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ACCURATE SOLUTION AND GRADIENT COMPUTATION FOR ELLIPTIC INTERFACE PROBLEMS WITH VARIABLE COEFFICIENTS.

Zhilin Li1, Haifeng Ji2, Xiaohong Chen3

  • 1Center for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA.

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|October 7, 2017
PubMed
Summary
This summary is machine-generated.

A novel augmented method accurately solves elliptic interface problems with variable coefficients. This approach achieves second-order accurate solutions and gradients on both sides of the interface.

Keywords:
Elliptic interface problemM-matrixaccurate gradient computationconvergence proofdiscrete Green functioninterfacevariable coefficient with discontinuity

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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations

Background:

  • Elliptic interface problems with piecewise variable coefficients present challenges in achieving accurate solutions and gradients.
  • Existing methods often struggle with discontinuities at interfaces.

Purpose of the Study:

  • To develop a new augmented method for solving elliptic interface problems.
  • To achieve second-order accuracy for both the solution and its gradient from each side of the interface.

Main Methods:

  • Introducing the jump in the normal derivative as an augmented variable.
  • Reformulating the problem as a new PDE with a leading Laplacian operator.
  • Employing upwind-type finite difference discretization for irregular grids.
  • Utilizing a multi-grid solver and GMRES iterative method.

Main Results:

  • The augmented method provides second-order accurate solutions and gradients.
  • The method's accuracy is independent of the coefficient jump magnitude.
  • Numerical examples validate the theoretical analysis and method efficiency.

Conclusions:

  • The proposed augmented method is effective for elliptic interface problems.
  • It offers a robust approach for obtaining accurate solutions and gradients at interfaces.
  • The method demonstrates significant efficiency and theoretical soundness.