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Accurate and efficient numerical solutions for elliptic obstacle problems.

Philku Lee1, Tai Wan Kim2, Seongjai Kim3

  • 1Department of Mathematics, Sogang University, Ricci Building R1416, 35 Baekbeom-ro, Mapo-gu, Seoul, 04107 South Korea.

Journal of Inequalities and Applications
|February 21, 2017
PubMed
Summary
This summary is machine-generated.

This study introduces improved finite difference (FD) methods for solving elliptic obstacle problems. The enhanced successive over-relaxation (SOR) method and subgrid FD techniques significantly accelerate convergence and reduce numerical errors.

Keywords:
elliptic obstacle problemgradient-weighting methodobstacle relaxationsubgrid finite difference (FD)successive over-relaxation (SOR) method

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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations
  • Computational Mathematics

Background:

  • Elliptic obstacle problems involve finding superharmonic solutions or minimal surfaces over obstacles using inequality constraints.
  • Existing finite difference (FD) methods face accuracy challenges, particularly near free boundaries where mesh grids do not align.

Purpose of the Study:

  • To develop and analyze efficient iterative algorithms for solving elliptic obstacle problems using FD methods.
  • To introduce subgrid FD methods to mitigate accuracy loss near free boundaries.
  • To present a gradient-weighting method for nonlinear obstacle problems.

Main Methods:

  • Investigation of iterative algorithms based on the successive over-relaxation (SOR) method.
  • Development of subgrid finite difference (FD) methods for improved accuracy.
  • Application of a gradient-weighting method for nonlinear obstacle problems.
  • Analysis of iterative algorithm convergence for linear and nonlinear cases.
  • Strategy for optimizing the relaxation parameter in SOR.

Main Results:

  • The obstacle SOR iteration with an optimal parameter shows convergence approximately one order faster than state-of-the-art methods.
  • Subgrid FD methods reduce numerical errors by one order of magnitude in most cases.
  • The proposed methods are numerically verified through various examples.

Conclusions:

  • The enhanced SOR and subgrid FD methods provide significant improvements in speed and accuracy for elliptic obstacle problems.
  • The gradient-weighting approach enhances efficiency for nonlinear variants.
  • The findings offer a more effective computational strategy for these types of problems.