Construction of a Practical Computation Code for Heat Convection Problems with Slow Flow

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(1) We have built a finite element scheme for solving numerically heat convection problems with slow flow like Earth's mantle convection in geophysics and melting glass convection in glass product furnaces. We have shown unconditional stability of the scheme and the convergence rate of the finite element solutions. These problems are modeled by Rayleigh-Benard equations with infinite Prandtl number, whose viscosity is strongly dependent on temperature. The obtained scheme is practically useful for three-dimensional problems. In order to reduce computation load we have employed the tetrahedral linear element for every unknown functions, velocity, pressure and temperature, and used stabilized finite element method.(2) We have constructed a computation code for the scheme mentioned above and implemented it on parallel computers. The Earth's mantle convection problem is solved in a spherically symmetric domain. By virtue of this property we have divided the domain into the union of congrue … More nt subdomains, which have allowed us to keep only stiffness matrices in a representative subdomain in solving Stokes equations by a preconditioned iterative method. As a result the required memory has reduced drastically. We could get speeding up of about 20 times in using 24 CPUs of Fujitsu GP7000, a shared memory type computer at Computing and Communications Center, Kyushu University. Using this code, we have studied the viscosity ratio dependency of stationary temperature fields and flow patterns. When the ratio increases, the heads of plumes flatten and the number of plumes increases.(3) We have presented a numerical verification method for solutions of the Navier-Stokes equations, and succeeded in the verification for low Reynolds number problems. Performing accuracy guaranteed computation, we have given a computer aided proof to the existence of bifurcation branches for two-dimensional heat convection problems.(4) Using a code for the convection in a three-dimensional sphere, we have studied the relation between the existence of continents and mantle convection. We have shown numerically that plumes arive under continents in some tens of billion years. Less