Finite element analysis for 3-dimensional large scale problem by using domain decomposition methods

A.大型磁场问题的有限元数值计算：首先，我们开发了时间简谐涡流问题和非线性静磁问题的计算程序。为了使代码高效，我们考虑了降低其计算量，以及增强应用问题，例如永磁体或移动障碍物。B.具有对称系数矩阵的线性方程组的数值求解器：我们考虑了基于残差最小化的MRTR方法的收敛。C.减法在离散Stokes方程中的应用：我们建立了一个用于离散Stokes方程的平衡预条件算子的粗空间。D.热对流问题有限元方法的误差估计：我们建立了一类求解变系数热对流问题的有限元方法的误差估计。E.用于大型结构分析的高效BDD方法的实现：我们发展了不完全BDD方法，提高了其并行效率，并实现了约1000万自由度的非定常结构分析。

A.Numerical computations of large scale magnetic problems by finite element method:First, we have developed computational codes for time-harmonic eddy current problems and nonlinear magnetostatic problems. To make the codes efficient, then we have considered the reduction of their computational costs, and the enhancement of applicable problems, for example, a permanent magnet or a moving obstacle.B.Numerical solver for linear systems with symmetric coefficient matrices:We have considered the convergence of MRTR method based on the minimization of residuals. Moreover, we have developed several precondisioners, which come from robust incomplete Cholesky factorization, and have shown their efficiency by appling to some real problems.C.Application of a substracturing method to discretized Stokes equations:We have established a coarse space of a balancing precondisioner for applying to discretized Stokes equations.D.Error estimates of finite element methods for thermal convection problems:We have established error estimates of a class of finite element methods for thermal convection problems with variable coefficients.E.Realization of highly efficient BDD methods for large scale structural analysis:We have developed incomplete BDD methods, improved its parallel efficiency, and realized nonstationary structural analysis with about 10 million DOF problems.