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Exponential Convergence of $hp$-FEM for the Integral Fractional Laplacian on cuboids

Björn Bahr, Markus Faustmann, Carlo Marcati, Jens Markus Melenk, Christoph Schwab

Abstract

For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product $hp$-finite element approximations on $(0,1)^3$, for forcing $f$ that is analytic in $[0,1]^3$. Exploiting analytic regularity estimates in weighted Sobolev spaces, we prove for $hp$-GLL interpolation approximations with $N$ degrees of freedom the energy norm error bound $\lesssim \exp(-b\sqrt[6]{N})$. Tensor product mesh families which are geometrically refined towards all sides of $(0,1)^3$ are used. Numerical experiments with $hp$-Galerkin FEM confirm the bound.

Exponential Convergence of $hp$-FEM for the Integral Fractional Laplacian on cuboids

Abstract

For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product -finite element approximations on , for forcing that is analytic in . Exploiting analytic regularity estimates in weighted Sobolev spaces, we prove for -GLL interpolation approximations with degrees of freedom the energy norm error bound . Tensor product mesh families which are geometrically refined towards all sides of are used. Numerical experiments with -Galerkin FEM confirm the bound.
Paper Structure (15 sections, 4 theorems, 39 equations, 4 figures)

This paper contains 15 sections, 4 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction and Main Result
  2. Integral Fractional Diffusion
  3. $hp$-FEM discretization on geometrically refined meshes
  4. Main Result
  5. Weighted Analytic Regularity
  6. Partition of $\Omega$
  7. Weighted analytic regularity
  8. Exponential convergence of $hp$-FEM
  9. Embedding into weighted integer order space
  10. Definition of the $hp$-interpolation operator $\Pi_q^L$
  11. Mesh layers and cutoff function
  12. Estimates for elements at the boundary
  13. Proof of \ref{['eq:hpExpConv']}
  14. Numerical example
  15. Conclusion

Key Result

theorem 1

Let $\Omega := (0,1)^3$, $f$ be analytic in $\overline{\Omega}$ and $u$ solve eq:weakform. Then, the Galerkin approximations $u_N \in V_N := W_q^L$ of eq:GalV with $W^L_q$ defined in eq:S^q_0 with $L\sim q \sim N^{1/6}$ converge exponentially to $u$, i.e., there are constants $b$, $C >0$ (depending

Figures (4)

  • Figure 1: Geometric mesh $\mathcal{T}^{L}_{\mathrm{geo},\sigma}$ with $\sigma=1/2$ and $L=4$.
  • Figure 2: Notation near a vertex $\mathbf{v}$. Left: top view of the vertex cone. Right: side view of the vertex cone.
  • Figure 3: Notation near an edge $\mathbf{e}$ with two faces $\mathbf{f},\mathbf{f}'$ meeting at the edge and no vertex close by. Left: front view (edge collapses to point). Right: side view.
  • Figure 4: Exponential convergence of $hp$-FEM for \ref{['eq:FracLap']} with $f\equiv 1$, $L=1,2,3$ and $\sigma\in \{(\sqrt{2}-1)^2 = 0.172...,0.25,0.5,0.75\}$.

Theorems & Definitions (7)

  • theorem 1
  • theorem 2
  • remark 1
  • lemma 1: approximation on cuboids
  • lemma 2
  • proof
  • proof : of Theorem \ref{['thm:hpExpConv']}