On the Increase of Convergence Rates of Relaxation Procedures for Elliptic Partial Difference Equations

ABSTRACT

Occasionally in the numerical solution of elliptic partial differential equations the rate of convergence of relaxation methods to the solution is adversely affected by the relative proximity of certain points in the grid. It has been proposed that the removal of the unknown functional values at these points by Gaussian elimination may accelerate the convergence. By application of the Perron-Frobenius theory of non-negative matrices it is shown that the rates of convergence of the Jacobi-Richardson and Gauss-Seidel iterations are not decreased and could be increased by this elimination. Although this may indicate that the elimination could improve the convergence rate for overrelaxation, it is still strictly an unsolved problem.

REFERENCES

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.8 Partial Differential Equations
          Subjects: Elliptic equations

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Convergence and stability

General Terms:
Performance, Theory, Verification