A structure preserving H-curl algebraic multigrid method for the eddy current equations

TL;DR

This paper introduces a new algebraic multigrid method for eddy current equations that preserves key mathematical properties without relying on simple interpolation, enabling more efficient and mesh-independent solutions.

Contribution

The paper presents a structure preserving H-curl algebraic multigrid method that enforces commuting relationships using energy minimization, improving upon previous methods that relied on sub-optimal interpolation.

Findings

Achieves mesh independent convergence rates.

Enables use of sophisticated nodal interpolation operators.

Demonstrates effectiveness on various test problems.

Abstract

A new algebraic multigrid method (AMG) is presented for solving the linear systems associated with the eddy current approximation to the Maxwell equations. This AMG method extends an idea proposed by Reitzinger and Schoberl. The main feature of the Reitzinger and Schoberl algorithm (RSAMG) is that it maintains null-space properties of the Curl-Curl operator throughout all levels of the AMG hierarchy. It does this by enforcing a commuting relationship involving grid transfers and the discrete gradient operator. This null-space preservation property is critical to the algorithm's success, however enforcing this commuting relationship is non-trivial except in the special case where one leverages a piece-wise constant nodal interpolation operator. For this reason, mesh independent convergence rates are generally not observed for RSAMG due to its reliance on sub-optimal piece-wise constant interpolation. We present a new AMG algorithm that enforces the same commuting relationship. The main advance is that the new structure preserving H-curl algorithm (SpHcurlAMG) does not rely on piece-wise constant interpolation and can leverage fairly general and more sophisticated nodal interpolation operators. The key idea is to employ energy minimization AMG (EAMG) to construct edge interpolation grid transfers and to enforce the commuting relationship by embedding it as constraints within an EAMG procedure. While it might appear that solving such a constrained energy minimization is costly, we illustrate how this is not the case in our context. Numerical results are then given demonstrating mesh independent convergence over a range of test problems.