[Paper Review] On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Markus Faustmann, Jens Markus Melenk+1 more|arXiv (Cornell University)|Dec 19, 2019

Advanced Numerical Methods in Computational Mathematics54 references15 citations

TL;DR

This paper establishes endpoint stability of Scott-Zhang type operators in Besov spaces, enabling a multilevel decomposition for adaptively refined meshes via newest vertex bisection. It introduces a local multilevel diagonal preconditioner for the fractional Laplacian that achieves uniformly bounded condition numbers, ensuring optimal convergence for iterative solvers on locally refined meshes with optimal eigenvalue bounds.

ABSTRACT

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from $H^{3/2}$ into $B^{3/2}_{2,\infty}$; for elementwise polynomials these are bounded from $H^{1/2}$ into $B^{1/2}_{2,\infty}$. As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.

Motivation & Objective

• To establish endpoint stability of Scott-Zhang type operators in B3/22,∞ and B1/22,∞ spaces for globally continuous and discontinuous piecewise polynomials.
• To develop a multilevel decomposition based on modified Scott-Zhang operators on hierarchies of meshes generated by newest vertex bisection.
• To design a local multilevel diagonal preconditioner for the integral fractional Laplacian on adaptively refined meshes with optimal eigenvalue bounds.
• To ensure uniformly bounded condition numbers for the preconditioned system, independent of mesh refinement level, even under strong local refinement.

Proposed method

• Prove stability of quasi-interpolation operators in endpoint Besov norms using interpolation theory and K-functional estimates.
• Construct a modified Scott-Zhang operator that preserves consistency across mesh hierarchies, particularly on the finest common coarsening of two meshes.
• Establish norm equivalence for multilevel decompositions on adaptively refined, shape-regular meshes via a new inverse estimate in fractional Sobolev norms.
• Derive a strengthened Cauchy-Schwarz inequality using weighted inverse estimates and scaling arguments to bound off-diagonal terms.
• Apply additive Schwarz framework with diagonal scaling, leveraging the multilevel decomposition to bound extremal eigenvalues of the preconditioned stiffness matrix.
• Implement a local multilevel diagonal preconditioner by restricting diagonal entries to nodal supports on hierarchical patches, ensuring optimal conditioning.

Experimental results

Research questions

• RQ1 Is the Scott-Zhang operator stable in the endpoint Besov space B3/22,∞ for globally continuous piecewise polynomials?
• RQ2 Can a multilevel decomposition with equivalent norms be constructed on adaptively refined meshes using Scott-Zhang-type operators?
• RQ3 Does a local multilevel diagonal preconditioner achieve uniformly bounded condition numbers for the fractional Laplacian on locally refined meshes?
• RQ4 How do the eigenvalues of the preconditioned system behave as the mesh is refined, especially under strong local refinement?
• RQ5 Can the preconditioner be constructed independently of the finest mesh while preserving optimal conditioning?

Key findings

• The Scott-Zhang operator is bounded from H3/2 into B3/22,∞ and from H1/2 into B1/22,∞, establishing endpoint stability in Besov norms.
• A multilevel decomposition with equivalent norms is constructed on meshes generated by newest vertex bisection, valid up to the endpoint case.
• The proposed local multilevel diagonal preconditioner ensures that the condition number of the preconditioned system remains uniformly bounded, independent of the mesh size and refinement ratio.
• For s ∈ (0, 1), the preconditioner achieves optimal eigenvalue bounds for piecewise linear discretizations in S1,10(T), and for s ∈ (0, 1/2) in S0,0(T).
• Numerical experiments confirm that the condition number of the preconditioned system grows at most logarithmically, while the unpreconditioned system exhibits a growth rate of O(N2s/dℓ).
• The preconditioner is structurally similar to those used in boundary element methods and can be efficiently realized, as confirmed by numerical results.