2001 Fiscal Year Final Research Report Summary
Construction of a Practical Computation Code for Heat Convection Problems with Slow Flow
Project/Area Number 
11554003

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  展開研究 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KYUSHU UNIVERSITY 
Principal Investigator 
TABATA Masahisa Faculty of Mathematics, Kyushu University, Prof. > 九州大学, 大学院・数理研究院, 教授 (30093272)

CoInvestigator(Kenkyūbuntansha) 
FUKUMOTO Yasuhide Faculty of Mathematics, Kyushu University, Assoc. Prof., 大学院・数理研究院, 助教授 (30192727)
HONDA Satoru Hiroshima Univ., Graduate School of Science, Prof., 大学院・理学研究科, 教授 (00219239)
NAKAO Mitsuhiro Faculty of Mathematics, Kyushu University, Prof., 大学院・数理研究院, 教授 (10136418)
SUZUKI Atsushi Faculty of Mathematics, Kyushu University, Assistant, 大学院・数理研究院, 助手 (60284155)
YAMAMOTO Nobito Univ. of ElectroCommunications, Assoc. Prof., 電気通信学部, 助教授 (30210545)

Project Period (FY) 
1999 – 2001

Keywords  Finite element method / Parallel computation / Earth's mantle convection / Melting glass convection / Temperature dependent viscosity / Heat convection problems / Error estimation constant / Accuracy guaranteed computation 
Research Abstract 
(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 RayleighBenard equations with infinite Prandtl number, whose viscosity is strongly dependent on temperature. The obtained scheme is practically useful for threedimensional 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
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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 NavierStokes 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 twodimensional heat convection problems. (4) Using a code for the convection in a threedimensional 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

Research Products
(18 results)