Fixed-Point L-scheme for Nonlinear PDEs

• Fixed-point iteration (L-scheme) is a robust method that replaces complex nonlinear operators with stabilized linear approximations to solve degenerate elliptic PDEs.
• It iteratively solves a Poisson-type problem using a carefully chosen scalar parameter that dominates the spectral properties of the original nonlinear operator.
• The method excels in handling challenges like non-convexity and mesh refinement, offering mesh-independent convergence and practical efficiency, notably for the Monge–Ampère equation.

Fixed-point iteration, commonly referred to as the L-scheme in the context of nonlinear and degenerate PDEs, is a robust, linearization-based iterative framework for solving nonlinear equations and operator equations. The designation “L-scheme” reflects the introduction of a scalar lumping or stabilization parameter, frequently denoted by Λ or L, which replaces genuinely nonlinear, solution-dependent operators by fixed linear approximations. This approach offers strong advantages in terms of robustness, especially in cases where standard Newton-type linearizations are either ill-posed, sensitive to the initial guess, or fail in the presence of degeneracy or loss of convexity. The L-scheme has found effective application to problems such as the elliptic Monge–Ampère equation with Dirichlet boundary conditions, which arises in optimal transport, geometric optics, and differential geometry. The scheme is characterized by solving, at each step, a linear problem—typically a Poisson equation—with a right-hand side formed from a nonlinear residual and a Laplacian weighted by a carefully chosen parameter that dominates the spectral properties of the nonlinear (often degenerate) operator.

1. Principles and Motivation of the L-scheme

The L-scheme is motivated by the intrinsic difficulties associated with Newton’s method for fully nonlinear, degenerate elliptic PDEs, particularly those involving the Monge–Ampère operator. In these problems, the discrete or continuous linearization depends on derivative (or Hessian) information of the current iterate; the resulting system may lose ellipticity if convexity or coercivity is not preserved, leading to divergence or instability. Rather than linearizing around the current iterate and retaining all variable-dependent coefficients, the L-scheme freezes or lumps these coefficients by replacing, for example, the Hessian cofactor matrix $\operatorname{cof}(D^2 u^i)$ with a scalar multiple of the identity, $\Lambda^i I$. As a result, each iteration step solves

$$\Lambda^i \Delta v^{i+1} = -\rho(u^i) \quad \text{in } \Omega,$$

where the residual $\rho(u^i)$ captures the deviation from the nonlinear target (e.g., $\det D^2 u^i - f(x, \nabla u^i)$ in Monge–Ampère (Köhle et al., 31 Aug 2025)).

This process ensures that the linear problem in each iteration is always uniformly elliptic for sufficiently large $\Lambda^i$, independently of the Hessian spectrum of $u^i$. Thus, even in the presence of degenerate or non-convex intermediate iterates, the method remains well defined and coercive.

2. Algorithmic Structure and Weight Selection

At the core of the L-scheme for the elliptic Monge–Ampère equation lies the following sequence:

1. Residual computation: Set $u^i$ to the current iterate and compute

$\rho(u^i) = \det(D^2 u^i) - f(x, \nabla u^i).$

1. Update problem: Solve

$$\Lambda^i \Delta v^{i+1} = -\rho(u^i) \text{ in } \Omega, \quad v^{i+1} = \gamma - u^i \text{ on } \partial\Omega,$$

where $\gamma$ is the prescribed boundary condition.

1. Iterate update: Set $u^{i+1} = u^i + v^{i+1}$.

The scalar parameter $\Lambda^i$ is chosen to overestimate the largest eigenvalue of the Hessian of $u^i$, to the appropriate power $(d-1)$ in $d$ dimensions: $$\Lambda^i \geq \|\lambda^{i}_{\max}(\cdot)\|_{L^\infty(\Omega)}^{d-1}.$$ This guarantees that the “lumped” linear operator dominates the spectrum of the original solution-dependent operator, yielding stability and contraction.

3. Convergence Analysis and Theoretical Guarantees

The convergence analysis splits into classical (smooth) and generalized settings:

• Classical (smooth) solutions: For smooth and convex $u$, under the assumption that $f$ is Lipschitz in $\nabla u$ with constant $\mu_f$ small enough, the sequence of errors $e^i = u^i - u$ satisfies

$$\|\Delta e^{i+1}\|_{L^2(\Omega)} \leq q^i\,\|\Delta e^i\|_{L^2(\Omega)},$$

where the contraction factor $q^i$ is explicitly

$$q^i = 1 - \frac{\xi}{\Lambda^i} + \frac{C_E \mu_f}{\Lambda^i},$$

with $\xi$ encoding information on the minimal Hessian eigenvalue and $C_E$ the elliptic regularity constant.

• Generalized (viscosity or Aleksandrov) solutions: Contraction in the $L^\infty$ norm is proven under analogous conditions, again relying on convexity of the error and an appropriate choice of the weight $\Lambda^i$.

This mechanism guarantees linear convergence in either the $H^2$ or $L^\infty$ norm, with the contraction factor directly controlled by the ratio of the problem’s geometric and analytic parameters to the stabilization weight.

4. Acceleration Techniques and Solver Strategies

Every L-scheme iteration reduces to solving a Poisson-type equation with fixed (scalar) coefficients. Because of this, a variety of fast linear solvers and preconditioners can be leveraged:

• Green’s function methods: In simple domains (e.g., rectangles) with homogeneous boundary conditions, the update $v^{i+1}$ can be represented as

$$v^{i+1}(x_0) = -\frac{1}{\Lambda^i} \int_{\Omega} G_p(x, x_0)\,\rho(u^i(x))\,dx,$$

where $G_p$ is the Laplacian Green’s function. This approach allows explicit solution formulas with high accuracy on coarse grids, but is less efficient for large-scale or fine meshes due to dense matrix operations.

• Finite difference or finite element with preconditioned iterative solvers: Discretization by standard methods yields sparse linear systems (e.g., arising from a 5-point stencil in 2D). Preconditioned conjugate gradient (PCG) methods can be used, with preconditioners such as incomplete LU or, more attractively, algebraic multigrid (AMG). Numerical evidence demonstrates that AMG-preconditioned PCG within the L-scheme achieves iteration counts and runtimes that are essentially independent of mesh size, outperforming direct and purely Green’s function-based solvers in practical scenarios.

5. Comparison to Newton-Type and Alternative Methods

The L-scheme directly addresses key limitations of Newton-type approaches:

• Initialization and convexity sensitivity: Standard Newton methods for Monge–Ampère require that all iterates $u^i$ are (strictly) convex to guarantee that the linearizations (involving $\operatorname{cof}(D^2 u^i)$) are elliptic. If intermediate $u^i$ lose convexity—an event likely on coarse or poor initial guesses, or for degenerate data—the method can become non-coercive or diverge. In contrast, the L-scheme is well-posed for any initial $u^0$, convex or not, as long as the stabilization parameter is sufficiently large.
• Mesh refinement robustness: Newton’s method may perform well close to the true solution, but its convergence (and even basic well-posedness) can deteriorate for finer discretizations or more oscillatory right-hand-sides, sometimes diverging altogether. The L-scheme shows mesh-independent iteration counts and superior global robustness.
• Practical efficiency: Although Newton may require fewer iterations when convergence is achieved, the L-scheme compensates by leveraging rapid Poisson solvers; in numerical experiments, it demonstrates consistently lower total computational cost in regimes where Newton struggles or fails.

6. Numerical Performance and Implementation Considerations

Benchmarks with smooth (“Gaussian”) and highly oscillatory right-hand sides confirm that the L-scheme outperforms Newton in speed and stability for the elliptic Monge–Ampère equation (Köhle et al., 31 Aug 2025). The method demonstrates:

• Stable contraction regardless of initial iterate or mesh refinement.
• Iteration count approximately independent of discretization granularity.
• Enhanced performance with AMG preconditioning compared to dense solvers or unpreconditioned conjugate gradient.
• Capability to handle challenging, degenerate, or non-convex intermediate states in nonlinear PDEs.

Implementation best practices recommend monitoring the eigenvalues of the Hessian in each iteration to ensure $\Lambda^i$ is updated so that robust contraction is maintained, and exploiting problem structure to maximize solver efficiency.

7. Extensions and Theoretical Context

The L-scheme generalizes to a broad range of nonlinear operator and PDE settings where direct linearization is problematic. Its stabilization principle aligns with themes in monotone operator theory, robust fixed-point methods, and numerically safe iterations for degenerate or non-smooth problems.

The methodology leverages the Banach fixed-point principle in metric and order-theoretic perspectives (as refined, for instance, by monotone convergence analyses in cone-ordered spaces), effectively linking analytic contraction to computational safety and flexibility. This suggests future research in operator selection, weight adaptation strategies, and cross-application to fully nonlinear variational problems.

Summary Table: L-scheme for Monge–Ampère Equation (as described in (Köhle et al., 31 Aug 2025))

Step	Operation	Parameter Choice
Residual Update	$\rho(u^i) = \det(D^2u^i) - f(x, \nabla u^i)$	—
Linear Solve	$\Lambda^i \Delta v^{i+1} = -\rho(u^i)$ in $\Omega$	$\Lambda^i \geq \|\lambda^i_{\max}\|^{d-1}$
Iterate Update	$u^{i+1} = u^i + v^{i+1}$	—

This structure provides a robust, efficient, and provably convergent method for nonlinear, degenerate PDEs where standard linearizations fail or are unreliable; it also supports fast solution by exploiting the reduced complexity of the linear (Poisson) step and advanced linear solving techniques.