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

Markus Faustmann, Jens Markus Melenk, Maryam Parvizi

TL;DR

As an application, 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 is obtained.

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.

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

TL;DR

As an application, 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 is obtained.

Abstract

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from into ; for elementwise polynomials these are bounded from into . 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.

Paper Structure

This paper contains 24 sections, 24 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Stability of (quasi-)interpolation operators in Besov spaces
  4. Multilevel decomposition based on mesh hierarchies generated by NVB
  5. The finest common coarsening
  6. Adapted Scott-Zhang operators
  7. The multilevel decomposition
  8. A realization of an optimal multilevel preconditioner for the fractional Laplacian
  9. A local multilevel diagonal preconditioner for adaptively refined meshes
  10. A local multilevel diagonal preconditioner using a finest common coarsening mesh hierarchy
  11. Stability of Scott-Zhang type operators
  12. Proof of Theorem \ref{['thm:stability-discontinuous']}
  13. Some generalizations and applications
  14. Multilevel decomposition based on NVB mesh-hierarchy
  15. Properties of the finest common coarsening (${\tt fcc}$)
  16. ...and 9 more sections

Key Result

Theorem 2.2

Fix $m \in \{1,2\}$ and $p \in {\mathbb N}_0$ with $p \ge m-1$. Let an operator $I_h^m$ satisfying Assumption assumption:I-discontinuous be given. Then, for all $u \in H^{m-1/2}(\Omega)$, we have where the constant $C > 0$ depends solely on $\Omega$, $d$, $m$ ,$p$, and the $\gamma$-shape regularity of ${\mathcal{T}}$. If the mesh ${\mathcal{T}}$ is additionally quasi-uniform, then, for all $u \i

Figures (2)

  • Figure 5.1: Adaptively generated NVB mesh on the L-shaped domain.
  • Figure 5.2: Estimated condition numbers for the unpreconditioned matrix $\mathbf{A}^L$ and the preconditioned matrix $\mathbf{P}_{AS}^{L}$ left: $s=0.25$, right: $s=0.75$.

Theorems & Definitions (54)

  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4: adapted Scott-Zhang operators
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • Proposition 3.1: karkulik-melenk15
  • ...and 44 more