Well-conditioned spectral discretizations of the biharmonic operator

Abstract

When standard spectral methods are applied to the biharmonic equation the condition number of the resulting algebraic system of equations is very large and grows as 0(N ⁸) where N is the degree of the polynomial expansion in each coordinate direction. When direct methods are used to solve these systems we fail to achieve machine precision for the solution due to rounding errors. An improved method which reduces the condition number to 0(N ⁴) is described. Numerical results are presented demonstrating the exponential convergence of the approximation.

Keywords:

• Biharmonic operator
• spectral method
• tau method
• condition number

C.R. CATEGORIES::

• G.1.8
• G.1.3