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Approximation of higher-order powers of the spectral fractional Laplacian via polyharmonic extension

Enrique Otárola, Abner J. Salgado

Abstract

We use the polyharmonic extension approach to develop a numerical technique for discretizing higher-order powers of the spectral fractional Laplacian $(-Δ)^s$ with $s \in (1,2)$.

Approximation of higher-order powers of the spectral fractional Laplacian via polyharmonic extension

Abstract

We use the polyharmonic extension approach to develop a numerical technique for discretizing higher-order powers of the spectral fractional Laplacian with .
Paper Structure (5 sections, 4 theorems, 27 equations)

This paper contains 5 sections, 4 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Polyharmonic extensions
  3. Higher powers of the spectral fractional Laplacian
  4. Regularity and truncation
  5. Discretization

Key Result

Theorem 1

Let $s \in (1,2)$ and $f \in {\mathcal{H}}^{-s}_{\mathcal{L}}$. If ${\mathfrak u} \in H^2_{{\mathcal{L}}}(y^{\mathfrak b}, {\mathbbm R}_+; {\mathcal{H}})$ solves then $u \coloneqq {\mathfrak u}(0) \in {\mathcal{H}}^s_{\mathcal{L}}$ satisfies ${\mathcal{L}}^s u = f$. Here, $d_s \coloneqq 2^{\mathfrak b} \frac{\Gamma(2-s)}{\Gamma(s)}$. Moreover, if where $K_s$ denotes the modified Bessel function

Theorems & Definitions (9)

  • Theorem 1: extension
  • Proposition 1: exponential decay
  • proof
  • Proposition 2: exponential approximation
  • proof
  • Theorem 2: best approximation
  • proof
  • Remark 1: regularity and rate of approximation
  • Remark 2: nonconforming methods