Wavlet-Galerkin solution of Partial differential equations
Patel Dhaneshkumar (University Baroda)
Discrete elliptic operators are used in the
approximation of solution of a uniformly elliptic and
possibly variable coefficient differential equations.
In computations, the sparsity and small condition
numbers of the discrete operators are the key to
efficiency. Sparsity enhances the speed of iterations,
while the small condition number guarantees rapid
convergence of such iterations. The matrices that we
obtain using finite difference methods are sparse.
However, they have large condition numbers.
Using the Galerkin method with Fourier system, we can
obtain a bounded condition number but the matrix is no
longer sparse. In the Galerkin method with a wavelet
base, we obtain both the advantages. In the talk I
will cover the following things:
(1)Error estimates to show the advantage of
wavelet-Galerkin method over finite difference method
and Fourier-Galerkin method not only in terms of fast
computation and rapid convergence but by
obtaining better accuracy.
(2)Use of different Wavelets to Solve PDE
[A] H.Resnikoff and R.Wells, Wavelet Analysis,
Springer-Verlag, New York, 1998
[B] K.Amaratunga and Williams J., Wavelet-Galerkin
solutions for one dimensional partial differential
equations, International J. Num.Methods in Eng.