有限元特征值计算中的子空间二次解耦算法

ALGORITHMS OF DOUBLE-DECOUPLING SUBSPACES FOR SOLVING FEM EIGEN-PROBLEMS

解线性方程组预条件子算法已在求解偏微分方程(PDE)的离散代数系统的高性能计算中取得巨大成功.相比之下,PDE特征值问题本身的高效快速并行的潜力目前远未发挥.根据代数基本定理可知,通过因式分解,任意一个一元n次实特征多项式可分解为若干个低次实多项式(如二次)或一次实多项式的乘积,因此,利用PDE方程的特征变换(如Fourier变换等)作预变换有可能把离散的高阶广义特征值问题直接解耦分解为一批低阶广义矩阵特征值的并行计算.本文以三次Hermite插值有限元为例,提出求解一类离散椭圆PDE广义特征值的二次解耦算法。新算法不但降低了常规算法(先把广义特征值问题化为普通特征值问题,再分解为n个一次多项式乘积)的计算复杂度,性能提升明显,而且能有效判别与防止伪特征值的出现(Spurious free无伪解).

By using so-called preconditioning algorithms,it has been successful to solve PDE discrete algebraic systems in high performance computations.However,the potential of PDE eigen-problem solver is still far from expected.It is well-known that from the fundamental algebraic theorem,any n-degree polynomial can be decomposed into a product with several lower degree,such as quadratic polynomials,and linear terms.Therefore,by using some characteristic transforms,such as discrete Fourier,it is possible to decouple the original high order eigen-problem into some lower eigen-problems in parallel.In this paper,we take a cubic Hermite-interpolation as an example,an algorithm of Double-Decoupling Subspaces for Solving FEM Eigen-problems is proposed for solving a class of discrete elliptic generalized eigen-problems.Performance has been raised obviously,comparing with the traditional algorithm:turn to ordinary eigen-problem first,then do diagonalization at once.Moreover,the new algorithm can judge spurious-free and prevent from spurious eigenvalues.