Space-Time Coupled Least-Squares Spectral Element Methods for Parabolic Problems*

Abstract

We study a space-time coupled least-squares spectral element method for parabolic initial boundary value problems using parallel computers. The method is spectrally accurate in both space and time. The spectral element functions are polynomials of degree q in the time variable and of degree p in each of the space variables and are non-conforming in both space and time. There is no need therefore to consider C⁰ element expansions at the inter element boundaries and consequently it is unnecessary to express the equation as an equivalent set of first-order equations by introducing an additional independent variable. Instead, we consider the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The normal equations obtained from the least-squares formulation can be solved by a preconditioned conjugate gradient method (PCGM) in matrix-free form. The method is implemented on a parallel computer using message-passing-interface (MPI). We take q proportional to p² in the p-version of the method, whereas $p=2q+1$ in the h-version. If the solution belongs to a certain Gevrey space, then the error decays exponentially in p in the p-version of the method and the error is $O(h^{2q-1})$ in the h-version. Further, the number of iterations, needed to obtain the approximate solution using PCGM, is $O({(\hspace{0.17em}\text{ln}\hspace{0.17em}p)}^2/h)$ per time step. These estimates are confirmed by the computational results.

Keywords:

• Sobolev spaces
• Gevrey space
• parabolic problem
• least-squares
• domain decomposition
• spectral element function
• parallel preconditioners

Acknowledgments

The authors would like to acknowledge the technical support from Center for Development of Advanced Computing (CDAC), Pune, India and would like to thank the reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.