Preconditioning the Deferred Correction Operators

Determine whether an invertible alternative \(\mathcal{L}_1\) operator can be defined that preconditions the consistent mass matrix \(\mathcal{M}\) more effectively than \(\mathcal{D}^{-1}\) for a given polynomial basis when used in the Neumann-series Deferred Correction iteration, with the aim of reducing the spectral radius and accelerating convergence.

Background

The paper reformulates the explicit Deferred Correction (DeC) iteration as a Neumann-series approximation involving A=I−D−1MA=I-\mathcal{D}^{-1}\mathcal{M}, where M\mathcal{M} is the consistent finite-element mass matrix and D\mathcal{D} is its row-sum lumped approximation. The convergence rate therefore depends on powers of AA and, in particular, on its spectral radius.

For high-order Bernstein bases, the spectral radius of AA can approach one as the polynomial degree increases, causing the theoretical number of correction steps to be insufficient in practice. The authors consequently identify preconditioning the DeC operators as a possible route to reducing the spectral radius and accelerating convergence, but do not develop such a preconditioner in the paper.

References

More specifically, we conjecture that it is possible to define a different \mathcal{L}_1 operator which is invertible but preconditions \mathcal{M} better than \mathcal{D}{-1} for a given polynomial basis when applied in the iteration eqn:NeumannDeC. The development of such a preconditioning strategy is beyond the scope of the present article.

eqn:NeumannDeC:

um,n(k+1)=un+Δt∑ℓ=0MΘm,ℓ∑p=0kApD−1f(uℓ,n(k−p)),u^{(\rm k+1)}_{\rm m,n} = u_n + \Delta t \sum_{\ell=0}^{\rm M} \Theta_{\rm m,\ell}\sum_{\rm p=0}^{\rm k} A^{\rm p}\mathcal{D}^{-1} f(u^{(\rm k-\rm p)}_{\ell,n}),

— A Deferred Correction, Continuous Galerkin Method for Curvilinear Staggered-Grid Lagrangian Hydrodynamics  (2610.00932 - Walton et al., 1 Oct 2026) in Section “Deferred Correction,” immediately after Proposition on the Neumann-series DeC update