Explicit Fixed-Point Subiterations in Nonlinear Solvers

• Explicit Fixed-Point Subiterations are iterative methods that decompose complex nonlinear problems into simpler explicit updates, enabling efficient handling of nonlinear PDEs.
• The approach leverages Anderson acceleration to optimize convergence speed, reducing iteration counts and computational cost in challenging numerical scenarios.
• Rigorous error and stability analyses ensure that these methods can attain accuracies comparable to implicit solvers under appropriate discretization and subiteration conditions.

Explicit fixed-point subiterations are iterative methodologies that decompose complex nonlinear or implicit problems into a series of tractable explicit updates. They are prevalent in numerical analysis, scientific computing, and applied mathematics, particularly for solving nonlinear partial differential equations or time-discretized systems where fully implicit methods are computationally intensive or intractable. Recent developments have focused on accelerating convergence, improving stability, and enhancing usability through techniques such as Anderson acceleration. The approach is notably effective when the cost of matrix assembly or the lack of a suitable preconditioner renders implicit systems expensive, as in highly convective fluid flows (Barnafi et al., 18 Oct 2025).

1. The Explicit Fixed-Point Subiteration Framework

At each time-step or nonlinear solve, the principal unknown (for instance, the state at time level $n$, denoted $u^n$) is updated via an iterative process. Starting from an initial guess $u^n(0)$, the algorithm generates a sequence $\{u^n(m)\}_{m=0,1,...}$ using

$u^n(m+1) = G(u^n(m))$

where $G$ is a nonlinear update operator often constructed to be fully explicit, i.e., avoiding the need to solve a coupled implicit system. The definition of $G$ is problem-dependent; for PDEs, it may represent an explicit time stepping, a nonlinear residual correction, or an approximate solution map based on the current iterate.

Convergence is typically declared when $\|u^n(m+1) - u^n(m)\|$ falls below a chosen tolerance. This structure allows for fine-grained control over both computational cost and solution accuracy.

2. Anderson Acceleration Integration

Convergence speed of fixed-point subiterations can be suboptimal, particularly near regions of strong nonlinearity or instability. Anderson acceleration overcomes these issues by forming the next iterate as an optimal linear combination of previous explicit updates: $$u^n(m+1) = \sum_{j=0}^m \alpha_j G(u^n(j)), \quad \sum_{j=0}^m \alpha_j = 1$$ where coefficients $\{\alpha_j\}$ are chosen to minimize the residual norm over the last $m+1$ iterates. This least-squares optimization leverages the information from multiple iterates, substantially reducing iteration count and improving robustness—even in cases where explicit-only methods stall or oscillate due to underlying nonlinear instabilities.

While Anderson acceleration introduces an overhead for the coefficient computation (a small least-squares problem at every subiteration), empirical results demonstrate an overall reduction in total computational cost due to faster convergence (Barnafi et al., 18 Oct 2025).

3. Error Analysis and Theoretical Properties

Rigorous error and stability analysis underpins the explicit fixed-point subiteration methodology. Under regularity assumptions on $G$—such as Lipschitz continuity or monotonicity—and an appropriately chosen time step (often subject to a CFL-type condition), the method satisfies classical consistency and stability properties. The error bound achieved after integration of spatial and temporal discretization typically conforms to

$\|u(t_n) - u^n\| \leq C (\Delta t^p + h^q)$

where $\Delta t$ and $h$ are the temporal and spatial discretization parameters, $p$ and $q$ are their associated accuracy orders, and $C$ encodes method-dependent constants. Notably, the error constants remain independent of discretization under stated conditions, ensuring scalability.

Theoretical support confirms that, with sufficient subiterations at each time-step, explicit fixed-point updates can achieve accuracy comparable to more traditional implicit solvers. Delayed treatment of implicit terms in time discretization is formalized and shown to be well-posed within this framework.

4. Applicability and Performance in Nonlinear Problems

Explicit fixed-point subiterations are particularly advantageous in settings where matrix assembly is costly or effective preconditioners are unavailable. The methodology is well-suited for nonlinear diffusion, convection-diffusion, and highly convective fluid flow problems, where the explicit update can efficiently handle steep gradients and strong nonlinear couplings.

Empirical benchmarks demonstrate the effectiveness of the approach: problems exhibiting highly convective dynamics, which are known to challenge implicit solvers and iterative linearization schemes, are reliably addressed using explicit subiteration coupled with Anderson acceleration. The explicit nature of updates facilitates parallelization and modular implementation, making the method robust and tractable for industrial-scale problems with complex domain geometries.

Representative Method Structure

Step	Operation	Termination Criterion
Initialize	$u^n(0)$	–
Iterate	$u^n(m+1) = G(u^n(m))$ or Anderson-accelerated	$\|u^n(m+1) - u^n(m)\| < \epsilon$
Advance time	Move to $u^{n+1}(0)$ with converged $u^n$	–

5. Usability and Scalability Considerations

The explicit subiteration methodology is simple to implement, requiring only repeated application of explicit updates and periodic Anderson acceleration. No complex nonlinear or linear system solves are necessary; this significantly lowers the barrier to entry for domain scientists requiring reliable solvers without extensive numerical infrastructure.

Modularity and the avoidance of global matrix factorizations make the approach highly amenable to parallelization. Subiterations can be performed independently across spatial domains or problem components, contributing to full scalability on high-performance computing platforms.

Parameter choices, such as the number of subiterations, tolerance levels, and the size of the Anderson acceleration window, affect both convergence and computational cost. Optimal tuning depends on problem characteristics, with larger acceleration histories often yielding faster convergence at minimal extra per-iteration expense.

6. Future Research Directions

Several avenues exist for further investigation of explicit fixed-point subiterations. Adaptive schemes that automatically adjust subiteration counts and acceleration parameters based on local error estimates promise enhancements in robustness and efficiency. Deeper error analysis for challenging nonlinear and multiphysics systems could extend theoretical guarantees and best practices.

Hybrid algorithms combining explicit subiterations with other nonlinear solution strategies, or incorporating dynamic preconditioning when available, have the potential to widen the domain of applicability. Extensions to broader classes of nonlinear PDEs, such as those with coupled multiphysics effects, are promising directions for advancing this framework in computational science (Barnafi et al., 18 Oct 2025).

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Explicit fixed-point subiterations, especially when combined with Anderson acceleration, represent a robust and efficient strategy for solving nonlinear time-discretized problems. The methodology circumvents the bottlenecks associated with implicit methods, provides theoretical guarantees, and delivers practical performance across a range of applications, establishing it as a pivotal approach in modern numerical computation for nonlinear systems.