A full approximation scheme multilevel method for nonlinear variational inequalities

TL;DR

This work presents FASCD, a multilevel solver for variational inequalities that fuses Brandt's full-approximation scheme with constraint decomposition, augmented by level-defect constraints and telescoping ideas to enable effective coarse corrections. The method is smoother-agnostic and achieves near mesh-independent convergence with a reduced-space Newton smoother, delivering near-optimal FMG performance on a variety of VI operators, including nonlinear and non-symmetric cases. The authors provide a detailed multilevel FE framework, rigorous construction of level-defect constraints, and an implementation that integrates with standard PDE software, demonstrating efficiency on problems ranging from classical obstacle problems to ice-sheet VI, with strong parallel scaling. Overall, FASCD offers a scalable, flexible, and open-source solution strategy for challenging VI systems arising in physics, geometry, and optimization contexts, with potential for broad extension to other VI formulations and smoother choices.

Abstract

We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a common extension of both the full approximation scheme (FAS) multigrid technique for nonlinear partial differential equations, due to A.~Brandt, and the constraint decomposition (CD) method introduced by X.-C.~Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain function space subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and full multigrid cycles are optimal solvers. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems.

Figures (9)

• Figure 1: A constraint decomposition (CD) for a unilateral obstacle problem on a two-point space $\Omega=\{x_1,x_2\}$, with $\mathcal{V}=\left\{v \,:\, \Omega \to \mathbb{R}\right\}$ and $\mathcal{K}=\{v\ge \psi\}$.
• Figure 1: The monotone injection operators $R^{\bm{\oplus}}$ and $R^{\bm{\ominus}}$ assign to a coarse degree of freedom (blue) the maximum and minimum respectively of the fine function over the patch of coarse cells sharing its vertex. The coarse mesh is in black and the fine in grey.
• Figure 1: For the finer-level constraints $\underline{\gamma}^1$ (blue) and $\overline{\gamma}^1$ (red) on the left, direct application of $R^{\bullet},R^{\bm{\ominus}},R^{\bm{\oplus}}$ to generate coarser-level constraints is problematic. Possibility \ref{['eq:fe:badmonotone']}, which generates an empty coarser-level set $\mathcal{K}^0$, is at right. Defect constraints \ref{['eq:fe:defectconstraints']} avoid this difficulty.
• Figure 1: Left: The FASCD V-cycle Algorithm \ref{['alg:fascd']} computes downward corrections $y_j \in \mathcal{D}^j$, but the upward corrections $z_j\in\mathcal{U}^j$ are in larger sets. Right: FMG Algorithm \ref{['alg:fascd-fmg']} generates the initial iterate on finer levels by injection and truncation (doubled edges).
• Figure 1: Coincidence sets for the ball and spiral solutions in Example \ref{['ex:results:classical']}.

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Lemma 2.4
• Proof 1
• Example 2.5
• Example 2.6
• Lemma 3.1
• Example 5.1
• Lemma 5.2