# Modifying AMG coarse spaces with weak approximation property to exhibit approximation in energy norm

## TL;DR

This paper introduces a modification to AMG coarse spaces that achieves approximation in energy norm using polynomial approximations, maintaining computational efficiency and demonstrating effectiveness through numerical experiments.

## Contribution

It proposes a novel polynomial-based modification of AMG coarse spaces to ensure energy norm approximation while preserving sparsity and computational feasibility.

## Key findings

- Modified coarse spaces provide energy norm approximation.
- Numerical results confirm theoretical predictions.
- Sparse matrices are achieved via polynomial approximation.

## Abstract

Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation (WAP) property which is necessary and sufficient condition for uniform two-grid convergence. This paper studies a modification of such coarse spaces so that the modified ones provide approximation in energy norm. Our modification is based on the projection in energy norm onto an orthogonal complement of original coarse space. This generally leads to dense modified coarse space matrices which is hence computationally infeasible. To remedy this, based on the fact that the projection involves inverse of a well-conditioned matrix, we use polynomials to approximate the projection and, therefore, obtain a practical, sparse modified coarse matrix and prove that the modified coarse space maintains computationally feasible approximation in energy norm. We present some numerical results for both, PDE discretization matrices as well as graph Laplacian ones, which are in accordance with our theoretical results.