Randolph E. Bank

Philip E. Gill

Michael Holst

Jennifer Trefftzs

by Randolph E. Bank, Michael J. Holst

Abstract:

We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly load-balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection. Only the initial fan-out of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution, namely, the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments that illustrate the effectiveness of this approach. The justification of the initial refinement prediction step, as well as the justification of skipping the two communication-intensive steps, is supported by some recent (J. Xu and A. Zhou, Math. Comp., to appear] and not so recent (J.~A. Nitsche and A.~H. Schatz, Math. Comp., 28 (1974), pp.~937--958; A.~H. Schatz and L.~B. Wahlbin, Math. Comp., 31 (1977), pp.~414--442; A.~H. Schatz and L.~B. Wahlbin, Math. Comp., 64 (1995), pp.~907--928) results on local a priori and a posteriori error estimation. This revision of the original article (R.~E. Bank and M.~J. Holst, SIAM J. Sci. Comput., 22 (2000), pp.~1411--1443) updates the numerical experiments, and reflects the knowledge we have gained since the original paper appeared.

April 1999