Finite element solution of thermal convection on a hypercube concurrent computer

Abstract

Numerical solutions to thermal convection flow problems are vital to many scientific and engineering problems. One fundamental geophysical problem is the thermal convection responsible for continental drift and sea floor spreading. The earth's interior undergoes slow creeping flow (~cm/yr) in response to the buoyancy forces generated by temperature variations caused by the decay of radioactive elements and secular cooling. Convection in the earth's mantle, the 3000 km thick solid layer between the crust and core, is difficult to model for three reasons: (1) Complex rheology -- the effective viscosity depends exponentially on temperature, on pressure (or depth) and on the deviatoric stress; (2) the buoyancy forces driving the flow occur in boundary layers thin in comparison to the total depth; and (3) spherical geometry -- the flow in the interior is fully three dimensional. Because of these many difficulties, accurate and realistic simulations of this process easily overwhelm current computer speed and memory (including the Cray XMP and Cray 2) and only simplified problems have been attempted [e.g. Christensen and Yuen, 1984; Gurnis, 1988; Jarvis and Peltier, 1982]. As a start in overcoming these difficulties, a number of finite element formulations have been explored on hypercube concurrent computers. Although two coupled equations are required to solve this problem (the momentum or Stokes equation and the energy or advection-diffusion equation), we will concentrate our efforts on the solution to the latter equation in this paper. Solution of the former equation is discussed elsewhere [Lyzenga, et al, 1988]. We will demonstrate that linear speedups and efficiencies of 99 percent are achieved for sufficiently large problems.