1. Numerical Solutions of Elliptic PDEs
We survey the
treatments of singularity problems of elliptic PDEs in [1] . All applicable
methods are discussed, including conformal transformation, local refinment,
singular function, and combined methods, etc. This paper also contains
338 related references on this subject.
Motz's problem
is the benchmark of the elliptic boundary value problem with singularity.
In [2] we use the conformal transformation method (CTM) to compute the
most accurate leading coefficients D0 of its series solution u .We
obtain :
D0 with 15999 significant digits,
D0 to D9 with near 2000 significant digits,
D0 to D499, where the last one has the worst 123
significant digits.
In [3] we use the numerical boundary approximation method to get the most accurate global solution of Motz's problem with total error O(10^-97). This error is smaller than that from CTM. Many error analysis are done to guarantee this solution. We list here the coefficients of Motz's solution of
101
terms with total error O(10^-25),
201 terms
with total error O(10^-49),
301
terms with total error O(10^-73),
401
terms with total error O(10^-97).
We use symbolic
boundary approximation method under different divisions of the domain and
combine numerical and symbolic techniques to solve Motz's problem in Project
NSC-85-2121-M-110-013-C. Now we are preparing a paper to compare this method
with the numerical boundary approximation method.
[4] works on
the eigenvalue problem of an elliptic partial differential equations. We
combine numerical methods and analytic formulae to get the high accurate
eigenvalues and eigenfunctions. Related error analysis are also included.
2. Numerical Solutions of Parabolic PDEs
Iterates of the backward-difference scheme for the heat equation are shown to exhibit ultimate symmetry behavior in [5], while iterates of the forward and Crank-Nicolson schemes also exhibit the same behaviors under additional conditions. We have uncovered the underlying matrix theory for this phenomenon.\newline
3. Matrix Computation
In the computation of classical power method, we can get the ratio of the two largest eigenvalues from its rate of convergence, then the second largest eigenvalue can be approximated accordingly. This idea is explored in Project NSC-84-2121-M- 110-004-MS. Our new research shows that the third largest eigenvalues can also be obtained using the same technique. Similar idea applies for inverse power method.
4. Others
For the bisection algorithm to find roots of nonlinear equations. We give a counter-example in [6] to show the bisection method is not linearly convergent in this sense.
References
[1] Zi-Cai Li and Tzon-Tzer Lu (1999), Singularity and treatments
of elliptic boundary value equations. To appear in
Mathematical and Computer Modelling.
[2] Zi-Cai Li, Tzon-Tzer Lu and Lan-Chung Chen (1999), Very high
accurate solutions of Motz's problem , Part I:
The conformal transformation method,
preprint.
[3] Tzon-Tzer Lu and Zi-Cai Li (1999), Very high accurate solutions
of Motz's problem, Part II: The boundary
approximation method, preprint.
[4] Z.C. Li, T.T. Lu, D.J. Guan and C.B. Yang (1994), Boundary
approximation method for the eigenvalue problems
with interfaces, Technical report,
Department of Applied Mathematics, National Sun Yat-sen University.
[5] Tzon-Tzer Lu, Symmetry Formation in the Discrete Heat Equation,
in: S.S. Cheng, S. Elaydi, G. Ladas (Eds.), New
Developments in Difference Equations
and Applications: Proceedings of the 3rd International Conference on Difference
Equations and Applications, 281-288,
Gordon and Breach, 1999.
[6] Sui-Sun Cheng and Tzon-Tzer Lu (1985), The bisection algorithm
is not linearly convergent. College Math. J., Vol.16,
No.1, 56-57.