Optimal solutions employing an algebraic Variational Multiscale approach Part I: Steady Linear Problems

TL;DR

This paper develops an algebraic Variational Multiscale approach with an optimal projector for steady linear problems, demonstrating exponential convergence and employing a dual-mesh discretisation to closely approximate the continuous solution.

Contribution

It introduces a novel abstract framework for VMS methods with an optimal projector and a dual-mesh discretisation, extending previous work and providing error analysis and convergence results.

Findings

The proposed method achieves exponential convergence to the optimal projection.

Error analysis confirms convergence of the Greens' function approximation.

Implementation with MSEM demonstrates practical applicability.

Abstract

This work extends our previous study from S. Shrestha et al. (2024) by introducing a new abstract framework for Variational Multiscale (VMS) methods at the discrete level. We introduce the concept of what we define as the optimal projector and present a discretisation approach that yields a numerical solution closely approximating the optimal projection of the infinite-dimensional continuous solution. In this approach, the infinite-dimensional unresolved scales are approximated in a finite-dimensional subspace using the numerically computed Fine-Scale Greens' function of the underlying symmetric problem. The proposed approach involves solving the VMS problem on two separate meshes: a coarse mesh for the full PDE and a fine mesh for the symmetric part of the continuous differential operator. We consider the 1D and 2D steady advection-diffusion problems in both direct and mixed formulations as the test cases in this paper. We first present an error analysis of the proposed approach and show that the projected solution is achieved as the approximate Greens' function converges to the exact one. Subsequently, we demonstrate the working of this method where we show that it can exponentially converge to the chosen optimal projection. We note that the implementation of the present work employs the Mimetic Spectral Element Method (MSEM), although, it may be applied to other Finite/Spectral Element or Isogeometric frameworks. Furthermore, we propose that VMS should not be viewed as a stabilisation technique; instead, the base scheme should be inherently stable, with VMS enhancing the solution quality by supplementing the base scheme.