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WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS.

Lin Mu1, Junping Wang, Guowei Wei

  • 1Department of Applied Science, University of Arkansas at Little Rock, Little Rock, AR 72204 ( lxmu@ualr.edu ).

Journal of Computational Physics
|September 28, 2013
PubMed
Summary
This summary is machine-generated.

A new Weak Galerkin finite element method (WG-FEM) effectively solves elliptic partial differential equations with complex interfaces. This method achieves high-order convergence, even for solutions with singularities, outperforming existing numerical schemes.

Keywords:
Finite element methodslow solution regularitynonsmooth interfacesecond order elliptic interface problemsweak Galerkin method

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Weak Galerkin methods offer flexibility in handling boundary and interface conditions for partial differential equations (PDEs).
  • Solving elliptic PDEs with discontinuous coefficients and interfaces presents challenges, particularly when solutions exhibit singularities due to non-smooth interfaces.

Purpose of the Study:

  • To develop and validate a Weak Galerkin finite element method (WG-FEM) for solving elliptic PDEs with discontinuous coefficients and interfaces.
  • To demonstrate the capability of WG-FEM in achieving high-order convergence for interface problems.
  • To investigate the performance of WG-FEM for solutions with low regularity.

Main Methods:

  • Development of a Weak Galerkin finite element method (WG-FEM) for elliptic PDEs.
  • Theoretical analysis proving the potential for high-order schemes using high-order polynomials.
  • Extensive numerical experiments on second-order elliptic interface problems.

Main Results:

  • High-order convergence confirmed in L2 and L∞ norms for piecewise linear WG-FEM.
  • WG-FEM demonstrates effectiveness for interface problems with non-smooth solutions.
  • Lowest-order (piecewise constant) WG-FEM achieves O(h) to O(h^2) accuracy in the L∞ norm for challenging interface problems, surpassing existing methods.

Conclusions:

  • The developed WG-FEM is a robust and effective numerical method for elliptic PDEs with interfaces and discontinuous coefficients.
  • WG-FEM shows significant promise for problems with low solution regularity, offering improved accuracy over established schemes.
  • The method's flexibility and high-order convergence make it a valuable tool for computational science and engineering.