[Paper Review] A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data

Andrew T. Corrigan, John Wallin+1 more|ArXiv.org|Jan 26, 2008

Numerical methods in engineering13 references3 citations

TL;DR

This paper extends sampling inequalities for fractional order Sobolev semi-norms by incorporating arbitrary-order derivative data and enabling optimal bounds for non-integer smoothness. The method improves convergence rates in Schaback’s unsymmetric meshless framework, particularly for higher-order Sobolev norms and inhomogeneous boundary value problems.

ABSTRACT

To improve convergence results obtained using a framework for unsymmetric meshless methods due to Schaback (Preprint Göttingen 2006), we extend, in two directions, the Sobolev bound due to Arcangéli et al. (Numer Math 107, 181-211, 2007), which itself extends two others due to Wendland and Rieger (Numer Math 101, 643-662, 2005) and Madych (J. Approx Theory 142, 116-128, 2006). The first is to incorporate discrete samples of arbitrary order derivatives into the bound, which are used to obtain higher order convergence in higher order Sobolev norms. The second is to optimally bound fractional order Sobolev semi-norms, which are used to obtain more optimal convergence rates when solving problems requiring fractional order Sobolev spaces, notably inhomogeneous boundary value problems.

Motivation & Objective

• To improve convergence rates in unsymmetric meshless methods by extending existing sampling inequalities for Sobolev semi-norms.
• To incorporate discrete samples of arbitrary-order derivatives into sampling bounds, addressing instability in higher-order test discretizations.
• To optimally bound fractional-order Sobolev semi-norms, enabling better convergence estimates for problems in fractional Sobolev spaces.
• To enable higher-order convergence in stronger norms by modifying Schaback’s framework with higher-order test discretizations.

Proposed method

• Generalizes the sampling inequality of Arcangéli et al. to allow fractional-order Sobolev semi-norms on the left-hand side, replacing integer-order norms.
• Introduces a new parameter μ to represent the order of derivative data used in the sampling inequality, generalizing prior work where μ = 0.
• Derives a bound of the form |u|_{l,q,Ω} ≤ C(d^{r-l-n(1/p-1/q)_+}|u|_{r,p,Ω} + d^{n/γ - l}||u|_b||_κ), with improved decay via α(s) = s^{m̃-m} for fractional m.
• Applies the new inequality within Schaback’s framework for unsymmetric meshless methods, using strong testing with higher-order test discretizations.
• Establishes convergence rates of order O(h^{(m̃-m)+(μ₁-m₁)}) in Sobolev norms, with μ₁ denoting the test discretization order.
• Uses inverse estimate factors γ(r) = r^{m- m̃} to balance trial and test discretization parameters, ensuring proportional refinement.

Experimental results

Research questions

• RQ1 Can sampling inequalities be extended to bound fractional-order Sobolev semi-norms while maintaining optimal convergence rates?
• RQ2 How can higher-order derivative data be incorporated into sampling inequalities to stabilize higher-order test discretizations in meshless methods?
• RQ3 What is the optimal balance between trial and test discretization parameters when using higher-order test spaces in Schaback’s framework?
• RQ4 Does the inclusion of higher-order derivative data lead to improved convergence rates in higher-order Sobolev norms compared to zero-order testing?
• RQ5 Can the constant α(s) in the sampling inequality be improved for fractional-order norms to avoid excessive refinement of the test discretization?

Key findings

• The new sampling inequality achieves optimal decay rate α(s) = s^{m̃-m} for fractional-order Sobolev semi-norms, avoiding the suboptimal α(s) = s^{m̃-⌈m⌉} found in prior work.
• Incorporating higher-order derivative data (via parameter μ) enables higher-order convergence in stronger Sobolev norms, with Table 1 showing improved rates for μ₁ ≥ 1.
• For two- or three-dimensional Poisson problems, using μ₁ = 2 or higher in test discretization yields convergence order O(h^{(m̃-m)-2}) in H⁶(Ω), outperforming μ₁ = 0.
• The order of convergence is limited by the test space’s regularity: to maintain uniform stability, μₖ must be less than the test space order by at least nₖ/2.
• Convergence in L²(Ω) cannot be concluded suboptimally from higher-order norms due to the requirement that U must have regularity at least H^{m+2}(Ω), implying m ≥ 2 + n/2.
• The modified Schaback framework achieves convergence rates of O(h^{(m̃-m)+(μ₁-m₁)}), demonstrating that higher-order test discretizations improve convergence in higher-order norms.