Preconditioning spectral element schemes for definite and indefinite problems

Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution

Towards a robust Terra

In this work mantle convection simulation with Terra is investigated from a numerical point of view, theoretical analysis as well as practical tests are performed. The stability criteria for the numerical formulation of the physical model will be made clear. For the incompressible case and the Terra specific treatment of the anelastic approximation, two inf-sup stable grid modifications are presented, which are both compatible with hanging nodes. For the Q1hQ12h element pair a simple numeric test is introduced to prove the stability for any given grid. For the Q1h Pdisc 12h element pair and 1-regular refinements with hangig nodes an existing general proof can be adopted. The influence of the slip boundary condition is found to be destabilizing. For the incompressible case a cure can be adopted from the literature. The necessary conditions for the expansion of the stability results to the anelastic approximation will be pointed out. A numerical framework is developed in order to measure the effect of different numerical approaches to improve the handling of strongly varying viscosity. The framework is applied to investigate how block smoothers with different block sizes, combination of different block smoothers, different prolongation schemes and semi coarsening influence the multigrid performance. A regression-test framework for Terra will be briefly introduced

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on April 2-7, 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The vibrancy and diversity in this field are amply expressed in these important papers, and the collection clearly shows the continuing rapid growth of the use of multigrid acceleration techniques