MULTISCALE ANALYSIS: THE FINE-SCALE GREENS' FUNCTION, PROJECTION, OPTIMIZATION, LOCALIZATION, AND STABILIZED METHODS, SIAM Journal of Numerical Analysis, 45(2), 2007] dealing with the explicit computation of the Fine-Scale Green's function. The original approach chooses a set of functionals associated with a projector to compute the Fine-Scale Green's function. The construction of these functionals, however, does not generalise to arbitrary projections, higher dimensions, or Spectral Element methods. We propose to generalise the construction of the required functionals by using dual functions. These dual functions can be directly derived from the chosen projector and are explicitly computable. We show how to find the dual functions for both the $L^2$ and the $H^1_0$ projections. We then go on to demonstrate that the Fine-Scale Green's functions constructed with the dual basis functions consistently reproduce the unresolved scales removed by the projector. The methodology is tested using one-dimensional Poisson and advection-diffusion problems, as well as a two-dimensional Poisson problem. We present the computed components of the Fine-Scale Green's function, and the Fine-Scale Green's function itself. These results show that the method works for arbitrary projections, in arbitrary dimensions. Moreover, the methodology can be applied to any Finite/Spectral Element or Isogeometric framework.

Figures (26)

• Figure 1: Nodal Lagrange and corresponding Edge polynomials constructed over 1D mesh with 4 GLL points
• Figure 2: Dual nodal and edge polynomials constructed over 1D mesh with 4 GLL points
• Figure 3: Primal edge basis functions of polynomial degree $p = 2$ over 2 elements and their corresponding $L^2$ duals over $x \in \Omega \in [0, 1]$
• Figure 4: Primal edge basis functions of polynomial degree $p = 3$ over 2 elements and their corresponding $L^2$ duals over $x \in \Omega \in [0, 1]$
• Figure 5: Primal nodal basis function of polynomial degree $p = 1$ over 2 elements and its corresponding $H_0^1$ dual over $x \in \Omega \in [0, 1]$

Theorems & Definitions (7)

• Remark 3.1
• Definition 3.1: frankel
• Definition 3.2: frankel
• Definition 3.3
• Remark 3.2
• Remark 3.3
• Remark 5.1