Adaptive multi-domain spectral methods

When pseudo-spectral methods are used with domain decomposition procedures in the numerical solution of partial differential equations, the use of multiple domains can significantly effect the accuracy of the approximation. If large gradients occur near the boundaries of the domain then the accuracy can be enhanced, while if the rapid variations occur in the interior of the subdomains then the accuracy can be degraded. We have developed an adaptive multi-domain method. In this method we employ functionals, defined within each sub-domain, which measure the error in the pseudo-spectral approximation. Using polynomial interpolation, these functionals can be evaluated for arbitrary location of the interfaces. The location of the interfaces can then be determined so as to minimize the maximum error in all of the subdomains, or to equalize the errors within the subdomains. We have implemented an adaptive multi-domain pseudo-spectral method for the solution of one-dimensional wave equations. Computed results demonstrate that the use of adaptive multi-domain methods can result in significantly enhanced accuracy for a fixed number of collocation points.