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Exact subgrid interface correction schemes for elliptic interface problems.

Jae-Seok Huh1, James A Sethian

  • 1Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA.

Proceedings of the National Academy of Sciences of the United States of America
|July 19, 2008
PubMed
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We present a novel nonconforming finite-element method for elliptic interface problems with discontinuous coefficients. This method accurately resolves solution jumps using a singular correction function, simplifying implementation and enhancing stability.

Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Elliptic interface problems often feature discontinuous coefficients and sources.
  • These discontinuities can lead to jumps in the solution or its derivative, complicating numerical solutions.
  • Existing methods struggle to accurately capture these subgrid-scale phenomena.

Purpose of the Study:

  • To develop a nonconforming finite-element method for second-order elliptic interface problems.
  • To accurately resolve jump discontinuities in solutions and their derivatives caused by interface properties.
  • To provide a robust and stable numerical approach for problems with discontinuous coefficients and singular sources.

Main Methods:

  • Construction of a singular correction function using closest point extension and signed distance functions.

Related Experiment Videos

  • Implicit interface representation for accurate subgrid resolution.
  • Development of a two-level iteration strategy for solving the resulting system, akin to iterative preconditioning.

    • Main Results:

    • The method accurately captures discontinuities without instability, even with discontinuous coefficients.
    • Regularization by a singular function simplifies implementation, with effects localized to the right-hand side.
    • A normalization step leads to a saddle-point-like problem formulation for general cases.

    Conclusions:

    • The proposed nonconforming finite-element method effectively handles elliptic interface problems with challenging discontinuities.
    • The singular correction function provides accurate subgrid resolution and simplifies the numerical treatment.
    • The method offers a stable and efficient approach for a wide range of interface problems.