# Conditioning of implicit Runge-Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

## TL;DR

This paper analyzes how the conditioning of implicit Runge-Kutta methods for finite element diffusion problems is affected by mesh anisotropy and proposes bounds that incorporate mesh and diffusion properties, with effective preconditioning strategies.

## Contribution

It provides explicit bounds on the condition number for implicit RK methods on anisotropic meshes, considering different solution strategies and preconditioning effects, with geometric interpretations.

## Key findings

- Condition number bounds depend on mesh size, nonuniformity, and diffusion matrix.
- Diagonal preconditioning effectively mitigates mesh nonuniformity effects.
- Numerical examples validate the theoretical bounds and strategies.

## Abstract

The conditioning of implicit Runge-Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution (the system is solved as a whole) and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into smaller systems using the Jordan normal form of the RK matrix and then solve them successively. For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that . If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite. The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Numerical examples are given.