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

(1)建立了求解地球物理学中地幔对流和玻璃制品炉中熔融玻璃对流等慢流热对流数值问题的有限元格式。我们证明了该格式的无条件稳定性和有限元解的收敛速度。这些问题是用具有无限普朗特数的瑞利-贝纳德方程来模拟的，该方程的粘度对温度有很强的依赖性。所得格式对三维问题具有实际意义。为了减少计算量，对速度、压力、温度等未知函数均采用四面体线性单元，并采用稳定有限元法。(2)我们构建了上述方案的计算代码，并在并行计算机上实现。地幔对流问题是在球对称域内解决的。利用这一性质，我们将该域划分为若干子域的并集，这使得我们在用预条件迭代法求解Stokes方程时，在一个有代表性的子域中只保留刚度矩阵。因此，所需的内存大大减少了。九州大学计算与通信中心的共享内存型计算机富士通GP7000使用24个cpu，速度可以提高20倍左右。利用这个程序，我们研究了固定温度场和流型对粘度比的依赖关系。当该比值增大时，羽流的头部变平，羽流的数量增加。(3)提出了Navier-Stokes方程解的数值验证方法，并成功地验证了低雷诺数问题的解。通过精度保证计算，给出了二维热对流问题分岔分支存在性的计算机辅助证明。(4)利用三维球面对流代码，研究了大陆的存在与地幔对流的关系。我们已经用数字表明，羽流在大约几百亿年内到达大陆下面。少

(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