Approximating solutions of linear elliptic PDE's on a smooth manifold using local kernel

Faheem Gilani, John Harlim

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


A mesh-free numerical method for solving linear elliptic PDE's using the local kernel theory that was developed for manifold learning is proposed. In particular, this novel approach exploits the local kernel theory which allows one to approximate the Kolmogorov operator associated with Itô diffusion processes on compact Riemannian manifolds without boundary or with Neumann boundary conditions using an integral operator. Theoretical justification for the convergence of this numerical technique is provided under the standard conditions for the existence of the weak solutions of the PDEs. Numerical results on various instructive examples, ranging from PDE's defined on flat and non-flat manifolds with known and unknown embedding functions show accurate approximation with error on the order of the kernel bandwidth parameter.

Original languageEnglish (US)
Pages (from-to)563-582
Number of pages20
JournalJournal of Computational Physics
StatePublished - Oct 15 2019

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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