49
Views
3
CrossRef citations to date
0
Altmetric
Original Articles

Error Analysis of Adaptive Finite Difference Methods Using Stretching Functions for Polar Coordinate Form of Poisson-Type Equation

, &
Pages 17-44 | Published online: 31 Aug 2006

References

  • Bramble , J. H. and Hubbard , B. E. 1962 . On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation . Numer. Math. , 4 : 313 – 327 .
  • Cao , W. , Huang , W. and Russell , R. D. 1999 . A study of monitor functions for two dimensional adaptive mesh generation . SIAM J. Sci. Comput. , 20 : 1978 – 1994 .
  • Chen , X. , Matsunaga , N. and Yamamoto , T. 1998 . “ Smoothing Newton methods for nonsmooth Dirichlet problems ” . In Reformulation—Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods Edited by: Fukushima , M. and Qi , L. 56 – 79 . Dordrecht : Kluwer .
  • Costa Mouro , A. P. On the use of nonuniform meshes in the numerical solution of singularly perturbed boundary value problems in two dimensions . Abstract of ICCAM 2000 . July 2000 , Leuven.
  • Fang , Q. , Matsubara , T. , Shogenji , Y. and Yamamoto , T. 2001 . Convergence of inconsistent finite difference scheme for Dirichlet problem whose solution has singular derivatives at the boundary . Information , 4 : 161 – 170 .
  • Fang , Q. , Shogenji , Y. and Yamamoto , T. “ Convergence analysis of adaptive finite difference methods using stretching functions for boundary value problems with singular solutions ” . (submitted)
  • Gorenflo , R. and Über , S. 1973 . “ Gerschgorins Methode der Fehlerabschatzung bei Differenzenzenverfahren ” . Lecture Notes in Math. 333 128 – 143 . Springer .
  • Huang , W. , Ren , Y. and Russell , R. D. 1994 . Moving mesh partial differential equations based on the equidistribution principle . SIAM J. Numer. Anal. , 31 : 709 – 730 .
  • Huang , W. and Sloan , D. M. 1994 . A simple adaptive grid method in two dimensions . SIAM J. Sci. Comput. , 15 : 776 – 797 .
  • Ishiguro , M. Numerical errors and condition number for two-dimensional Laplace's equation by stretching functions . Abstract of JSIAM meeting 2000 . September 2000 . pp. 470 – 471 . (in Japanese)
  • Mackenzie , J. A. and Robertson , M. L. 2000 . The numerical solution of one-dimensional phase change problems using an adaptive moving mesh method . J. Comput. Phys. , 161 : 537 – 557 .
  • Matsunaga , N. and Yamamoto , T. 1999 . Convergence of Swartztrauber-Sweet's approximation forthe Poisson-type equation on a disk . Numer. Funct. Anal. and Optimz. , 20 : 917 – 928 .
  • Matsunaga , N. and Yamamoto , T. 2000 . Superconvergence of the Shortley-Weller approximation for Dirichlet problems . J. Comp. Appl. Math. , 116 : 263 – 273 .
  • Mulholland , L. S. , Qiu , L. and Sloan , D. M. 1997 . Solution of evolutionary partial differential equations using adaptive finite differences with pesudespectralpost-processing . J. Comp. Phys. , 131 : 280 – 298 .
  • Qiu , Y. and Sloan , D.M. 1999 . Analysis of difference approximations to a singular perturbed two-point boundary value problem on an adaptively generated grid . J. Comput. App. Math. , 101 : 1 – 25 .
  • Qiu , Y. , Sloan , D. M. and Tang , T. 2000 . Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: analysis of convergence . J. Comput. Appl. Math. , 116 : 121 – 143 .
  • Tang , T. , Xue , W. and Zhang , P. 2001 . Analysis of moving mesh methods based on geometrical variables . J. Comput. Math. , 19 : 41 – 54 .
  • Thompson , J. F. , Thames , J. F. and Mastin , C. W. 1974 . Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensonal bodies . J. Comp. Phys. , : 15
  • Thompson , J. F. , Warsi , Z. U. A. and Mastin , C.W. 1985 . Numerical Grid Generation New York, Amsterdam, , North-Holland
  • Tsuboi , K. 2000 . Condition number of tridiagonal matrix in stretching coordinatesystem (in Japanese) . JSIAM , 10 : 133 – 144 .
  • Varga , R. S. 2000 . “ Matrix terative analysis ” . In Springer Series in Computational Mathematics. , 2nd ed. Vol. 27 , New York : Springer-Verlag .
  • Yamamoto , T. 2002 . Convergence of consistent and inconsistent finite difference schemes and an acceleration technique . J. Comput. Appl. Math. , 140 : 849 – 866 .
  • Yamamoto , T. , Fang , Q. and Chen , X. 2001 . Superconvergence and nonsuperconvergence of the Shortley-Weller approximations for Dirichlet problems . Numer. Funct. Anal. Optimiz. , 22 : 161 – 170 .
  • Young , D. M. 1971 . Iterative Solution of Large Linear Systems New York, London : Academic Press .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.