Spectral element methods for parabolic problems (Q880206)

A spectral element method for parabolic initial value problems on smooth domains using parallel computers is presented. The space domain is divided into a number of regular quadrilaterals of size \(h\) and the time step is proportional to \(h^2\). At each time step we minimize a least-squares functional in different Sobolev norms and a term which measures the jump in the function and its derivatives across the inter-element boundaries. The Sobolev spaces used are of different order in space and time. A preconditioner for the minimization problem is defined which allows the problem to decouple. Error estimates are obtained for both the \(h\) and \(p\) versions of the method.